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3Great Clarendon Street, Oxford OX2 6DPOxford University Press is a department of the University of Oxford.It furthers the University’s objective of excellence in research, scholarship,and education by publishing worldwide inOxford New YorkAuckland Cape Town Dar es Salaam Hong Kong KarachiKuala Lumpur Madrid Melbourne Mexico City NairobiNew Delhi Shanghai Taipei TorontoWith offices inArgentina Austria Brazil Chile Czech Republic France GreeceGuatemala Hungary Italy Japan Poland Portugal SingaporeSouth Korea Switzerland Thailand Turkey Ukraine VietnamOxford is a registered trade mark of Oxford University Pressin the UK and in certain other countriesPublished in the United Statesby Oxford University Press Inc., New Yorkc○ André Nies 2009The moral rights of the author have been assertedDatabase right Oxford University Press (maker)First Published 2009All rights reserved. No part of this publication may be reproduced,stored in a retrieval system, or transmitted, in any form or by any means,without the prior permission in writing of Oxford University Press,or as expressly permitted by law, or under terms agreed with the appropriatereprographics rights organization. Enquiries concerning reproductionoutside the scope of the above should be sent to the Rights Department,Oxford University Press, at the address aboveYou must not circulate this book in any other binding or coverand you must impose the same condition on any acquirerBritish Library Cataloguing in Publication DataData availableLibrary of Congress Cataloging in Publication DataData availableTypeset by Newgen Imaging Systems (P) Ltd., Chennai, IndiaPrinted in Great Britainon acid-free paper byBiddles Ltd., Kings Lynn, NorfolkISBN 978–0–19–923076–11 3 5 7 9 10 8 6 4 2

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PrefaceThe complexity and randomness aspects of sets of natural numbers are closelyrelated. Traditionally, computability theory is concerned with the complexityaspect. However, computability theoretic tools can also be used to introducemathematical counterparts for the intuitive notion of randomness of a set. Recentresearch shows that, conversely, concepts and methods originating from randomnessenrich computability theory.This book is about these two aspects of sets of natural numbers and abouttheir interplay. Sets of natural numbers are identified with infinite sequences ofzeros and ones, and simply called sets.Chapters 1 and 6 are mostly about the complexity aspect. We introduce lownessand highness properties of sets.Chapters 2, 3, and 7 are mostly about the randomness aspect. Firstly we studyrandomness of finite objects. Then we proceed to sets. We establish a hierarchyof mathematical randomness notions. Each notion matches our intuition ofrandomness to some extent.In Chapters 4, 5, and 8 we mainly study the interplay of the computabilityand randomness aspects. Section 6.3 also touches upon this interplay. Chapter 9looks at analogs of results from the preceding chapters in higher computabilitytheory.In the area or research connecting complexity and randomness, several times,properties of sets were studied independently for a while, only to be shownto coincide later. Some important results in this book show such coincidences.Other results separate properties that are conceptually close. Even if propertiesintroduced in different ways coincide, we still think of them as conceptuallydistinct.This book can be used in various ways: (1) as a reference by researchers; (2)for self-study by students; and (3) in courses at the graduate level.Such a course can lean towards computability (Chapter 1, some of Chapters4 and 6), randomness (Chapters 2, 3, 7, and 1 to the extent needed), or theinterplay between the two (Chapters 4, 5, 8, and as much as needed from otherchapters).Figure 1 displays major and minor dependencies between chapters. The latterare given by dashed lines; the labels indicate the section which depends on thepreceding chapter.The book contains many exercises and a number of problems. Often the exercisesextend the material given in the main text in interesting ways. They shouldbe attempted seriously by the student before looking at the solutions at the backof the book. The problems are currently open, possibly only because no one hastried.

viiiPreface123646.38.1, 8.45.1588.39.379Fig. 1. Major and minor dependencies between chapters.Notation is listed from page 416 on for reference. The absolute value of anumber r ∈ R is denoted by abs(r). The cardinality of a set X is denoted by#X. We use the bracket notation in sums as explained in Knuth (1992). Forinstance, ∑ n n−2 [[n is odd] denotes 1 + 1/9+1/25 + ...= π 2 /8.The following conventions on variables apply.n, m, k, l natural numbersx, y, z, v, w binary strings (often identified with numbers)σ, ρ, τ binary strings when seen as descriptions or oracle stringsA, . . . E, V,...,Z subsets of Nf,g,hfunctions N → NA, B,... classes.This book would not exist without the help of my colleagues and friends. Specialthanks to Santiago Figueira, Noam Greenberg, Bjørn Kjos-Hanssen, AntonínKučera, Antonio Montalbán, Joseph Miller, Selwyn Ng, Alex Raichev, and JanReimann. Substantial help was also provided by George Barmpalias, DavidBelanger, Laurent Bienvenue, Helen Broome, Peter Cholak, Barbara Csima,David Diamondstone, Nick Hay, Greg Hjorth, Bart Kastermans, Steffen Lempp,Ken Harris, Chris Porter, Richard Shore, Stephen Simpson, Sebastiaan Terwijn,Paul Vitanyi, and Liang Yu. I am grateful to the University of Auckland, andespecially to the department of computer science. I gratefully acknowledge supportby the Marsden fund of the Royal Society of New Zealand. I thank OxfordUniversity Press, and in particular Dewi Jackson, for their support and patience.Auckland, July 2008.

Contents1 The complexity of sets 11.1 The basic concepts 3Partial computable functions 3Computably enumerable sets 6Indices and approximations 71.2 Relative computational complexity of sets 8Many-one reducibility 8Turing reducibility 9Relativization and the jump operator 10Strings over {0, 1} 12Approximating the functionals Φ e , and the use principle 13Weak truth-table reducibility and truth-table reducibility 14Degree structures 161.3 Sets of natural numbers 161.4 Descriptive complexity of sets 18∆ 0 2 sets and the Shoenfield Limit Lemma 18Sets and functions that are n-c.e. or ω-c.e. 19Degree structures on particular classes ⋆ 20The arithmetical hierarchy 211.5 Absolute computational complexity of sets 24Sets that are low n 26Computably dominated sets 27Sets that are high n 281.6 Post’s problem 29Turing incomparable ∆ 0 2-sets 30Simple sets 31A c.e. set that is neither computable nor Turing complete 32Is there a natural solution to Post’s problem? 34Turing incomparable c.e. sets 351.7 Properties of c.e. sets 37Each incomputable c.e. wtt-degree contains a simple set 38Hypersimple sets 39Promptly simple sets 40Minimal pairs and promptly simple sets 41Creative sets ⋆ 431.8 Cantor space 45Open sets 47Binary trees and closed sets 47Representing open sets 48

xContentsCompactness and clopen sets 48The correspondence between subsets of N and real numbers 49Effectivity notions for real numbers 50Effectivity notions for classes of sets 52Examples of Π 0 1 classes 55Isolated points and perfect sets 56The Low Basis Theorem 56The basis theorem for computably dominated sets 59Weakly 1-generic sets 611-generic sets 63The arithmetical hierarchy of classes 64Comparing Cantor space with Baire space 671.9 Measure and probability 68Outer measures 68Measures 69Uniform measure and null classes 70Uniform measure of arithmetical classes 71Probability theory 732 The descriptive complexity of strings 74Comparing the growth rate of functions 752.1 The plain descriptive complexity C 75Machines and descriptions 75The counting condition, and incompressible strings 77Invariance, continuity, and growth of C 79Algorithmic properties of C 812.2 The prefix-free complexity K 82Drawbacks of C 83Prefix-free machines 83The Machine Existence Theorem and a characterization of K 86The Coding Theorem 912.3 Conditional descriptive complexity 92Basics 92An expression for K(x, y) ⋆ 932.4 Relating C and K 94Basic interactions 94Solovay’s equations ⋆ 952.5 Incompressibility and randomness for strings 97Comparing incompressibility notions 98Randomness properties of strings 993 Martin-Löf randomness and its variants 1023.1 A mathematical definition of randomness for sets 102Martin-Löf tests and their variants 104Schnorr’s Theorem and universal Martin-Löf tests 105

ContentsxiThe initial segment approach 1053.2 Martin-Löf randomness 106The test concept 106A universal Martin-Löf test 107Characterization of MLR via the initial segment complexity 107Examples of Martin-Löf random sets 108Facts about ML-random sets 109Left-c.e. ML-random reals and Solovay reducibility 113Randomness on reals, and randomness for bases other than 2 115A nonempty Π 0 1 subclass of MLR has ML-random measure ⋆ 1163.3 Martin-Löf randomness and reduction procedures 117Each set is weak truth-table reducible to a ML-random set 117Autoreducibility and indifferent sets ⋆ 1193.4 Martin-Löf randomness relative to an oracle 120Relativizing C and K 121Basics of relative ML-randomness 122Symmetry of relative Martin-Löf randomness 122Computational complexity, and relative randomness 124The halting probability Ω relative to an oracle ⋆ 1253.5 Notions weaker than ML-randomness 127Weak randomness 128Schnorr randomness 129Computable measure machines 1313.6 Notions stronger than ML-randomness 133Weak 2-randomness 1342-randomness and initial segment complexity 1362-randomness and being low for Ω 140Demuth randomness ⋆ 1414 Diagonally noncomputable functions 1444.1 D.n.c. functions and sets of d.n.c. degree 145Basics on d.n.c. functions and fixed point freeness 145The initial segment complexity of sets of d.n.c. degree 147A completeness criterion for c.e. sets 1484.2 Injury-free constructions of c.e. sets 150Each ∆ 0 2 set of d.n.c. degree bounds a promptly simple set 151Variants of Kučera’s Theorem 152An injury-free proof of the Friedberg–Muchnik Theorem ⋆ 1544.3 Strengthening the notion of a d.n.c. function 155Sets of PA degree 156Martin-Löf random sets of PA degree 157Turing degrees of Martin-Löf random sets ⋆ 159Relating n-randomness and higher fixed point freeness 160

xiiContents5 Lowness properties and K-triviality 1635.1 Equivalent lowness properties 165Being low for K 165Lowness for ML-randomness 167When many oracles compute a set 168Bases for ML-randomness 170Lowness for weak 2-randomness 1745.2 K-trivial sets 176Basics on K-trivial sets 176K-trivial sets are ∆ 0 2 177The number of sets that are K-trivial for a constant b ⋆ 179Closure properties of K 181C-trivial sets 182Replacing the constant by a slowly growing function ⋆ 1835.3 The cost function method 184The basics of cost functions 186A cost function criterion for K-triviality 188Cost functions and injury-free solutions to Post’s problem 189Construction of a promptly simple Turing lower bound 190K-trivial sets and Σ 1 -induction ⋆ 192Avoiding to be Turing reducible to a given low c.e. set 193Necessity of the cost function method for c.e. K-trivial sets 195Listing the (ω-c.e.) K-trivial sets with constants 196Adaptive cost functions 1985.4 Each K-trivial set is low for K 200Introduction to the proof 201The formal proof of Theorem 5.4.1 2105.5 Properties of the class of K-trivial sets 215A Main Lemma derived from the golden run method 215The standard cost function characterizes the K-trivial sets 217The number of changes ⋆ 219Ω A for K-trivial A 221Each K-trivial set is low for weak 2-randomness 2235.6 The weak reducibility associated with Low(MLR) 224Preorderings coinciding with LR-reducibility 226A stronger result under the extra hypothesis that A ≤ T B ′ 228The size of lower and upper cones for ≤ LR ⋆ 230Operators obtained by relativizing classes 231Studying ≤ LR by applying the operator K 232Comparing the operators S LR and K ⋆ 233Uniformly almost everywhere dominating sets 234∅ ′ ≤ LR C if and only if C is uniformly a.e. dominating 235

Contentsxiii6 Some advanced computability theory 2386.1 Enumerating Turing functionals 239Basics and a first example 239C.e. oracles, markers, and a further example 2406.2 Promptly simple degrees and low cuppability 242C.e. sets of promptly simple degree 243A c.e. degree is promptly simple iff it is low cuppable 2446.3 C.e. operators and highness properties 247The basics of c.e. operators 247Pseudojump inversion 249Applications of pseudojump inversion 251Inversion of a c.e. operator via a ML-random set 253Separation of highness properties 256Minimal pairs and highness properties ⋆ 2587 Randomness and betting strategies 2597.1 Martingales 260Formalizing the concept of a betting strategy 260Supermartingales 261Some basics on supermartingales 262Sets on which a supermartingale fails 263Characterizing null classes by martingales 2647.2 C.e. supermartingales and ML-randomness 264Computably enumerable supermartingales 265Characterizing ML-randomness via c.e. supermartingales 265Universal c.e. supermartingales 266The degree of nonrandomness in ML-random sets ⋆ 2667.3 Computable supermartingales 268Schnorr randomness and martingales 268Preliminaries on computable martingales 2707.4 How to build a computably random set 271Three preliminary Theorems: outline 272Partial computable martingales 273A template for building a computably random set 274Computably random sets and initial segment complexity 275The case of a partial computably random set 2777.5 Each high degree contains a computably random set 279Martingales that dominate 279Each high c.e. degree contains a computably randomleft-c.e. set 280A computably random set that is not partial computablyrandom 281A strictly computably random set in each high degree 283A strictly Schnorr random set in each high degree 285

xivContents7.6 Varying the concept of a betting strategy 288Basics of selection rules 288Stochasticity 288Stochasticity and initial segment complexity 289Nonmonotonic betting strategies 294Muchnik’s splitting technique 295Kolmogorov–Loveland randomness 2978 Classes of computational complexity 3018.1 The class Low(Ω) 303The Low(Ω) basis theorem 303Being weakly low for K 3052-randomness and strong incompressibility K 308Each computably dominated set in Low(Ω) is computable 309A related result on computably dominated sets in GL 1 3118.2 Traceability 312C.e. traceable sets and array computable sets 313Computably traceable sets 316Lowness for computable measure machines 318Facile sets as an analog of the K-trivial sets ⋆ 3198.3 Lowness for randomness notions 321Lowness for C-null classes 322The class Low(MLR, SR) 323Classes that coincide with Low(SR) 326Low(MLR, CR) coincides with being low for K 3288.4 Jump traceability 336Basics of jump traceability, and existence theorems 336Jump traceability and descriptive string complexity 338The weak reducibility associated with jump traceability 339Jump traceability and superlowness are equivalent for c.e. sets 341More on weak reducibilities 343Strong jump traceability 343Strong superlowness ⋆ 3468.5 Subclasses of the K-trivial sets 348Some K-trivial c.e. set is not strongly jump traceable 348Strongly jump traceable c.e. sets and benign cost functions 351The diamond operator 3568.6 Summary and discussion 361A diagram of downward closed properties 361Computational complexity versus randomness 363Some updates 3649 Higher computability and randomness 3659.1 Preliminaries on higher computability theory 366Π 1 1 and other relations 366

ContentsxvWell-orders and computable ordinals 367Representing Π 1 1 relations by well-orders 367Π 1 1 classes and the uniform measure 369Reducibilities 370A set theoretical view 3719.2 Analogs of Martin-Löf randomness and K-triviality 372Π 1 1 Machines and prefix-free complexity 373A version of Martin-Löf randomness based on Π 1 1 sets 376An analog of K-triviality 376Lowness for Π 1 1-ML-randomness 3779.3 ∆ 1 1-randomness and Π 1 1-randomness 378Notions that coincide with ∆ 1 1-randomness 379More on Π 1 1-randomness 3809.4 Lowness properties in higher computability theory 381Hyp-dominated sets 381Traceability 382Solutions to the exercises 385Solutions to Chapter 1 385Solutions to Chapter 2 389Solutions to Chapter 3 391Solutions to Chapter 4 393Solutions to Chapter 5 395Solutions to Chapter 6 399Solutions to Chapter 7 400Solutions to Chapter 8 402Solutions to Chapter 9 408References 410Notation Index 418Index 423

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1The complexity of setsWe study the complexity of sets of natural numbers. There are two interrelatedtypes of complexity.Computational.Descriptive.Informally, we ask how much (or little) the set knows.We ask how well the set can be described.In both cases, to understand the complexity of sets, we introduce classes ofsimilar complexity, namely classes of sets sharing a certain complexity property.For both types of complexity, the smallest class we will consider is the class ofcomputable sets.Classes of computational complexity. To give a mathematical definition for theintuitive notion of a computable function f : N → N, a formal model of computationis used, for instance Turing machines. The model can be extended toallow queries to an “oracle set” Z during the computations, thereby definingwhat it means for a function to be computable with oracle Z. In most cases, aclass of computational complexity is given by a condition indicating the strengthof Z as an oracle. For instance, Z is low if deciding whether a computation usingZ converges is no harder than deciding whether a computation without anoracle converges. Z is computably dominated if each function computed by Z isbounded by a computable function, and Z is high if some function computed byZ grows faster than each computable function. These classes will be studied inSection 1.5.Classes of descriptive complexity. One introduces description systems. The descriptionsare finite objects, such as first-order formulas or Turing programs,which can be encoded by natural numbers in an effective way. Formally, a descriptionsystem is simply a function F : I → P(N) where I ⊆ N. IfF (e) =Z,then e is a description of Z in that system, and the class of descriptive complexitygiven by F is the range of F . Examples include the computably enumerable sets,where F (e) =W e is the set of inputs on which the e-th Turing program halts, thearithmetical sets, and the Π 1 1 sets, a high-level analog of the c.e. sets where theenumeration takes place at stages that are computable ordinals. (C.e. sets areintroduced in Definition 1.1.8, arithmetical sets in 1.4.10, and Π 1 1 sets in 9.1.1.)Since descriptions can be encoded by natural numbers, all classes of descriptivecomplexity are countable. (In other areas of logic it can also be the case thatdescriptions are infinite, though they should be simpler than the objects they describe.In descriptive set theory, say, certain functions from N to N, called Borelcodes, describe Borel sets of real numbers. In model theory, for some complete

2 1 The complexity of setsfirst-order theories T , sequences of ordinals can describe countable models of Tup to isomorphism.)The classes of descriptive complexity we consider usually form an almost linearhierarchy given by the inclusion of classes. (The class of Π 0 2 singletons, Definition1.8.61, is one of the few exceptions to this rule.) This contrasts with thecase of computational complexity. For instance, the only sets that are low andcomputably dominated are the computable sets. Another difference is that beingin a class of descriptive complexity means the set is well-behaved in a particularsense. In contrast, classes of computational complexity will be of two types: theones consisting of sets that know little, and the ones consisting of sets that knowa lot. Classes of the first type are given by lowness properties, such as being low,or computably dominated, while classes of the second type are given by highnessproperties, such as being high.The counterpart of knowing a lot in descriptive complexity might be beinghard to describe. We will see that this is one aspect of the intuitive notionof randomness for sets. The other, related, aspect is not satisfying any exceptionalproperties (in the sense of the uniform measure on Cantor space 2 N ). InChapters 3, 7 and 9 we will introduce various classes capturing the degree ofrandomness of a set. A central one is the class of Martin-Löf random sets.So far we have only discussed the absolute complexity of a set Z, by lookingat its membership in certain classes. The relative computational complexity ismeasured by comparing Z to other sets. To do so, one introduces preorderings ≤ ron sets, called reducibilities: X ≤ r Y means that X is no more complex than Y inthe sense of ≤ r . Traditionally, a reducibility specifies a way to determine whethern ∈ X with the help of queries of the form “k ∈ Y ?” Such a method to compute Xfrom Y is called a reduction procedure. We also study weak reducibilities, whichcan be used to compare the computational complexity of sets even if there is noreduction procedure; see Section 5.6 and page 339.There are a few examples of preorderings ≤ r used to compare the relativedescriptive complexity of sets, such as enumeration reducibility (Odifreddi, 1999,Ch. XIV) and the reducibility ≤ K (5.6.1). Furthermore, some preorderings ≤ rhave been introduced where A ≤ r B expresses in some way that B is at least asrandom as A, for instance ≤ K again, ≤ S on the left-c.e. sets (3.2.28), and ≤ vLon the Martin-Löf random sets (5.6.2). No general theory has emerged so far.One of the aims of this chapter is to give a brief, but self-contained introductionto computability theory, focussing on the material that is needed later on. Topicsleft out here can often be found in Soare (1987) or Odifreddi (1989, 1999).We will rely on important meta-concepts: uniformity, relativization, and universality.Uniformity is discussed in Remark 1.1.4, and relativization before Proposition1.2.8 on page 10.We will return to the complexity of sets in Sections 1.2, 1.4, and 1.5 of thischapter, as well as in Chapters 5, 8, and 9.

1.1 The basic concepts1.1 The basic concepts 3We review fundamental concepts of computability theory: partial computablefunctions, computably enumerable sets, and computable sets. We provide animportant tool, the Recursion Theorem.Partial computable functionsOne main achievement of mathematical logic is a formal definition for the intuitiveconcept of a computable function. We mostly consider functions f : N k ↦→ N(where k ≥ 1). This is an inessential restriction since other finite objects thatcould be considered as inputs, say finite graphs, can be encoded by natural numbersin some efficient way. One obtains a mathematical definition of computablefunctions by introducing Turing machines (Turing, 1936). Such a machine has ktapes holding the inputs (say in binary), one output tape, and several internalwork tapes. A Turing machine reads and writes symbols from a finite alphabet(which includes the symbols 0 and 1) on these tapes. The input tapes areread-only, while the output tape is write-only. The behavior of the machine isdescribed by a finite sequence of instructions, called a Turing program, which iscarried out in a step-wise fashion one instruction at a time. See Odifreddi (1989)for details. The function f is computable if there is a Turing program P for themachine model with k input tapes which, for all inputs x 0 ,...,x k−1 , halts withf(x 0 ,...,x k−1 ) on the output tape.Of course, for certain inputs, a Turing program may run forever. There is noalgorithm to decide whether a program halts even on a single input, let alone onall. Thus it is natural to include partial functions in our mathematical definitionof computability.1.1.1 Definition. Let ψ be a function with domain a subset of N k and range asubset of N. We say that ψ is partial computable if there is a Turing program Pwith k input tapes such that ψ(x 0 ,...,x k−1 )=y iff P on inputs x 0 ,...,x k−1outputs y. We write ψ(x 0 ,...,x k−1 )↓ if P halts on inputs x 0 ,...,x k−1 .Wesaythat ψ is computable if ψ is partial computable and the domain of ψ is N k .Many other formal definitions for the intuitive notion of a computable functionwere proposed. All turned out to be equivalent. This lends evidence to theChurch–Turing thesis which states that any intuitively computable function iscomputable in the sense of Definition 1.1.1. More generally, each informally givenalgorithmic procedure can be implemented by a Turing program. We freely usethis thesis in our proofs: we give a procedure informally and then take it forgranted that a Turing program implementing it exists.Fix k and an effective listing of the Turing programs for k inputs. Let Pek bethe program for k inputs given by the e-th program. Let Φ k e denote the partialcomputable function with k arguments given by Pe k .IfΦ = Φ k e then e is calledan index for Φ. Often there is only one argument, and instead of Φ 1 e we writeΦ e . (1.1)

4 1 The complexity of setsThe following notation is frequently used when dealing with partial functions.Given expressions α, β,α ≃ βmeans that either both expressions are undefined, or they are defined with thesame value. For instance, √ (r + 1)(r − 1) ≃ √ r 2 − 1 for r ∈ R.A universal Turing program. The function Ξ(e, x) ≃ Φ e (x) is partial computablein the intuitive sense. The informal procedure is: on inputs e, x, fetchthe e-th Turing program P 1 e and run it on input x. If its computation halts withoutput y then give y as an output. By the Church–Turing thesis, Ξ is partial computable.A Turing program computing Ξ is called a universal Turing program.The following theorem states that Ξ can emulate partial computable functionsin two arguments whenever the Turing programs are listed in an appropriateeffective way. For details see Odifreddi (1989).1.1.2 Theorem. (Parameter Theorem) For each partial computable function Θin two variables there is a computable strictly increasing function q such that∀e ∀x Φ q(e) (x) ≃ Θ(e, x).An index for q can be obtained effectively from an index for Θ.Proof idea. Given a Turing program P for Θ, we obtain the program Pq(e) 1 bymaking the first input e part of the program code.✷For a formal proof, one would need to be more specific about the effectivelisting of Turing programs. The same applies to the next result.1.1.3 Lemma. (Padding Lemma) For each e and each m, one may effectivelyobtain e ′ >msuch that the Turing program P e ′ behaves exactly like P e .Proof idea. We obtain the program P e ′ from P e by adding sufficiently muchcode which is never executed.✷1.1.4 Remark. (Uniformity) In subsequent results we will often make statementslike the one in the last line of Theorem 1.1.2: we do not merely assert theexistence of an object, but actually that its description (within some specifieddescription system) can be computed from descriptions of the given objects. Theformal version of the last line in Theorem 1.1.2 is: there is a computable functionh such that if Θ = Φ 2 e, then ∀i ∀x Φ q(i) (x) ≃ Θ(i, x) holds for q = Φ 1 h(e) .One says that the construction is uniform, namely, there is a single procedure toobtain the desired object from the ingredients which works for each collection ofgiven objects. Proofs of basic results are usually uniform. More complex proofscan be nonuniform. We will at times be able to show this is necessarily so, forinstance in Proposition 5.5.5.The Recursion Theorem is an important technical tool. It was proved by Kleene(1938) in a paper on ordinal notations. Informally, it asserts that one cannotchange in an effective way the input/output behavior of all Turing programs.

1.1 The basic concepts 51.1.5 Recursion Theorem. Let g : N ↦→ N be computable. Then there is an esuch that Φ g(e) = Φ e . We say that e is a fixed point for g.Proof. By the Parameter Theorem 1.1.2 there is a computable function q suchthat Φ q(e) (x) ≃ Φ g(Φe(e))(x) for all e, x. Choose an i such that q = Φ i , thenSo e = Φ i (i) =q(i) is a fixed point.Φ q(i) = Φ Φi(i) = Φ g(Φi(i)). (1.2)We obtained the index i for q effectively from an index for g, by the uniformityof the Parameter Theorem. Thus, if g is a computable function of two arguments,we can compute a fixed point e = f(n) for each function g n given by g n (e) =g(e, n). Taking the uniformity one step further, note that an index for f can beobtained effectively from an index for g. This yields an extended version:1.1.6 Recursion Theorem with Parameters. Let g : N 2 ↦→ N be computable.Then there is a computable function f, which can be obtained effectivelyfrom g, such that Φ g(f(n),n) = Φ f(n) for each n.✷The incompleteness theorem of Gödel (1931) states that for each effectively axiomatizablesufficiently strong consistent theory T in the language of arithmetic one canfind a sentence ɛ which holds in N but is not provable in T . Peano arithmetic is anexample of such a theory. The incompleteness theorem relies on a fixed point lemmaproved in a way analogous to the proof of the Recursion Theorem. One represents aformula σ in the language of arithmetic by a natural number σ. This is the analog ofrepresenting a partial computable function Ψ by an index e, in the sense that Ψ = Φ e.Notice the “mixing of levels” that is taking place in both cases: a partial computablefunction of one argument is applied to a number, which can be viewed as an index fora function. A formula in one free variable is evaluated on a number, which may be acode for a further formula. The fixed point lemma says that, for each formula Γ(x) inone free variable, one can determine a sentence ɛ such thatT ⊢ ɛ ↔ Γ(ɛ).Informally, ɛ asserts that it satisfies Γ itself. Roughly speaking, if Γ(x) expresses thatthe sentence x is not provable from T , then ɛ asserts of itself that it is not provable,hence ɛ holds in N but T ⊬ ɛ.In the analogy between Gödel’s and Kleene’s fixed point theorems, Γ plays the roleof the function g. Equivalence of sentences under T corresponds to equality of partialcomputable functions. One obtains the fixed point ɛ as follows: the map F (σ) =σ(σ),where σ is a formula in one free variable, is computable, and hence can be representedin T by a formula ψ in two free variables (here one uses that T is sufficiently strong;we skip technical details). Hence there is a formula α expressing “Γ(F (σ))”, or moreprecisely ∃y[ψ(σ,y)&Γ(y)]. Thus, for each formula σT ⊢ α(σ) ↔ Γ(σ(σ)).Forming the sentence σ(σ) is the analog of evaluating Φ e(e), and α is the analog ofthe function q. Now let ɛ be α(α). Since α is the analog of the index i for q, ɛ (that is,the result of evaluating α on its own code number) is the analog of Φ i(i) (the resultof applying q to its own index). As in the last line (1.2) of the proof of the RecursionTheorem, one obtains that T ⊢ ɛ ↔ α(α) ↔ Γ(α(α)).✷

6 1 The complexity of sets1.1.7 Exercise. Extend the Recursion Theorem by showing that each computablefunction g has infinitely many fixed points. Conclude that the function f in Theorem1.1.6 can be chosen one-one.Computably enumerable sets1.1.8 Definition. We say that a set A ⊆ N is computably enumerable (c.e., forshort) if A is the domain of some partial computable function.The reason for choosing this term will become apparent in 1.1.15. LetW e = dom(Φ e ). (1.3)Then (W e ) e∈N is an effective listing of all c.e. sets. A sequence of sets (S e ) e∈Nsuch that {〈e, x〉: x ∈ S e } is c.e. is called uniformly computably enumerable. Anexample of such a sequence is (W e ) e∈N .The characteristic function f of a set A is given by f(x) =1ifx ∈ A andf(x) = 0 otherwise; A and f are usually identified. A is called computable if itscharacteristic function is computable; otherwise A is called incomputable.1.1.9 Proposition. A is computable ⇔ A and N − A are c.e.Proof. ⇒: If A is computable, there is a program Q 0 that halts on input x iffx ∈ A, and a program Q 1 that halts on input x iff x ∉ A.⇐: By the Church–Turing thesis it suffices to give an informal procedure forcomputing A. Fix programs Q 0 ,Q 1 such that Q 0 halts on input x iff x ∈ A, andQ 1 halts on input x iff x ∉ A. To decide whether x ∈ A, run the computationsof Q 0 on x and of Q 1 on x in parallel until one of them halts. If Q 0 halts first,output 1, otherwise output 0.✷We may obtain a c.e. incomputable set denoted ∅ ′ by a direct diagonalization.We define ∅ ′ in such a way that N −∅ ′ differs from W e at e: let∅ ′ = {e: e ∈ W e }.The reason for choosing this notation becomes apparent in 1.2.9. The set ∅ ′ iscalled the halting problem, since e ∈∅ ′ iff program P 1 e halts on input e. (It is oftendenoted by K, but we reserve this letter for prefix-free Kolmogorov complexity.)1.1.10 Proposition. The set ∅ ′ is c.e. but not computable.Proof. ∅ ′ is c.e. since ∅ ′ = dom(J), where J is the partial computable functiongiven by J(e) ≃ Φ e (e). If ∅ ′ is computable then there is e such that N −∅ ′ = W e .Then e ∈∅ ′ ↔ e ∈ W e ↔ e ∉∅ ′ , contradiction. (This is similar to Russell’sparadox in set theory.)✷The sequence (W e ) e∈N is universal for uniformly c.e. sequences.1.1.11 Corollary. For each uniformly c.e. sequence (A e ) e∈N there is a computablefunction q such that A e = W q(e) for each e.

1.2 Relative computational complexity of sets 9If X is computable, Y ≠ ∅, and Y ≠ N, then X ≤ m Y : choose y 0 ∈ Y andy 1 ∉ Y . Let f(n) =y 0 if n ∈ X, and f(n) =y 1 otherwise. Then X ≤ m Y via f.Thus, disregarding ∅ and N, the computable sets form the least many-one degree.For each set Y the class {X : X ≤ m Y } is countable. In particular, there is nogreatest many-one degree. However, ∅ ′ is the most complex among the c.e. setsin the sense of ≤ m :1.2.2 Proposition. A is c.e. ⇔ A ≤ m ∅ ′ .An index for the many-one reduction as a computable function can be obtainedeffectively from a c.e. index for A, and conversely.Proof. ⇐: If A ≤ m ∅ ′ via h, then A = dom(Ψ) where Ψ(x) ≃ J(h(x)) (recallthat J(e) ≃ Φ e (e)). So A is computably enumerable.⇒: We claim that there is a computable function g such that{{e} if n ∈ A,W g(e,n) =∅ else.For let Θ(e, n, x) converge if x = e and n ∈ A. By a three-variable versionof the Parameter Theorem 1.1.2, there is a computable function g such that∀e, n, x [Θ(e, n, x) ≃ Φ g(e,n) (x)]. By Theorem 1.1.6, there is a computable functionh such that W g(h(n),n) = W h(n) for each n. Thenn ∈ A ⇒ W h(n) = {h(n)} ⇒ h(n) ∈∅ ′ , andn ∉ A ⇒ W h(n) = ∅ ⇒ h(n) ∉∅ ′ .The uniformity statements follow from the uniformity of Theorem 1.1.6.1.2.3 Definition. A c.e. set C is called r-complete if A ≤ r C for each c.e. set A.Usually ≤ m implies the reducibility ≤ r under consideration. Then, since ∅ ′ is m-complete, a c.e. set C is r-complete iff ∅ ′ ≤ r C. An exception is 1-reducibility,which is more restricted than ≤ m : we say that X ≤ 1 Y if X ≤ m Y via a one-onefunction f.Exercises.1.2.4. The set ∅ ′ is 1-complete. (This will be strengthened in Theorem 1.7.18.)1.2.5. (Myhill) X ≡ 1 Y ⇔ there is a computable permutation p of N such thatY = p(X). (For a solution see Soare 1987, Thm. I.5.4.)Turing reducibilityMany-one reducibility is too restricted to serve as an appropriate measure forthe relative computational complexity of sets. Our intuitive understanding of “Yis at least as complex as X” is: X can be computed with the help of Y (or, “Xcan be computed relative to Y ”). If X ≤ m Y via h, then this holds via a veryparticular type of relative computation procedure: on input x, compute k = h(x)and output 1 (“yes”) if k ∈ Y , and 0 otherwise. To formalize more general waysof relative computation, we extend the machine model by a one-way infinite✷

10 1 The complexity of sets“oracle” tape which holds all the answers to oracle questions of the form “is kin Y ?”. The tape has a 1 in position k if k ∈ Y , otherwise it has a 0. To make thequery, the machine moves the head on the oracle tape to position k and checkswhether the entry at that position is 1.Extending the definitions at (1.1) to oracle Turing machines, from now on wewill view the effective listing (Φ e ) e∈N as a listing of partial functions depending ontwo arguments, the oracle set and the input. We write Φ Y e (n)↓ if the program P ehalts when the oracle is Y and the input is n; we write Φ e (Y ; n), or Φ Y e (n) for thisoutput. We also use the notation Φ Y e (n) ↑ for the negation of Φ Y e (n) ↓. The Φ eare called Turing functionals. Extending (1.3), we letW Y e = dom(Φ Y e ). (1.4)In this context we call W e a c.e. operator. Turing functionals will be studied inmore detail in Section 6.1, and c.e. operators in Section 6.3.1.2.6 Definition. A total function f : N ↦→ N is called Turing reducible to Y ,orcomputable relative to Y ,orcomputable in Y , if there is an e such that f = Φ Y e .We denote this by f ≤ T Y . We also say that Y computes f. ForasetA, wewrite A ≤ T Y if the characteristic function of A is Turing reducible to Y .Sometimes we also consider Turing reductions to total functions g. Then f ≤ T gmeans that f is Turing reducible to the graph of g, that is, to {〈n, g(n)〉: n ∈ N}.1.2.7 Exercise. Verify that ≤ m and ≤ T are preorderings of the subsets of N.Relativization and the jump operatorThe process of extending definitions, facts, and even proofs from the case involvingplain computations to the case of computations relative to an oracle is calledrelativization. For instance, in Definition 1.2.6, we relativized the notion of acomputable function to obtain the notion of a function computable in Y . Recallthat W Y e = dom(Φ Y e ). A set A is c.e. relative to Y (or c.e. in Y )ifA = W Y e forsome e. Any notation introduced for the unrelativized case will from now on beviewed as a shorthand for the oracle version where the oracle is ∅. For instance,we view Φ e as a shorthand for Φ ∅ e.The relativization of Proposition 1.1.9 is as follows.1.2.8 Proposition. A is computable in Y ⇔ A and N − A are c.e. in Y .It is proved by viewing the proof of Proposition 1.1.9 relative to an oracle. (Notethat we now assume a version of the Church–Turing thesis with oracles.)Relativizing the halting problem to Y yields its Turing jump Y ′ . This importantoperation was introduced by Kleene and Post (1954).1.2.9 Definition. We write J Y (e) ≃ Φ Y e (e). The set Y ′ = dom(J Y )istheTuring jump of Y . The map Y → Y ′ is called the jump operator.By the oracle version of the Church–Turing thesis, J is a Turing functional.

1.2 Relative computational complexity of sets 11When relativizing, special care should be applied that every computation involvedbecomes a computation with oracle Y . However, in some lucky cases aproof designed for the computable case and yielding some computable objectactually works for all oracles, and the resulting object is always computable. Forinstance, in the relativized proof of the Parameter Theorem 1.1.2, the function qis computable, and independent of the oracle. Thus, for each functional Θ thereis a computable function q such that, for each oracle Y and each pair of argumentse, x, wehaveΦ Y q(e) (x) ≃ ΘY (e, x). As a consequence one obtains a versionof the Recursion Theorem 1.1.6 for Turing functionals.1.2.10 Theorem. For each computable binary function g there is a computablefunction f such that Φ Y g(f(n),n) = ΦY f(n)for each set Y and each number n.✷The proof of Proposition 1.2.2 (that the halting problem is m-complete) uses theRecursion Theorem. So we obtain a version relative to an oracle, but still withunrelativized m-reducibility:1.2.11 Proposition. A is c.e. in Y iff A ≤ m Y ′ . ✷Relativizing Proposition 1.1.10 (the halting problem is c.e. but incomputable),we obtain that the jump produces a set that is c.e. relative to the given set andnot Turing below it.1.2.12 Proposition. For each Y , the set Y ′ is c.e. relative to Y . Also, Y ≤ m Y ′and Y ′ ≰ T Y , and therefore Y< T Y ′ .Proof. Y ′ is c.e. in Y since Y ′ = dom(J Y ). As Y is c.e. relative to itself, byProposition 1.2.11 Y ≤ m Y ′ .IfY ′ ≤ T Y then there is e such that N − Y ′ = W Y e .Then e ∈ Y ′ ↔ e ∈ W Y e ↔ e ∉ Y ′ , contradiction. ✷1.2.13 Definition. We define Y (n) inductively by Y (0) = Y and Y (n+1) =(Y (n) ) ′ .ThusY< T Y (1) < T Y (2) < T ... by Proposition 1.2.12.The following relates the reducibilities ≤ m and ≤ T via the jump operator.1.2.14 Proposition. For each Y,Z, we have Y ≤ T Z ⇔ Y ′ ≤ m Z ′ .Proof. ⇒: The set Y ′ is c.e. in Y and hence c.e. in Z. Therefore Y ′ ≤ m Z ′ byProposition 1.2.11.⇐: By Proposition 1.2.8, Y and N − Y are c.e. in Y .SoY,N − Y ≤ m Y ′ ≤ m Z ′ ,whence both Y and N−Y are c.e. in Z by Proposition 1.2.8 again. Hence Y ≤ T Z.✷We will frequently use the fact that the jump is a universal Turing functional:1.2.15 Fact. From a Turing functional Φ = Φ e one can effectively obtain acomputable strictly increasing function p, called a reduction function for Φ, suchthat ∀Y ∀x Φ Y (x) ≃ J Y (p(x)).

12 1 The complexity of setsProof. Let Θ Y (x, y) ≃ Φ Y (x) (an index for Θ is obtained effectively). By theoracle version of the Parameter Theorem, there is a computable strictly increasingfunction p such ∀Y ∀y Φ Y p(x) (y) ≃ ΘY (x, y) ≃ Φ Y (x). Letting y = p(x) weobtain J Y (p(x)) = Φ Y p(x) (p(x)) = ΦY (x).✷We often apply this fact without an oracle. Thus, from a partial computablefunction α = Φ ∅ e one can effectively obtain a reduction function p such that∀x α(x) ≃ J(p(x)).Fact 1.2.15 yields a machine independent characterization of the jump: if Ĵ isa further universal Turing functional, there is a computable permutation π ofN such that Ĵ Y (x) ≃ J Y (π(x)) for each Y,x. This is proved in the same wayas Myhill’s Theorem (Exercise 1.2.5). An example of such an alternative jumpoperator is Ĵ X (y) ≃ Φ X e (n) where y = 〈e, n〉.We provide a further useful variant of the Recursion Theorem due to Smullyan.Given computable functions g and h one may obtain a pair of fixed points.1.2.16 Double Recursion Theorem. Given computable binary functions g, h,one can effectively obtain numbers a, b such that Φ Y g(a,b) = ΦY a & Φ Y h(a,b) = ΦY bfor each Y .Proof. By the Recursion Theorem with Parameters 1.2.10, there is a computablefunction f such that for each Y and each n we have Φ Y g(f(n),n) = ΦY f(n) .Now apply the Recursion Theorem to the function λn.h(f(n),n) in order toobtain a fixed point b, and let a = f(b). Then a, b is a pair as required. ✷Strings over {0, 1}To proceed we need some more terminology and notation. An element of {0, 1}is called a bit. Finite sequences of bits are called strings.String notationThe set of all strings is denoted by {0, 1} ∗ . The letters σ, ρ, τ, x, y, z will usuallydenote strings. The following notation is fairly standard.στ concatenation of σ and τσa σ followed by the symbol aσ ≼ τ σ is a prefix of τ, that is, ∃ρ [σρ = τ]σ | τ σ, τ are incompatible i.e. neither of σ, τ is a prefix of the otherσ < L τ σ is to the left of τ, that is, ∃ρ [ρ0 ≼ σ & ρ1 ≼ τ]|σ| the length of σ∅ the empty string, that is, the string of length 0.We picture {0, 1} ∗ as a tree growing upwards, with σ0 to the left of σ1. Therelation < L is a linear order called the lexicographical order. For a set Z,Z ↾ n denotes the string Z(0)Z(1) ...Z(n − 1).The notations σ ≼ Z , σ < L Z, etc., are used in the obvious senses. For sets Y,Zwe write Y< L Z if ∃ρ [ρ0 ≺ Y & ρ1 ≺ Z].

1.2 Relative computational complexity of sets 13Identifying strings with natural numbersFrequently strings, not merely natural numbers, are the bottom objects. In thiscase we want to apply notation developed for natural numbers in the setting ofstrings, so it will be useful to identify strings with natural numbers. We couldidentify σ with a strong index for the nonempty finite set {|σ|} ∪ {i: σ(i) =1},in other words, with the number that has binary representation 1σ. The onlyproblem with this is that 0 does not correspond to a string. To avoid this,we identify σ ∈ {0, 1} ∗ with n ∈ N s.t. the binary representation of n +1 is 1σ.For instance, the string 000 is identified with the number 7, since the binaryrepresentation of 8 is 1000. Also, the string 10 is identified with the number 5.The empty string ∅ is identified with 0. If we want to make the identificationexplicit, we writen = number(σ) and σ = string(n). (1.5)Note that number(0 i )=2 i − 1 and number(1 i )=2 i+1 − 2. Thus, the interval[2 i −1, 2 i+1 −1) is identified with the strings of length i. The length-lexicographicalorder where the more significant bits are on the left is the linear ordering onbinary strings given by {〈x, y〉: number(x) < number(y)}.LogarithmFor n ∈ N + , we let log n = max{k ∈ N: 2 k ≤ n}. Then n ≥ 2 log n >n/2.If σ = string(n), then |σ| = log(n + 1). For instance, if n =2 i − 1, then σ =string(n) =0 i , and |σ| = log 2 i = i; ifn =2 i+1 − 2, then σ = string(n) =1 i , and|σ| = log(2 i+1 − 1) = i. (The usual real-valued logarithm is denoted by log 2 x.)Approximating the functionals Φ e , and the use principleA general convention when we are dealing with an approximation at a stage s isthat all numbers that matter to a computation at stage s should be less than s.For instance, when approximating a functional Φ e , at stage s we only allow oraclequestions less than s. We extend Definition 1.1.13 to oracle computations.1.2.17 Definition. We write Φ Y e,s(x) =y if e, x, y < s and the computation ofprogram P e on input x yields y in at most s computation steps, with all oraclequeries less than s. We write Φ Y e,s(x) ↓ if there is y such that Φ e,s (x) =y, andΦ Y e,s(x)↑ otherwise. Further, we let W Y e,s = dom(Φ Y e,s).The use principle is the fact that a terminating oracle computation only asksfinitely many oracle questions. Hence (Φ Y e,s) s∈N approximates Φ Y e , namely,Φ Y e (x) =y ↔ ∃s Φ Y e,s(x) =y.1.2.18 Definition. The use of Φ Y e (x), denoted use Φ Y e (x), is defined if Φ Y e (x)↓,in which case its value is 1+the largest oracle query asked during this computation(and 1 if no question is asked at all). Similarly, use Φ Y e,s(x) is 1+the largestoracle question asked up to stage s.

14 1 The complexity of setsWe writeΦ σ e (x) =yif Φ F e (x) yields the output y, where F = {i

1.2 Relative computational complexity of sets 15⇐: Consider the following procedure relative to an oracle Z: on input n, firstcompute D g(n) .Ifσ ≼ Z for some σ ∈ D g(n) , output 1, otherwise output 0. Bythe oracle version of the Church–Turing thesis, a Turing program Pe 1 formalizingthe procedure exists, and Φ e is the corresponding functional. Clearly Φ Z e is totalfor each oracle Z.(ii) Let Φ e be the Turing functional of the previous paragraph. Then for each Zthe function use Φ Z e (n) is bounded by max{|σ|: σ ∈ D g(n) }.✷By (i), the truth-table reduction procedures correspond to computable functionsg. Thus, our effective listing of the partial computable functions yields aneffective listing of reduction procedures which includes all the tt-reductions.1.2.22 Proposition. f ≤ tt A ⇔ there is a Turing functional Φ and a computablefunction t such that f = Φ A and the number of steps needed to computeΦ A (n) is bounded by t(n).Proof. ⇒: Suppose f = Φ A and Φ Z is total for each oracle Z. Let g be as inthe proof of implication “⇒” of Proposition 1.2.21(i). Then t(n) = max{|σ|: σ ∈D g(n) } bounds the running time of the computation Φ Z (n) for each oracle Z.⇐: Let ˜Φ be the Turing functional such that ˜Φ Z (n) =Φ Z t(n)(n) if the latter isdefined, and ˜Φ Z (n) = 0 otherwise. Then ˜Φ Z is total for each Z and ˜Φ A = Φ A .Hence f ≤ tt A.✷Note that, by the proof of “⇒”, every function f ≤ tt A is bounded from aboveby a computable function: for each n, f(n) ≤ max{Φ σ |σ|(n): σ ∈ D˜g(n)}.Clearly X ≤ m Y implies X ≤ tt Y (as an exercise, specify a computable functiong as in Proposition 1.2.21). To summarize, the implications between ourreducibilities are≤ m ⇒≤ tt ⇒≤ wtt ⇒≤ T .None of the converse implications hold. In fact, the classes of complete sets differ.(Recall from Definition 1.2.3 that a c.e. set C is r-complete if A ≤ r C for each c.e. set A.)A hypersimple set (1.7.5 below) can be Turing-complete, but is never wtt-completeby 4.1.15. A simple set (1.6.2 below) can be tt-complete, but is never m-completeby Odifreddi (1989). A natural example of a set which is wtt- but not tt-completecomes from algorithmic randomness: the set of (binary) rationals in [0, 1] that are lessthan Chaitin’s halting probability Ω. See Section 3.2 for definitions, Proposition 3.2.30for wtt-completeness of this set and Theorem 4.3.9 for its tt-incompleteness. A directconstruction of such a set is also possible, but cumbersome (Odifreddi, 1989, III.9).Exercises. The effective disjoint union of sets A and B isA ⊕ B = {2n: n ∈ A} ∪ {2n +1: n ∈ B}.1.2.23. (i) Show that A, B ≤ m A ⊕ B.(ii) Let ≤ r be one of the reducibilities above. Then, for any set X,A, B ≤ r X ↔ A ⊕ B ≤ r X.1.2.24. Let C = A 0 ∪A 1 where A 0,A 1 are c.e. and A 0 ∩A 1 = ∅. Then C ≡ wtt A 0 ⊕A 1.1.2.25. Show that ∃Zf ≤ tt Z ⇔ there is a computable h such that ∀nf(n) ≤ h(n).

16 1 The complexity of setsDegree structuresOne can abstract from the particularities of a set and only consider its relativecomputational complexity, measured by a reducibility ≤ r . The equivalenceclasses of the equivalence relation given byX ≡ r Y↔ X ≤ r Y ≤ r Xare called r-degrees. The r-degree of a set X is denoted by deg r (X). The r-degrees form a partial order denoted by D r . We state some basic facts about D rfor a reducibility ≤ r between ≤ m and ≤ T . In the case of many-one reducibilitywe disregard the sets ∅ and N. For proofs see for instance Odifreddi (1989).A structure (U, ≤, ∨) isanuppersemilattice if (U, ≤) is a partial order and, foreach x, y ∈ U, x ∨ y is the least upper bound of x and y.1.2.26 Fact.(i) The least element of D r is 0 = deg r ({0}), the degree consisting of thecomputable sets.(ii) D r is an uppersemilattice, where the least upper bound of the degrees ofsets A and B is given by the degree of A ⊕ B.(iii) For each a ∈ D r the set {b: b ≤ a} is countable.(iv) D r has cardinality 2 ℵ0 .✷By 1.2.12, D r has no maximal elements: for each A, wehaveA ′ > r A. By 1.2.14,the jump operator induces a map ′ : D T ↦→ D T , given by deg T (X) ↦→ deg T (X ′ ),called the Turing jump. This map is monotonic: for each pair x, y ∈ D T we havex ≤ y → x ′ ≤ y ′ . Note that 0 < 0 ′ < 0 ′′ x such that x ′ = y ′ .1.2.27 Definition. Let (U, ≤, ∨) be an uppersemilattice, and let I ⊆ U benonempty. We say I is an ideal of U if I is closed downward, and x, y ∈ I impliesx ∨ y ∈ I. For instance I = {x: ∃n x ≤ 0 (n) } is an ideal in D T , called the idealof arithmetical degrees. The ideal I is called principal if I = {b: b ≤ a} for somea ∈ U.1.3 Sets of natural numbersSets of natural numbers are important objects of study in computability theory.They can be identified with infinite sequences of bits, and also with the realnumbers r such that in 0 ≤ r

1.3 Sets of natural numbers 17for more on this). Strings are important for us because they represent the finiteinitial segments of sets. The global view is to think of the set as a single entity.In the ideal case we would like to give a description of the set.Let us step back and consider some alternatives to sets of natural numbers.How essential is the use of the natural numbers for the indexing of bits? Onecan also use other effectively given domains D, such as the tree {0, 1} ∗ of finitestrings over {0, 1}, or the rationals. Instead of subsets of N, one now studiessubsets of {0, 1} ∗ , or subsets of Q.1. The local view changes, since we have a different perception of what thefinite “parts” of a subset of D are. For instance, if D = {0, 1} ∗ , a finite partmight consist of the labeled finite tree of the bits up to level n. IfD = Q it isunclear how one would define a notion of finite part taking the order structureof D into account. The following definition provides a reasonable notion of finitepart for a subset of an arbitrary domain D. Afinite assignment for D is asequence α =(〈d 0 ,r 0 〉,...,〈d k−1 ,r k−1 〉) where all d i ∈ D are distinct, k ∈ N,and r i ∈ {0, 1}. IfZ : D → {0, 1}, we think of α as a part of Z if Z(d i )=r i foreach i

18 1 The complexity of setspreference to sets. In particular, the theory of algorithmic randomness relies onthe uniform measure and therefore only works for sets.We could be slightly more general and work with functions N → {0,...,b− 1}for some fixed b ∈ N − {0, 1}. They form a compact space on which the uniformmeasure can be defined, and can be identified with the real numbers r such thatin 0 ≤ r

1.4 Descriptive complexity of sets 191.4.2 Lemma. (Shoenfield Limit Lemma)Z is ∆ 0 2 ⇔ Z ≤ T ∅ ′ . The equivalence is uniform.Proof. ⇐: Fix a Turing functional Φ e such that Z = Φ ∅′ . Then the requiredcomputable approximation is given by Z s = {x x: Z s (x) ≠ Z s−1 (x)} for each x.(ii) If Z s (s − 1) = 0 for each s>0 and b(x) can be chosen constant of value n,then we say Z is n-c.e.Thus, Z is 1-c.e. iff Z is c.e., and Z is 2-c.e. iff Z = A − B for c.e. sets A, B.1.4.4 Proposition. Z is ω-c.e. ⇔ Z ≤ wtt ∅ ′ ⇔ Z ≤ tt ∅ ′ .The equivalences are effective.Proof. First suppose that Z ≤ wtt ∅ ′ via a functional Φ e with computable usebound f. To show that Z is ω-c.e., as before let Z s = {x

20 1 The complexity of setslisting of all such (possibly partial) truth-table reduction procedures defined oninitial segments of N. Then Proposition 1.4.4 yields an indexing of the ω-c.e. setsthat includes computable approximations:1.4.5 Definition. The ω-c.e. set with index e is V e = {x : Θ ∅′e (x) =1}. A computableapproximation of V e is given by V e,s = {x : Θe ∅′ (x)[s] =1}.Thus, if Θ e (x) is undefined then V e,s (x) = 0 for each s. By 1.2.2, for each e weuniformly have a many-one reduction of W e to ∅ ′ . Hence there is a computablefunction g such that W e = V g(e) for each e.The hierarchy of descriptive complexity classes introduced so far iscomputable ⊂ c.e. ⊂ 2-c.e. ⊂ 3-c.e. ⊂ ...⊂ ω-c.e. ⊂ ∆ 0 2. (1.6)It is proper even when one considers the Turing degrees of sets at the variouslevels (see Odifreddi 1989). For instance, there is a 2-c.e. set Z such that no setY ≡ T Z is c.e., and there is a ∆ 0 2 set Z such that no set Y ≡ T Z is ω-c.e.The definitions of ∆ 0 2 sets and ω-c.e. sets can be extended to functions g : N → N.1.4.6 Definition. A function g is ∆ 0 2 if there is a binary computable functionλx, s.g s (x) such that ∀xg(x) = lim s g s (x). Moreover, g is ω-c.e. if if there is, inaddition, a computable bound b such that b(x) ≥ #{s >x: g s (x) ≠ g s−1 (x)}for each x.One can extend the Limit Lemma and the first equivalence of Proposition 1.4.4 tofunctions. The second equivalence in 1.4.4 fails for functions by Exercise 1.2.25 andsince an ω-c.e. function need not be bounded by a computable function.1.4.7 Exercise. Let g : N → N be a function.(i) g is ∆ 0 2 ⇔ g ≤ T ∅ ′ .(ii) g is ω-c.e. ⇔ g ≤ wtt ∅ ′ .1.4.8 Exercise. (Mohrherr, 1984) Let E ≥ tt ∅ ′ . Then Z ≤ wtt E implies Z ≤ tt E.The following will be needed later.1.4.9 Fact. There is a binary function q ≤ T ∅ ′ with the following property: for eachω-c.e. function g there is an e such that ∀ng(n) =q(e, n).Proof. On inputs e, n, use ∅ ′ to determine whether ∃s ∀σ [|σ| = s → Φ σ e (n) ↓]. If so,let q(e, n) =Φ ∅′e (n)[s], otherwise q(e, n) =0.Ifg ≤ tt ∅ ′ via Φ e as in Definition 1.2.20,then ∀ng(n) =q(e, n).✷Degree structures on particular classes ⋆Recall from the beginning of this chapter that the three approaches to measure thecomplexity of sets are via the descriptive complexity, the absolute computational, andthe relative computational complexity. These approaches are related in several ways.One connection is that classes of computational complexity can actually comprise theleast degree of a reducibility; see Section 5.6. Here we consider another type of connection.For a reducibility ≤ r, one can study the r-degrees of the sets in a particularclass of similar complexity C. In this way, one arrives at interesting degree structures,

1.4 Descriptive complexity of sets 21for instance R T , the Turing degrees of c.e. sets, and D T (≤ 0 ′ ), the Turing degreesof the ∆ 0 2 sets. This is a bit more natural when C is closed downward under ≤ r,say,when C is the class of ∆ 0 2 sets and ≤ r is Turing reducibility. Given n, the class of n-c.e.sets is merely closed downward under ≤ m. Recall from Fact 1.2.26 that 0 is the degreeconsisting of the computable sets. By the results in Section 1.2 and the Limit Lemma,all the degree structures discussed here have a greatest degree, the degree of the haltingproblem, denoted by 1. All degree structures are uppersemilattices because the relevantclasses are closed under ⊕.In the following we list some properties of the Turing degree structures on classes inthe hierarchy (1.6). Like most theorems on degree structures, they can be expressed inthe first-order language of partial orders. For details on these often difficult results seeOdifreddi (1989, 1999) or Soare (1987).In D T (≤ 0 ′ ), and even in the Turing degrees of ω-c.e. sets, there is a minimal element,that is, there is a degree a > 0 such that x < a implies x = 0 (Sacks, 1963b). The n-c.e.degrees do not have minimal elements (Lachlan; see Odifreddi, 1999, XI.5.9b).R T is dense, i.e., for each a < b there is c such that a < c < b (Sacks, 1964).The structures of n-c.e. and of ω-c.e. degrees have a maximal incomplete element,i.e., there is a < 1 such that a < x implies x = 1 (Cooper, Harrington, Lachlan, Lemppand Soare, 1991). In contrast, D T (≤ 0 ′ ) has no maximal incomplete element.The arithmetical hierarchyUp to now we have defined classes of descriptive complexity via computableapproximations, possibly with extra conditions. This led to the hierarchy (1.6)on page 20. To obtain more powerful description systems, we will replace thisdynamic way of describing a set by descriptions using the first-order language ofarithmetic (with signature containing the symbols +, ×). Computable relationsare first-order definable in the language of arithmetic (Kaye, 1991, Thm. 3.3), sowe may as well suppose that, for k ≥ 1 and each computable relation on N k , thesignature contains a k-place relation symbol. For each description system, we useas descriptions the first-order formulas in this extended language satisfying certainsyntactic conditions. In this way we will also obtain alternative, equivalentdescription systems for the computable, the c.e., and the ∆ 0 2 sets.1.4.10 Definition. Let A ⊆ N and n ≥ 1.(i) A is Σ 0 n if x ∈ A ↔∃y 1 ∀y 2 ...Qy n R(x, y 1 ,...,y n ), where R is a symbol fora computable relation, Q is “∃” ifn is odd and Q is “∀” ifn is even.(ii) A is Π 0 n if N − A is Σ 0 n, that is x ∈ A ↔ ∀y 1 ∃y 2 ...Qy n S(x, y 1 ,...,y n ),where S is a symbol for a computable relation, Q is “∀” ifn is odd and Q is “∃”if n is even.(iii) A is arithmetical if A is Σ 0 n for some n.One can show that these classes are closed under finite unions and intersections.A bounded quantifier is one of the form “∃x < n”or“∀x < n”. Westill obtain the same classes if we intersperse bounded quantifiers of any typein the quantifier part of expressions above, or replace single quantifiers Q bywhole blocks of quantifiers of the same type as Q. For instance, the expression

22 1 The complexity of sets∃y∀z 1.(i) First suppose A is Σ 0 n, namely x ∈ A ⇔∃y 1 ∀y 2 ...Qy n R(x, y 1 ,...,y n ) forsome computable relation R. Then the setB = {〈x, y 1 〉: ∀y 2 ...Qy n R(x, y 1 ,...,y n )}is Π 0 n−1, and A is c.e. relative to B. By (ii) for n − 1wehaveB ≤ m N −∅ (n−1) .So A is c.e. relative to ∅ (n−1) .Now suppose A is c.e. relative to ∅ (n−1) . Then there is a Turing functional Φsuch that A = dom(Φ ∅(n−1) ). By the use principle,x ∈ A ⇔ ∃η,s[Φ η s(x)↓ & ∀i

1.4 Descriptive complexity of sets 23Σ 0 1 Σ 0 2 Σ 0 3⊂⊂∆ 0 1 ∆ 0 2 ∆ 0 3⊂⊂⊂⊂···⊂⊂⊂⊂⊂⊂Π 0 1 Π 0 2 Π 0 3Fig. 1.1. The arithmetical hierarchy.1.4.14 Proposition. A is ∆ 0 2 ⇔ A is both Σ 0 2 and Π 0 2.Proof.A ∈ ∆ 0 2 ⇔ A ≤ T ∅ ′ by the Limit Lemma 1.4.2⇔ A and N − A are c.e. in ∅ ′ by Proposition 1.2.8⇔ A ∈ Σ 0 2 ∩ Π 0 2 by Theorem 1.4.13. ✷The following is therefore consistent with Definition 1.4.1.1.4.15 Definition. We say that A is ∆ 0 n if A is both Σ 0 n and Π 0 n.The computable sets coincide with the ∆ 0 1 sets by Fact 1.4.12. The hierarchyof classes introduced in Definitions 1.4.10 and 1.4.15 is called the arithmeticalhierarchy (Figure 1.1).1.4.16 Proposition. Let n ≥ 1. Then A is ∆ 0 n ⇔ A ≤ T ∅ (n−1) .Proof. By Theorem 1.4.13, A is ∆ 0 n ⇔ A and N − A are c.e. in ∅ (n−1) .ByProposition 1.2.8, this condition is equivalent to A ≤ T ∅ (n−1) .✷The Σ 0 2-sets Z can still be reasonably described by a suitable computable sequenceof finite sets (Z s ) s∈N .1.4.17 Proposition. Z is Σ 0 2 ⇔ there is a computable sequence of strongindices (Z s ) s∈N such that Z s ⊆ [0,s) and x ∈ Z ↔ ∃s∀t ≥ sZ t (x) =1.The equivalence is uniform.Proof. ⇒: By Theorem 1.4.13(i) there is a Turing functional Φ such thatZ = dom(Φ ∅′ ). Now let Z s = {x

24 1 The complexity of setsExercises. Show the following.1.4.19. ∅ ′ is not an index set.1.4.20. (i) The set {e: W e ≠ ∅} is Σ 0 1 complete.(ii) The set {e: W e finite} is Σ 0 2-complete. In fact, for each Σ 0 2 set S there is a uniformlyc.e. sequence (X n) n∈N of initial segments of N such that ∀n [n ∈ S ↔ X n finite], andthis sequence itself is obtained effectively from a description of S.(iii) The set Tot = {e: dom(Φ e)=N} = {e: W e = N} is Π 0 2-complete.(iv) Both {e: W e cofinite} and {e: W e computable} are Σ 0 3-complete.1.4.21. The set {e: dom Φ ∅′e= N} is Π 0 3-complete.1.4.22. Let S be a class of c.e. sets [ω-c.e. sets] containing all the finite sets. Supposethe index set of S is Σ 0 3. Then S is uniformly c.e. [uniformly ω-c.e.]1.4.23. ⋄ Let S be a class of c.e. sets closed under finite variants that contains thecomputable sets but not all the c.e. sets. If the index set of S is Σ 0 3 then it is Σ 0 3-complete.1.4.24. Let X ⊆ N. (i) Each relation R ≤ T X is first-order definable in the structure(N, +, ·,X). (ii) The index set {e: W e ≤ T X} is Σ 0 3(X).1.4.25. A is ∆ 0 n ⇔∀xA(x) = lim k1 lim k2 ...lim kn−1 g(x, k 1,...,k n−1) for some computable{0, 1}-valued function g.1.5 Absolute computational complexity of setsAt the beginning of this chapter we discussed classes of similar complexity. A lownessproperty of a set specifies a sense in which the set is computationally weak.Usually this means that it is not very useful as an oracle. Naturally, we requirethat such a property be closed downward under Turing reducibility; in particularit only depends on the Turing degree of the set. If a set is computable thenit satisfies any lowness property. A set that satisfies a lowness property can bethought of as almost computable in a specific sense.Highness properties say that the set is computationally strong. They are closedupward under Turing reducibility. If a set satisfies a highness property it is almostTuring above ∅ ′ in a specific sense.Classes of computational complexity are frequently defined in terms of howfast the functions computed by the set grow. To compare the growth rate offunctions one can use the domination preordering on functions.1.5.1 Definition. Let f,g: N → R. We say that f dominates g if f(n) ≥ g(n)for almost every n.In this section we introduce two lowness properties and one highness propertyof a set A. We also consider some of their variants.(a) A is low if A ′ ≤ T ∅ ′ .(b) A is computably dominated if each function g ≤ T A is dominated by acomputable function.(c) A is high if ∅ ′′ ≤ T A ′ .

1.5 Absolute computational complexity of sets 25The classes given by (a) and (c) can be characterized by domination properties.For (c), by Theorem 1.5.19, A is high iff there is a function g ≤ T A dominatingeach computable function, which shows that highness is opposite to beingcomputably dominated. For (a) see Exercise 1.5.6.A general framework for lowness properties and highness properties will begiven in Section 5.6. We consider weak reducibilities ≤ W . Such a reducibilitydetermines a lowness property C ≤ W ∅ and a dual highness property C ≥ W ∅ ′ .For instance, we could define A ≤ W B iff A ′ ≤ T B ′ . Then the associated lownessand highness properties are (a) and (c) above. Table 8.3 on page 363 containsfurther examples of such dual properties given by a weak reducibility.It can be difficult to determine whether a lowness property is satisfied by morethan the computable sets, and whether a highness property applies to sets otherthan the sets Turing above ∅ ′ . We will need to introduce new methods to doso for the properties (a)–(c): the priority method (page 32) or basis theorems(page 56) for (a) and (b), and, for instance, pseudojump inversion (page 249)for (c). So far, we actually have not seen any example of a set that is neithercomputable nor Turing above ∅ ′ .A pair of lowness properties can be “orthogonal” in the sense that the onlysets that satisfy them both are the computable sets. For instance, the propertiesin (a) and (b) are orthogonal. In contrast, classes of descriptive complexity forman almost linear hierarchy, disregarding cases like the Σ 0 1 versus the Π 0 1 setswhere one class is simply obtained by taking the complements of the sets inthe other class. For a further difference between computational and descriptivecomplexity, the downward closed class given by (b), say, is uncountable, whileclasses of descriptive complexity are countable.Given a lowness property L, we will be interested in the question whether it isnull or conull (Definition 1.9.8), and whether there are sets A, B ∈ L such that∅ ′ ≤ T A ⊕ B. Similarly, we will be interested in whether a highness property His null or conull, and whether there are sets A, B ∈ H that form a minimal pair(only the computable sets are Turing below both A and B). Being conull meansthat the property is not very restrictive. Most of the properties we study will benull, including the ones in (a)–(c). If sets A, B as above exist then the propertyis not that close to being computable (for lowness properties), or being Turingabove ∅ ′ (for highness properties). We will return to the topic of minimal pairssatisfying a highness property on page 258.In this section we concentrate on the properties (a)–(c) and their variants.We introduce further properties in subsequent chapters, often using conceptsrelated to randomness. In Chapter 4 we study the conull highness property ofhaving diagonally noncomputable degree, and the stronger highness property ofhaving the same degree as a completion of Peano arithmetic (which is null byExercise 5.1.15). In Chapter 5 we consider lowness for Martin-Löf randomness.Lowness for other randomness notions is studied in Chapter 8. Figure 8.1 onpage 361 gives an overview of all the downward closed properties.

26 1 The complexity of setsSets that are low nRecall from 1.2.13 that C (n) is the result of n applications of the jump operator,beginning with the set C. A hierarchy of absolute computational complexity isobtained by considering C (n) within the Turing degrees, for n ≥ 0. Note thatC (n) ≥ T ∅ (n) by Proposition 1.2.14.1.5.2 Definition. Let n ≥ 0. We say that C is low n if C (n) ≡ T ∅ (n) .The most important among these classes is low 1 , the class of sets C such thatC ′ ≡ T ∅ ′ .IfC ∈ low 1 we simply say that C is low. Each low set is ∆ 0 2. Thus,such a set is computationally weak in the sense that ∅ ′ can determine whetherΦ C e (x) converges for each e, x, and in case it does find the output. We will see inTheorem 1.6.4 that an incomputable low c.e. set exists. In particular, the jumpis not a one-one map on the Turing degrees.Each class low n is closed downward under Turing reducibility, and containedin ∆ 0 n+1. The hierarchycomputable ⊂ low 1 ⊂ low 2 ⊂ ...⊂ {Z : Z ≱ T ∅ ′ } (1.7)is proper by Theorem 6.3.6 below.The following property due to Mohrherr (1986) will be important later on.1.5.3 Definition. C is superlow if C ′ ≡ tt ∅ ′ .It suffices to require that C ′ ≤ tt ∅ ′ , because ∅ ′ ≤ m C ′ for any C by 1.2.14.By 1.4.4, it is also equivalent to ask that C ′ be ω-c.e., namely, C ′ can be computablyapproximated with a computable bound on the number of changes.The class of superlow sets is closed downwards under Turing reducibility. Itlies strictly between the classes of computable and of low sets. In Theorem 1.6.5we build a c.e. incomputable superlow set, and Exercise 1.6.7 asks for a low butnot superlow c.e. set. Also see Remark 6.1.5.The following states that A ′ is as simple as possible compared to A.1.5.4 Definition. A is generalized low 1 ,orinGL 1 for short, if A ′ ≡ T A ⊕∅ ′ .Equivalently, ∅ ′ is Turing complete relative to A. Clearly the class GL 1 coincideswith low 1 on the ∆ 0 2 sets, and no set A ≥ T ∅ ′ is in GL 1 . However, a set in GL 1 isnot necessarily computationally weak. In fact, if B ≱ T ∅ ′ then there is A ≥ T Bsuch that A is in GL 1 by a result of Jockusch (1977); also see Odifreddi (1999,Ex. XI.3.11).We extend Definition 1.5.1. For functions f,ψ : N → N, where possibly ψ ispartial, we say that f dominates ψ if ∀ ∞ n [ψ(n)↓ → f(n) ≥ ψ(n)].Exercises. Show the following.1.5.5. If C is superlow, there is a computable function h such that Y ≤ T C impliesY ≤ tt ∅ ′ with use function bounded by h for each Y . (Here we view truth table reductionsas functions mapping inputs to truth tables; see before 1.4.5.)1.5.6. A is in GL 1 ⇔ some function f ≤ T A ⊕∅ ′ dominates each function that ispartial computable in A.

1.5 Absolute computational complexity of sets 271.5.7. If B is low 2 then the index set {e: W e ≤ T B} is Σ 0 3.1.5.8. B is Low 2 ⇔ Tot B = {e: Φ B e total} is Σ 0 3.Computably dominated setsWe study a lowness property of a set A stating that the functions computedby A do not grow too quickly.1.5.9 Definition. A is called computably dominated if each function g ≤ T A isdominated by a computable function.Exercise 1.5.17 shows that we cannot effectively determine the dominating functionfrom the Turing reduction of g to A, unless A is computable.We say that E ⊆ N is hyperimmune if E is infinite and p E is not dominated bya computable function, where p E is the listing of E in order of magnitude (alsosee Definition 1.7.1 below). The intuition is that E is a very sparse set.1.5.10 Proposition. A is not computably dominated ⇔ there is a hyperimmuneset E ≡ T A.Proof. ⇐: Immediate since p E ≤ T E.⇒: Suppose g ≤ T A is not dominated by a computable function. Let E =ran(h), where the function h is defined as follows: h(0) = 0, and for each n ∈ N,h(2n +1)=h(2n) +g(n) +1and h(2n +2)=h(2n +1)+p A (n) + 1. ClearlyE ≡ T h ≡ T A. Moreover g(n) ≤ h(2n + 1), so that h = p E is not dominated bya computable function.✷For this reason, a set that is not computably dominated is also called a set ofhyperimmune degree. In the literature, a computably dominated set is usuallycalled a set of hyperimmune-free degree. The study of hyperimmune-free degreeswas initiated by Martin and Miller (1968).1.5.11 Proposition. A is computably dominated ⇔ for each function f,f ≤ T A → f ≤ tt A.Proof. ⇒: Suppose f = Φ A . Let g(x) =µs Φ A s (x) ↓. Then g ≤ T A, so thereis a computable function t such that t(x) ≥ g(x) for each x. Thust bounds therunning time of Φ A , whence f ≤ tt A by Proposition 1.2.22.⇐: By the remark after Proposition 1.2.22, each function f ≤ tt A is dominatedby a computable function.✷Each computable set A is computably dominated. Are there others? We willanswer this question in the affirmative in Theorem 1.8.42. Here we observe thatthere are none among the incomputable ∆ 0 2 sets.1.5.12 Proposition. If A is ∆ 0 2 and incomputable, then A is not computablydominated.Proof. Let (A s ) s∈N be a computable approximation of A. Then the followingfunction g is total:

28 1 The complexity of setsg(s) ≃ µt ≥ s. A t ↾ s = A↾ s .Note that g ≤ T A. Assume that there is a computable function f such thatg(s) ≤ f(s) for each s. Then A is computable: for each n and each s>nwe haveA t (n) =A(n) for some t ∈ [s, f(s)), namely t = g(s). On the other hand, if sis sufficiently large then A u (n) =A s (n) for all u ≥ s. Thus, to compute A, oninput n determine the least s>nsuch that A u (n) =A s (n) for all u ∈ [s, f(s)).Then A s (n) =A(n), so the output A s (n) is correct.✷Exercises.1.5.13. (a) Strengthen Proposition 1.5.12 as follows: if C< T A ≤ T C ′ for some set C,then A is of hyperimmune degree. (b) Conclude that, if A is computably dominatedand C< T A, then C ′ < T A ′ .1.5.14. Strengthen Proposition 1.5.12 in yet another way: if A is Σ 0 2 and incomputablethen A is of hyperimmune degree.1.5.15. Show that if A is computably dominated then A ′′ ≤ T A ′ ⊕∅ ′′ . In particular,each computably dominated set is in GL 2 (Definition 1.5.20 below).1.5.16. ⋄ (Jockusch, 1969) Show that if X ≤ T A → X ≤ tt A for each set X, then Ais already computably dominated.1.5.17. ⋄ Let us call a set A uniformly computably dominated if there is a computablefunction r such that for each e, ifΦ A e is total then Φ r(e) is total and dominates Φ A e .Show that the only uniformly computably dominated sets are the computable ones.Sets that are high nRecall from Definition 1.5.2 that a set C is low n if C (n) ≡ T ∅ (n) , namely C (n) isas low as possible. How about having a complex n-th jump?1.5.18 Definition. Let n ≥ 0. A set C is high n if ∅ (n+1) ≤ T C (n) .Of course, C (n) could be even more complex. However, if C is ∆ 0 2 then C (n) ≤ T∅ (n+1) . So in that case, to be high n means that the n-th jump is as complex aspossible.All the classes high n are closed upward under Turing reducibility, so the complementaryclasses non-high n =2 N −high n are closed downward. We have refinedthe hierarchy (1.7):comp. ⊂ low 1 ⊂ low 2 ⊂ ...⊂ non-high 2 ⊂ non-high 1 ⊂ {Z : Z ≱ T ∅ ′ }. (1.8)This hierarchy of downward closed classes is a proper one as we will see in 6.3.6.Also, there is a c.e. set that is not in low n or high n for any n by 6.3.8.Of particular interest is the class high 1 = {C : ∅ ′′ ≤ T C ′ } (simply called thehigh sets), because such sets occur naturally in various contexts. For instance, atheorem of Martin (1966b) states that a c.e. set C is high iff C ≡ T A for somemaximal set A; also see Soare 1987, Thm. XI.2.3. Here a co-infinite c.e. set Ais called maximal if for each c.e. set W ⊇ A, either W is cofinite or W − A isfinite. For another example, C is high iff there is a computably random but notMartin-Löf random set Z ≡ T C by Theorem 7.5.9 below.

1.6 Post’s problem 29We are not yet in the position to show the existence of a high set C ≱ T ∅ ′ .Our first example will be Chaitin’s halting probability Ω relative to ∅ ′ (3.4.17).In Corollary 6.3.4 we prove that there is a high c.e. set C< T ∅ ′ .It is easy to define a function f ≤ T ∅ ′ that dominates all computable functions:the set {〈e, x〉: Φ e (x)↓} is c.e., and hence many-one reducible to ∅ ′ via acomputable function h. Let f(x) = max{Φ e (x): e ≤ x & h(〈e, x〉) ∈∅ ′ }.IfΦ e istotal then f(x) ≥ Φ e (x) for all x ≥ e. This property of ∅ ′ in fact characterizesthe high sets. Thus, being high is opposite to being computably dominated.1.5.19 Theorem. (Martin, 1966b) C is high ⇔ some function f ≤ T C dominatesall computable functions.Proof. ⇒: We define a function f ≤ T C that dominates each total Φ e , extendingthe argument for the case C = ∅ ′ just given. Note that {e: Φ e total}is Π 0 2, and hence {e: Φ e total} ≤ m N −∅ ′′ ≤ T C ′ . By the Limit Lemma1.4.2 there is a binary function p ≤ T C such that for each e, lim s p(e, s) exists,and lim s p(e, s) = 1 iff Φ e is total. To compute f(x) with oracle C, lets ≥ x be least such that for each e ≤ x, either Φ e,s (x) ↓ or p(e, s) = 0, and letf(x) = max{Φ e,s (x): e ≤ x & Φ e,s (x)↓}.If Φ e is total then there is s 0 ≥ e such that p(e, s) = 1 for all s ≥ s 0 , so thatf(x) ≥ Φ e (x) for all x ≥ s 0 .⇐: Suppose that f ≤ T C dominates all computable functions. We show thatN −∅ ′′ = {e: Φ ∅′e (e)↑} is Turing reducible to C ′ . Note that Φ ∅′e (e)↑ iff the computationis undefined at infinitely many stages, that is, no computation Φ ∅′e (e)[s]is stable. Thus Φ ∅′e (e)↑ iff the partial computable functiong(s) ≃ µt > s [Φ ∅′e (e)[t]↑]is total. In that case g is dominated by f, and thereforee ∉∅ ′′ ⇔∃n 0 ∀n ≥ n 0 ∃t [n ≤ t ≤ f(n) &Φ ∅′e (e)[t]↑].Since t is bounded by f(n) and f ≤ T C, this shows that N −∅ ′′ is Σ 0 2(C). Also∅ ′′ ∈ Σ 0 2 ⊆ Σ 0 2(C). Therefore ∅ ′′ ≤ T C ′ by Prop. 1.4.14 relative to C. ✷The following class contains the class GL 1 of Definition 1.5.4.1.5.20 Definition. A is generalized low 2 ,orGL 2 for short, if A ′′ ≡ T (A ⊕∅ ′ ) ′ .Equivalently, ∅ ′ is high relative to A.1.5.21 Exercise. Show that A is GL 2 ⇔ some function f ≤ T ∅ ′ ⊕ A dominates each(total) function g ≤ T A. (Compare this with Exercise 1.5.6 characterizing GL 1.)1.6 Post’s problemPost (1944) asked whether a c.e. set can be incomputable and Turing incomplete,that is, whether there is a c.e. set A such that ∅ < T A< T ∅ ′ . It took 12 years toanswer his question. Kleene and Post (1954) made a first step by building a pair

30 1 The complexity of setsof Turing incomparable ∆ 0 2 sets. To do so they introduced the method of finiteextensions. Post’s question was finally answered in the affirmative by Friedberg(1957b) and Muchnik (1956) independently. They built a pair of Turing incomparablesets that are also computably enumerable, strengthening the Kleene–Postresult. Their proof technique is nowadays called the priority method with finiteinjury. For more background on Post’s problem see Chapter III of Odifreddi(1989) and our discussion on page 34.Turing incomparable ∆ 0 2-setsFor sets Y,Z we write Y | T Z if Y ≰ T Z & Z ≰ T Y .1.6.1 Theorem. There are sets Y,Z ≤ T ∅ ′ such that Y | T Z.Proof idea. Note that Y | T Z is equivalent to the conjunction of the statementsR i for all i, whereR 2e : ∃n ¬Y (n) =Φ Z e (n)R 2e+1 : ∃n ¬Z(n) =Φ Y e (n).Thus we may divide the overall task that Y | T Z into subtasks, called requirements.Tomeet a requirement means to make its statement true.The construction of Y and Z is relative to ∅ ′ . We meet the requirements oneby one in the given order. We define sequences σ 0 ≺ σ 1 ≺ ... and τ 0 ≺ τ 1 ≺ ...,and let Y = ⋃ i σ i and Z = ⋃ i τ i. At stage i + 1 we meet R i by defining σ i+1and τ i+1 appropriately. Since we have ∅ ′ at our disposal as an oracle, we may askwhether Φ Z e (n) can be made defined for a particular number n. Then we maydefine Y (n) in such a way that it differs from Φ Z e (n). This method of providinga counterexample to an equality of sets is called diagonalization, and a numbersuch as n above is called a diagonalization witness.Construction. Let σ 0 = τ 0 = ∅.Stage i +1, i =2e. Let n = |σ i |. Using ∅ ′ as an oracle, check whether thereis τ ≻ τ i such that y = Φ τ e(n) ↓. (Note that this is a Σ 0 1 question, so it can beanswered by ∅ ′ .) If so, let τ i+1 = τ and σ i+1 = σ i x, where x = max(1 − y, 0).Otherwise, let σ i+1 = σ i 0 and τ i+1 = τ i 0 (merely to ensure the strings areextended at every stage).Stage i +1, i =2e +1. Similar, with the sides interchanged: let n = |τ i |. Using ∅ ′as an oracle, see if there is σ ≻ σ i such that y = Φ σ e (n)↓. If so, let σ i+1 = σ andτ i+1 = τ i x, where x = max(1 − y, 0). Otherwise, let σ i+1 = σ i 0 and τ i+1 = τ i 0.Verification. Clearly Y,Z ≤ T ∅ ′ . Each requirement R 2e is met due to the actionsat stage i + 1 where i =2e: if we cannot find an extension τ ≻ τ i such thatΦ τ e(n)↓ then by the use principle (see after 1.2.17) Φ Z e (n)↑, because Z extends τ i .Otherwise we ensure that Y (n) ≠ Φ Z e (n). Either way R 2e is met. The case of arequirement R 2e+1 is similar.✷

Simple sets1.6 Post’s problem 31We now turn our attention to the c.e. sets and their Turing degrees. The haltingproblem is a rather special example of an incomputable c.e. set. Here we investigatea whole class of incomputable sets, the co-infinite c.e. sets that meet eachinfinite c.e. set nontrivally.1.6.2 Definition. A c.e. set A is simple if N − A is infinite and A ∩ W ≠ ∅ foreach infinite c.e. set W .In particular, N − A is not c.e., so A is not computable. The intuition is that Ais so large that it meets each infinite c.e. set. To call such a set “simple” ismisleading, but it has been done so for decades and no one intends to change theterm. At least the halting problem ∅ ′ is not simple: by the Padding Lemma 1.1.3one can obtain an infinite c.e. set W of indices for the empty set. Then ∅ ′ ∩W = ∅.1.6.3 Theorem. There is a simple set.Proof, version 1.This argument is due to Post (1944). Let A = ran(ψ) whereψ(i) ≃ the first element ≥ 2i enumerated into W i .Since ψ is partial computable, A is c.e. If x0. For each ii.In the present construction, each S i acts at most once.After giving the formal construction one has to verify that it actually buildsobjects as desired:

32 1 The complexity of setsVerification. If W i has an element ≥ 2i then by construction W i ∩ A ≠ ∅. ThusS i is met. A number < 2e can only be enumerated by a requirement S i , iS 1 >L 1 >...Requirements further to the left are said to have stronger (or higher) priority.A requirement can only restrain requirements of weaker (or lower) priority duringthe construction. In the verification one shows by induction on descendingpriority that the action of each requirement is finitary, and hence each singlerequirement is not restrained from some stage on. The strategies for the requirementshave to be designed in such a way that they can live with finitely manydisturbances. For instance, S i has to cope with the restraints of finitely manystronger priority lowness requirements. So it needs an element of W i that is

1.6 Post’s problem 33larger than the eventual values of these restraints. There is no computable upperbound for the maximum of these eventual values, so the construction does notany longer make A effectively simple.An undesirable situation for L e is the following: it thought it had alreadysecured a computation J A (e) ↓, but then it is injured because a number x0.(1) For each e1, theninitialize the requirements S i for i>e. We say that L e acts.(2) For each ee: f(e, s) ≠ f(e, s − 1)} ≤ 2e +2.Hence A ′ is ω-c.e., whence A ′ ≤ tt ∅ ′ .✷

34 1 The complexity of setsRelativizing the proof of Theorem 1.6.4 yields for each oracle C a set A ≰ T Csuch that A is c.e. relative to C and (A ⊕ C) ′ = C ′ . Thus the jump fails to beone-one on the structure D T of all Turing degrees in the following strong sense.1.6.6 Corollary. For each c ∈ D T there is a ∈ D T such that a > c and a ′ = c ′ .1.6.7. ⋄ Exercise. Prove a further variant of Theorem 1.6.4: there is a low c.e. set Athat is not superlow. (A detailed solution requires some of the terminology on buildingTuring functionals introduced in Section 6.1 below.)Is there a natural solution to Post’s problem?Post may have hoped for a different kind of solution to the problem he posed, one thatis more natural. The meaning of the word “natural” in real life might be: somethingthat exists independent of us humans. In mathematics, to be natural an object mustbe more than a mere artifact of arbitrary human-made definitions (for instance, theparticular way we defined a universal Turing program). Natural properties should beconceptually easy. Being a simple set is such a property, satisfying the requirementsin the proof of Theorem 1.6.4 is not. In computability theory a natural class of setsshould be closed under computable permutations. With very few exceptions, classeswe study satisfy this criterion; see also Section 1.3. On the other hand, the class ofsets satisfying the lowness requirements in the proof of Theorem 1.6.4 may fail thiscriterion (depending on the particular choice of a universal Turing program). Also,what we put into the c.e. set A constructed depends on the particular way the sets W eare defined, even in which order they are enumerated. So the set A is an artifact ofthe way we specified the universal Turing program (defined before 1.1.2). Neither theproperty (satisfying the requirements) is natural, nor the set A for which the propertyholds.Let us say a Post property is a property of c.e. sets which is satisfied by some incomputableset and implies Turing incompleteness. Post was not able to define such aproperty; the closest he came was to show that each hypersimple set (Definition 1.7.5)is truth-table incomplete (also see 4.1.15). The first result in the direction of a naturalPost property was by Marchenkov (1976), who introduced a “structural” Post property(to be maximal relative to some c.e. equivalence relation, and also a left cut in a computablelinear order). Harrington and Soare (1991) found a Post property that is evenfirst-order definable in E, the lattice of c.e. sets under inclusion. While Marchenkov’sproperty relies on other effective notions, the Harrington–Soare property is based purelyon the interaction of c.e. sets given by the inclusion relation. However, in both cases,the construction showing that an incomputable set with the Post property exists ismore complex than the one in the proof of Theorem 1.6.4.In Section 5.2 we will encounter a further Post property, being K-trivial. The constructionof a c.e. incomputable K-trivial set only takes a few lines and has no injury torequirements. However, it is harder to show that each K-trivial set is Turing-incomplete.Kučera (1986) gave an injury-free solution to Post’s problem where the constructionto show existence is somewhat more difficult than in the case of a K-trivial set, but theverification that the set is Turing incomplete is easier. The Post property of Kučera isto be below a Turing incomplete Martin-Löf random set. No injury is needed to showthat there is a Turing incomplete (or even low) Martin-Löf random set, and Kučera’sconstruction produces a c.e. incomputable set Turing below it (see Section 4.2).

1.6 Post’s problem 35All these examples of c.e. incomputable sets with a Post property are still far fromnatural, because they depend, for instance, on our particular version of the universalTuring program. Perhaps the injury-free solutions are more natural, but the choice ofa universal program still matters since we have to diagonalize to make the set incomputable.Any reasonable solution W to Post’s problem should be relativizable to anoracle set X, so one would expect that X< T W X < T X ′ for each X. If the solutiondoes not depend on the choice of the universal program, it should also be degree invariant:if X ≡ T Y , then W X ≡ T W Y . The existence of such a degree invariant solutionto Post’s problem is a long-standing open question posed by Sacks (1963a).Turing incomparable c.e. setsThe priority method was introduced by Friedberg (1957b) and Muchnik (1956)when they extended the Kleene–Post Theorem 1.6.1 to the c.e. case: there areTuring incomparable c.e. sets A and B. There is only one type of requirementnow, but in two symmetric forms, one for A ≰ T B, and the other for B ≰ T A.The strategy combines elements from the strategies for simplicity and lownessrequirements in the proof of Theorem 1.6.4. For instance, a requirement to ensureA ≠ Φ B e enumerates into A and restricts the enumeration of B.1.6.8 Theorem. (Extends 1.6.1) There are c.e. sets A and B such that A | T B.Proof idea. We meet the same requirements as in Theorem 1.6.1, namelyR 2e : ∃n ¬A(n) =Φ B e (n)R 2e+1 : ∃n ¬B(n) =Φ A e (n).Here ¬A(n) =Φ B e (n) means that either Φ B e (n) ↑ or Φ B e (n) ↓≠ A(n). Again, wecannot use a construction relative to ∅ ′ , because we want to build computableenumerations of A and B. The strategy for R 2e is somewhat similar to the one inTheorem 1.6.1: it looks for an unused candidate n such that currently Φ B e (n) =0,and puts n into A; it also attempts to protect this computation by initializing therequirements of weaker priority. If later the B-enumeration of some requirement(necessarily of stronger priority) destroys the computation, then R 2e is initialized(in particular, declared unsatisfied), and has to start anew. In the verification, oneshows that R 2e acts only finitely often. Once it stops acting it is met: otherwise, ifactually A = Φ B e , there would be yet another candidate available, so it would actanother time. To ensure the candidate is not put into A by some other strategy,a requirement R i chooses its candidates from N [i] = {〈x, i〉: x ∈ N}.Construction of A and B. Let A 0 = B 0 = ∅. All requirements are initialized.Stage s>0. Let i be least such that R i is currently unsatisfied and, where rr,if i =2e + 1 then Φ A e (n) =0[s − 1] for the least n ∈ N [i] − B s−1 such that n>r.In the first case put n into A, in the second case n into B. Declare R i satisfiedand initialize all requirements of weaker priority. We say that R i acts. Note thatn

36 1 The complexity of setsClaim. Each requirement R i is initialized only finitely often, acts only finitelyoften, and is met.By induction, we may choose a stage t such that no requirement R k , k

1.7 Properties of c.e. sets 37showed that there are no superlow c.e. sets A 0,A 1 such that A 0 ⊕ A 1 is weak truthtablecomplete. On the other hand, one can achieve that A 0 ⊕ A 1 is Turing completeby Theorem 6.1.4.1.7 Properties of c.e. setsWe introduce three properties of a co-infinite c.e. set A and discuss how theyrelate to its relative computational complexity. Recall from 1.6.2 that such a set Ais simple if A ∩ W e ≠ ∅ for each e such that W e is infinite. We show that eachincomputable c.e. weak truth-table degree contains a simple set. Thus, simplicityhas no implication on the complexity beyond being incomputable, unless ourmeasure of relative complexity is finer than weak truth-table reducibility (seeRemark 1.7.4). The first two of the properties we introduce strengthen simplicity.(1) A is hypersimple if N − A is hyperimmune, namely, the function mapping nto the n-th element of N − A is not dominated by a computable function (seeafter 1.5.9). Such sets exist in each incomputable c.e. Turing degree, but nolonger in each c.e. weak truth-table degree.(2) A is promptly simple if for some computable enumeration, for each infinite W esome element of W e enters A with only a computable delay from the stage onwhen it entered W e . The Turing degrees of promptly simple sets form a propersubclass of the c.e. incomputable Turing degrees. In fact they coincide withthe c.e. degrees d that are non-cappable, namely ∀y ≠ 0 ∃b [0 < b ≤ d, y]holds in the partial order of c.e. Turing degrees. (Non-cappable Turing degreesare complex in the sense that they share incomputable knowledge with each c.e.degree y > 0.) This coincidence, due to Ambos-Spies, Jockusch, Shore and Soare(1984), implies that the promptly simple degrees are first-order definable in R T .We will only prove the easier implication, that each promptly simple degree isnon-cappable.(3) A is creative if it is incomputable in a uniform way, namely, there is a computablefunction p such that for each e, the number p(e) shows that W e is notthe complement of A. The intuition is that such sets are far from computable; infact we show that A is creative iff A is m-complete. By a result of Harrington,the class of creative sets can be characterized using only the Boolean operationson c.e. sets. Thus, being creative is first-order definable in the lattice E of c.e.sets under inclusion.Both the first-order definability of the promptly simple degrees in the c.e. Turingdegrees and the first-order definability of the creative sets in E are ratherunexpected because these properties were defined in terms of concepts thatappear to be external to the structure. The definability results show that theproperties are in fact intrinsic. Another example of this is the first-order definabilityin R T of the low n degrees for n ≥ 2, and of the high n degrees for n ≥ 1.These results are due to Nies, Shore and Slaman (1998).The properties (1) and (3) will only play a marginal role for us, but (2) will beimportant later on, because several strong lowness properties, such a being lowfor Martin-Löf randomness (Definition 5.1.7), hold for some promptly simple set.

38 1 The complexity of setsWe need the following notation throughout this section.1.7.1 Definition. (i) For a set B ⊆ N let B denote its complement N − B.(ii) For a set S ⊆ N, the function p S lists S in the order of magnitude. That is,p S (0)

1.7 Properties of c.e. sets 39coding of C into A. Hence p A(y) ≤ 3y for each y. Then A s ↾ 3y+1 = A↾ 3y+1 impliesC s (y) =C(y), so C ≤ wtt A. The rest is as before.✷1.7.4 Remark. The result cannot be strengthened to truth-table degrees: Jockusch(1980) proved that there is a nonzero c.e. truth-table degree which contains no simpleset. In fact, there is a c.e. incomputable set C such that no simple set A satisfies A ≤ tt Cand C ≤ wtt A.Hypersimple setsInformally, A is simple if the complement of A is thin in the sense that it does notcontain any infinite c.e. set. Stronger properties have been studied (see Odifreddi1999, pg. 393). They all state in some way that the complement of A is thin (whilestill being infinite). The strongest is being maximal: for each c.e. set W ⊇ A,either W is cofinite or W − A is finite. A moderate strengthening of being simpleis hypersimplicity, namely, the complement of A is hyperimmune (1.5.10).1.7.5 Definition. A is hypersimple if A is co-infinite and ∃ ∞ nf(n)

40 1 The complexity of setsIf X ⊆ A is infinite, then f(m) =p X(m) ≥ p A(m) for each m. Therefore A ≤ T X.Thus A is introreducible.1.7.7 Exercise. Show that each truth table-degree contains an introreducible set.1.7.8 Remark. We have seen two ways of obtaining a c.e. set. At first sight they seemto be very different.(a) The set can be given by a definition, “in one piece”. Examples are the first proofof Theorem 1.6.3 to obtain a simple set, and the proof of Proposition 1.7.6 (where thedefinition of a simple set is based on a computable enumeration of the given set C). Anexample of a simple weak truth-table complete set obtained by a definition is in 2.1.28.(b) We can build the set using a stage-by-stage construction. Examples are the secondproof of Theorem 1.6.3, and the proof of Theorem 1.7.3.Distinguishing these two ways is helpful for our understanding, even though formally,a construction of a c.e. set A is just a definition of a computable enumeration of Avia an effective recursion on stages. (In more complex constructions, like the onesencountered in Chapter 6, one also builds auxiliary objects such as Turing functionals.)For a construction, at stage s we need to know what happened at the previous stages,which is not the case for direct definitions (such as the first proof of 1.6.3).Constructions have the advantage of being flexible. For instance, we extended theconstruction in the second proof of Theorem 1.6.3 in order to obtain a low simple set.However, constructions also introduce more artifacts and thereby tend to make the setless natural. Definitions of c.e. sets seem to be more natural, but often they are notavailable.Let us briefly skip ahead to Kučera’s injury-free solution to Post’s problem (Section4.2) and the construction of a c.e. K-trivial set (Section 5.2), both already mentionedon page 34. These are constructions in that the action at stage s depends on thepast: we have to keep track of whether a requirement has already been satisfied. Onthe other hand, they are very close to direct definitions because the requirements donot interact. Both sets are obtained by applying a simple operation to a given object,similar to the proof of Proposition 1.7.6. This given object is a low Martin-Löf randomset for Kučera’s construction, and the standard cost function for the construction ofa K-trivial set.Promptly simple setsThe definition of simplicity is static: we are not interested in the stage when anelement of an infinite set W e appears in A, only in that it appears at all. However,since the sets W e are equipped with a computable enumeration (W e,s ) s∈N , onecan also consider a stronger, dynamic version of the concept, where an elementappearing in an infinite set W e at stage s is enumerated into A at the samestage or earlier. Given a computable enumeration (B s ) s∈N , for s > 0weletB at s = B s − B s−1 .1.7.9 Definition. A c.e. set A is promptly simple (Maass, 1982) if A is coinfiniteand, for some computable enumeration (A s ) s∈N of A, for each e,#W e = ∞ → ∃s>0 ∃x [x ∈ W e,at s ∩ A s ]. (1.11)

1.7 Properties of c.e. sets 41We would rather wish to say that x enters W e and A at the same stage, but this isimpossible: for each effective enumeration (A s) s∈N of a c.e. set A, there is an e suchthat W e = A but x ∈ A at s → x ∉ W e,s, that is, every element enters W e later than A.This is immediate if we assume a reasonable implementation of the universal Turingprogram. It has to simulate the enumeration of A, and each simulated step takes atleast as long as a step of the given enumeration of A.The following seemingly more general variant of Definition 1.7.9 can be found in theliterature: there is a computable enumeration (A s) s∈N of A and a computable function psuch that #W e = ∞ → ∃s∃x [x ∈ W e,at s ∩ A p(s) ]. However, this formulation isequivalent via the computable enumeration (A p(s) ∩ [0,s)) s∈N of A.With a minor modification, the construction of a low simple set A in Theorem1.6.4 produces a promptly simple set. This is so because, when a simplicityrequirement wants to enumerate a number into A that is greater than the restraintis has to obey, this wish can be granted without delay.1.7.10 Theorem. (Extends 1.6.4) There is a low promptly simple set.Proof. We modify the proof of Theorem 1.6.4. Instead of the simplicity requirements(S e ) in (1.9) we now meet the prompt simplicity requirementsPS e :#W e = ∞ ⇒ ∃s ∃x [x ∈ W e,at s & x ∈ A s ].In the construction on page 33 we replace (2) by the following:For each e

42 1 The complexity of setspromptly simple sets, a result of Ambos-Spies, Jockusch, Shore and Soare (1984).A further characterization of this class of c.e. Turing degrees from the same paperwill be provided in Theorem 6.2.2 below: A has promptly simple degree iff A islow cuppable, namely, there is a low c.e. set Z such that ∅ ′ ≤ T A ⊕ Z.1.7.14 Theorem. Let the set E be promptly simple. From an incomputable c.e.set C one can effectively obtain a simple set A such that A ≤ wtt C, E.Proof. Choose a computable enumeration (E s ) s∈N via which E is promptlysimple. We ensure that A ≤ wtt C by direct permitting (see the proof of Theorem1.7.2). For A ≤ wtt E we use a more general type of permitting called delayedpermitting:ifx enters A at stage s then E s−1 ↾ x ≠ E ↾ x , that is, E changes below xat some stage t ≥ s. Since E is c.e. this implies A ≤ wtt E.We make A simple by meeting the requirements S i in (1.9). We uniformly enumerateauxiliary sets G i to achieve the E-changes. By the Recursion Theorem wehave in advance a computable function g such that G i = W g(i) for each i. In moredetail, given a parameter r ∈ N, weletg be a computable function effectivelyobtained from r such that W g(i) = W r[i] for each i. Based on g, we enumeratea uniformly c.e. sequence (G i ) i∈N . Hence there is a computable function f suchthat W f(r) = ⋃ i G i ×{i}. Let r ∗ be such that W f(r∗ ) = W r ∗, then the function gobtained for parameter r ∗ is as required, because G i = W [i]f(r ∗ ) = W [i]r = W ∗ g(i)for each i.Suppose at stage s we are in the situation that we want to put some x into Ain order to meet S i , namely, x ∈ W i,s and x is permitted by C. We first put anumber y0. If there is i

1.7 Properties of c.e. sets 43a contradiction that S i is not declared satisfied. Since C is incomputable and Eis co-infinite, there are arbitrarily large stages s ≥ s 0 at which we attempt tomeet S i via numbers y

44 1 The complexity of setsFrom Myhill’s Theorem 1.2.5 we conclude the following.1.7.19 Corollary. For every pair of creative sets C 1 ,C 2 there is a computablepermutation p of N such that p(C 1 )=C 2 .✷Harrington proved that creativity is first-order definable in the lattice of c.e.sets under inclusion, a result that was first published in Soare (1987). The firstorderproperty defining creativity of C states that there is an auxiliary set Fsuch that, for each Z, there is a piece R which is incomputable as shown by F ,and on which C coincides with Z.For c.e. sets X and R such that X ⊆ R we write X ⊏ R if R − X is c.e.Note that if R is computable, then so is X. IfX ⊆ R but X ̸⊏ R then thereare infinitely many elements that first appear in R and later enter X (that is,#{x: ∃s [x ∈ R s−1 ∩X at s ]} = ∞), for otherwise R−X is c.e. as R−X is almostequal to the set {x: ∃t [x ∈ R t − X t ]}.1.7.20 Theorem. Let C be c.e. Then C is creative ⇔∃F ∀Z ∃R [R ∩ F ̸⊏ R & R ∩ C = R ∩ Z], (1.12)where the quantifiers range over c.e. sets.Proof idea. For the implication “⇒”, as all creative sets are computably isomorphicby 1.7.19, it suffices to provide a particular creative set satisfying (1.12).We can takeĈ = {〈x, e〉: 〈x, e〉 ∈W e },as we will verify below. It is harder to show the implication “⇐”. If C satisfies(1.12) then we want to define a computable function p via which C is creative.The condition (1.12) is designed to make the construction of such a p work. Weare given the c.e. set F and enumerate a c.e. set Z. Assume first that we actuallyknow a witness R for Z in (1.12). Note that R is infinite because R ∩ F ̸⊏ R.We have an infinite list of elements x for which we can dictate membership in Cbecause we control Z(x). We let p(e) =x be the e-th element in this list. At firstx ∉ Z and hence x ∉ C; ifx enters W e we put x into Z, so that x ∈ W e ∩ C.Thus C is creative via p.Actually we do not know such a witness R. So we play the strategy abovesimultaneously for all c.e. sets W i as possible witnesses. We will write R i insteadof W i to improve the readability. We define a partial computable function p ibased on the assumption that R i is the witness for Z in (1.12). At each stage weextend p i for the least i such that the condition R i ∩ C = R i ∩ Z looks correct sofar. When at stage s we define p i (e) for the next e, we need a value x ∈ R i suchthat x ∉ Z s−1 , so we have to be sure that x is not already taken at a previousstage as a value p i ′(e ′ ) for some i ′ ,e ′ . This is where the condition R i ∩ F ̸⊏ R icomes in: as explained above, there are infinitely many numbers that enter F ata stage s when they are already in R i,s−1 . We can only use elements x of this

1.8 Cantor space 45kind as values p i (e) =x. There is no conflict with the previous values becausethey are in F s−1 , nor with later values as they are not in F s .Proof details. ⇒: To see that the set Ĉ defined above is 1-complete, let e∗ bea c.e. index such that W e ∗ = ∅ ′ ×N. Then x ∈∅ ′ ↔〈x, e ∗ 〉∈Ĉ for each x. Next,(1.12) is satisfied via F = ∅ ′ ×N: given a c.e. set Z = W k , let R be the computableset N × {k}. Then R ∩ F = ∅ ′ × {k} ̸⊏ R, otherwise ∅ ′ is computable. For eachelement y = 〈x, k〉 of R, wehavey ∈ Ĉ ↔ y ∈ W k, so that R ∩ Ĉ = R ∩ Z.⇐: Suppose C satisfies (1.12) via F .Construction of Z and partial computable functions p i (i ∈ N).Let Z 0 = ∅, and declare p i,0 (e) undefined for each i, e.Stage s>0. Let i be least such that(a) if t

46 1 The complexity of setsan example of a Stone space. In this section we develop a bit of Stone duality,a correspondence between topological concepts in Stone spaces and concepts relatedto Boolean algebras. The dual algebra of a Stone space S is its Booleanalgebra B of clopen sets. The dual space of a Boolean algebra B is the spacewhere the points are the ultrafilters U, and the basic open sets are the sets ofthe form {U : x ∈ U} for x ∈ B (which are also closed). Open sets correspondto ideals, and closed sets to filters. The dual algebra of 2 N is a countable denseBoolean algebra. Such a Boolean algebra is unique up to isomorphism.We only develop Stone duality for 2 N , to the extent relevant to us, namely, forrepresenting open or closed sets. Instead of working with the countable denseBoolean algebra, we will restrict ourselves to {0, 1} ∗ . Filters then become binarytrees without dead branches. Thus each closed set is represented by such a tree.The advantage of this representation of closed sets is that we have “descendeddown one level”: we are looking at sets of strings rather than at classes. As aresult, we may apply the usual algorithmic notions developed for numbers (orstrings) as the atomic objects. For instance, we will study closed sets where therepresenting tree is a Π 0 1 set, called Π 0 1 classes. We prove some important existencetheorems, such as the Low Basis Theorem 1.8.37 that each nonempty Π 0 1class P contains a low set. The Π 0 1 classes are at the first level of the arithmeticalhierarchy for classes defined in 1.8.55.Given a set Z and a number n, we may regard Z ↾ n as a partial description of Zsince it specifies the first n bits. Classes will be used as a tool to study sets, becausethey constitute a more general type of partial descriptions: we may think of a class Cas a partial description of any set Z ∈ C. This enables us to switch to the global viewof a set, where the set is appreciated all at once (Section 1.3). We will be interestedin partial descriptions given by classes that have special topological properties, or areeasy to describe themselves, or both (like Π 0 1 classes).If the class C is small in a particular sense, it provides a close description of Z inthat sense. Usually we will take smallness in the sense of the uniform measure onCantor space (also called the product measure, and defined in 1.9.7). For instance, theclass corresponding to the partial description z = Z ↾ n of Z is the basic open cylinder{Y : z ≺ Y }. It has uniform measure 2 −n and therefore gets smaller as n increases.In Section 1.4 we defined computable approximations (Z r) r∈N of a ∆ 0 2 set Z, where Z ris contained in [0,r) (equivalently, one can let Z r be a string of length r). We think of Z ras a different type of partial description of Z. It also improves as r increases. The ideato approximate a set in stages can be generalized: we may weaken the effectivenesscondition on the approximation, or we can replace each Z r by a class as a partialdescription. In the proof of the Low Basis Theorem 1.8.37 we even do both: we define a∅ ′ -computable sequence (P r ) r∈N of Π 0 1 classes, where P 0 is the given class P . The lowset Z ∈ P is determined by {Z} = ⋂ r P r .In the introduction to Chapter 3 we will consider the idea that a random set is aset which does not even admit a close description. For instance, a Martin-Löf test (see3.2.1) is a further way to approximate a set in stages r by classes G r. Now the r-thpartial description G r is an open set which has a uniform measure of at most 2 −r andis uniformly c.e. in r. A set is Martin-Löf random if it is not in ⋂ rGr for any such test.Note that ⋂ rGr has uniform measure 0 but is usually not a singleton.

Open sets1.8 Cantor space 47For a string y, the class of infinite binary sequences extending y is denoted by[y] ={Z : y ≼ Z}.These classes are called basic open cylinders, or cylinders for short. Clearly[x] ⊇ [y] ↔ x ≼ y. The cylinders form a basis of a topology: R ⊆ 2 N is openif R is the union of the cylinders contained in R, or, in other words, if for eachZ ∈ 2 N we have Z ∈ R ↔∃n [Z ↾ n ] ⊆ R.A Turing functional Φ can be viewed as a partial map Φ :2 N → 2 N where Φ(Y )is defined iff Φ Y is total. Unless the domain of this map is empty, it can be consideredas a subspace of 2 N . One reason why we use the product topology on 2 Nis that we want this partial map to be continuous. This is the case because of theuse principle (see after 1.2.17) which states that a converging oracle computationonly depends on a finite initial segment of the oracle (Exercise 1.8.8).Binary trees and closed setsA subset P of a topological space is called closed if its complement is open. Werepresent closed sets in Cantor space by subtrees of {0, 1} ∗ .1.8.1 Definition. (i) A binary tree is a subset B of {0, 1} ∗ closed under takingprefixes. That is, x ∈ B and y ≼ x implies y ∈ B. (ii) Z is a path of B if Z ↾ n ∈ Bfor each n. The set of paths of B is denoted by Paths(B).For instance, T = {0 i : i ∈ N} ∪ {0 i 1: i ∈ N} is a binary tree such thatPaths(T )={0 ∞ }. Since binary trees are subtrees of {0, 1} ∗ , we may apply thevisual terminology of “above/left/right” introduced on page 12. The following isknown as König’s Lemma.1.8.2 Lemma. If B is an infinite binary tree then Paths(B) ≠ ∅.Proof. For each n let x n be the leftmost string x of length n such that B ∩{y : y ≽ x} is infinite. Then x n ≺ x n+1 for each n, and ⋃ n x n is a path of B.✷We say that x ∈ B is a dead branch of a binary tree B if B ∩ {y : y ≽ x} isfinite. By König’s Lemma, this is equivalent to the condition that no path of Bextend x.Paths(B) is a closed set: if Z ∉ Paths(B) then there is n such that Z ↾ n ∉ B,so [Z ↾ n ] ∩ Paths(B) =∅. However, there are “more” binary trees than closedsets. For instance, if we take the tree T above and cut off the dead branches, weobtain T ′ = {0 i : i ∈ N}. The trees T and T ′ have the same paths. Closed setscorrespond to trees B without dead branches, that is, x ∈ B implies x0 ∈ B orx1 ∈ B for each x ∈ {0, 1} ∗ .1.8.3 Fact. (Stone duality for closed sets)(i) If P is closed thenT P = {x: [x] ∩ P ≠ ∅}is a tree without dead branches such that Paths(T P )=P .

48 1 The complexity of sets(ii) If B is a tree without dead branches then B = T Paths(B) .Proof. (i) Clearly T P is closed under prefixes. Moreover, T P has no dead branchesbecause [x] =[x0] ∪ [x1]. Since 2 N − P is open,Z ∈ P ↔∀n [Z ↾ n ] ∩ P ≠ ∅↔Z ∈ Paths(T P ).(ii) Clearly T Paths(B) ⊆ B. For the converse inclusion, suppose that x ∈ B.Since B has no dead branches, {y ≽ x: y ∈ B} is infinite, so by König’sLemma 1.8.2 there is a set Z ∈ Paths(B) extending x.✷We will sometimes identify P and T P . For instance, when we say that x is on Pwe mean that x ∈ T P , or equivalently that [x] ∩ P ≠ ∅.Representing open setsIf R is open we let P =2 N − R and represent R by A R = {0, 1} ∗ − T P . Thus,A R = {x: [x] ⊆ R}.This set is closed under extensions of strings, namely x ∈ A R and x ≺ y impliesy ∈ A R . Further, for each x ∈ {0, 1} ∗ , x0 ∈ A R and x1 ∈ A R implies x ∈ A R .A subset of {0, 1} ∗ with these two properties is sometimes called an ideal (ofstrings). Ideals are the complements in {0, 1} ∗ of trees without dead branches.We occasionally identify R and A R and write x ∈ R for [x] ⊆ R.The open set generated by a set S ⊆ {0, 1} ∗ is[S] ≺ = {X ∈ 2 N : ∃y ∈ Sy≼ X}.There is a correspondence between ideals and open sets analogous to Fact 1.8.3.It is immediate from 1.8.3 by complementing:1.8.4 Fact. (Stone duality for open sets)(i) If R is open, then A R = {x: [x] ⊆ R} is an ideal and R =[A R ] ≺ .(ii) If C is an ideal of strings then C = A [C] ≺.The strings x in A R that are minimal under the prefix ordering form an antichainD =(x i ) i

1.8 Cantor space 49and compactness tells us that there is a set satisfying all the conditions. (We willuse this method for the first time in the proof of Theorem 1.8.37.)1.8.5 Proposition. If (P i ) i∈N is a descending sequence of nonempty closed sets,then ⋂ i P i ≠ ∅.Proof. Let v n be the leftmost string of length n on P n . For each e, asn growsv n ↾ e only moves to the right on {0, 1} ∗ ,soz e = lim n≥e (v n ↾ e ) exists.Let Z = ⋃ e z e. Fix n. For all e we have [z e ] ∩ P n ≠ ∅. Since P n is closed, thisimplies that Z ∈ P n .✷A subset of a topological space is called clopen if it is both open and closed.The clopen sets form a Boolean algebra with the usual union and intersectionoperations. The clopen sets in Cantor space play a role similar to the finitesubsets of N. For instance, we will provide computable approximations of c.e.open sets as effective unions of clopen sets. By the following, the strong indicesfor finite sets F ⊆ {0, 1} ∗ given by Definition 1.1.14 can also be used as indicesfor the clopen sets. They will be called strong indices for clopen sets.1.8.6 Proposition. C ⊆ 2 N is clopen ⇔ C =[F ] ≺ for somefinite set F ⊆ {0, 1} ∗ .Proof. ⇒: Since the cylinders form a basis of 2 N , there are sets D, E ⊆ {0, 1} ∗such that C =[D] ≺ = ⋃ σ∈D [σ] and 2N − C =[E] ≺ = ⋃ ρ∈E[ρ]. By the compactnessof 2 N there are finite sets F, G such that F ⊆ D, G ⊆ E and [F ] ≺ ∪[G] ≺ =2 N .Then C =[F ] ≺ ,soF is as required.⇐: Each cylinder [σ] is clopen since its complement is ⋃ {[ρ]: ρ ≠ σ & |ρ| = |σ|}.Thus each set of the form [F ] ≺ is clopen.✷Exercises.1.8.7. Let X, Y ∈ 2 N .IfX ≠ Y let d(X, Y )=2 −n where n is least such that X(n) ≠Y (n). Let d(X, X) = 0. Show that (2 N ,d)isametric space which induces the usualproduct topology on 2 N .A function F between topological spaces is continuous if the preimage under F ofeach open set is open. Consider a map F : D → 2 N where D is a subspace of 2 N . Sincethe basic open cylinders form a basis, F is continuous iff for each ρ ∈ {0, 1} ∗ such that[ρ] ∩ ran(F ) ≠ ∅ there is σ ∈ {0, 1} ∗ such that F ([σ] ∩ D) ⊆ [ρ].1.8.8 Exercise. We view a Turing functional Φ such that dom(Φ) ≠ ∅ as a map fromthe subspace D = dom(Φ) to 2 N . Show that Φ is continuous.Recall that A ⊕ B = {2n: n ∈ A} ∪ {2n +1: n ∈ B}.1.8.9 Exercise. Show that L: 2 N → 2 N is continuous ⇔ there is a Turing functionalΦ and a set A such that L(Z) =Φ(Z ⊕ A) for each Z ⊆ 2 N .The correspondence between subsets of N and real numbersCo-infinite subsets of N and real numbers in [0, 1) R are often identified. Whenwe make this identification we usually indicate it. We give the details.

50 1 The complexity of sets1.8.10 Definition. The mapF : {Z ∈ 2 N : Z is co-infinite} → [0, 1) R (1.13)is defined by F (Z) =0.Z = ∑ i∈Z 2−i−1 .We will determine the inverse G of F , thereby showing that F is a bijection.Each real number r ∈ [0, 1) R can be written in the form r = ∑ i≥0 r i2 −i−1where r i ∈ {0, 1}. We say 0.r 0 r 1 ... is a binary expansion of r. The set of dyadicrationals isQ 2 = {z2 −n : z ∈ Z,n∈ N}.A binary expansion of r is unique unless r ∈ Q 2 , in which case we give preferenceto the finite binary expansion. Thus we view 1/4as0.01 rather than as 0.00111 ...Via this binary expansion a real number r ∈ [0, 1) R can be identified with asequence Z = G(r) =r 0 r 1 ... that has infinitely many zeros, that is, with aco-infinite set Z. Clearly F and G are inverses.For a finite string y, we sometimes use the notation 0.y as a shorthand for 0.y000 ....Usually we identify a dyadic rational in (0, 1) with a finite string ending in 1.1.8.11 Remark. The bijection F maps the basic open cylinder [σ] (restricted to {Z ∈2 N : Z is co-infinite}) to the interval I(σ) =[0.σ, 0.σ +2 −|σ| ). For a linear order L withleast but no greatest element let Intalg L denote the Boolean algebra generated bythe⋃intervals [a, b) where a, b ∈ L. Thus, Intalg L consists of the sets of the form0≤i

1.8 Cantor space 51(ii) r is left-c.e. if {q ∈ Q 2 : qr}is c.e.(iii) Z ⊆ N is left-c.e. if the real number 0.Z is left-c.e., or, equivalently, if{σ : σ < L Z} is c.e. Similarly we define right-c.e. sets.(iv) r is difference left-c.e. if there are left-c.e. reals α, β ∈ R such that r = α−β.These classes of reals can be characterized via effective approximations by rationals.One may in fact require that the rationals be dyadic. The characterization(v) below is due to Ambos-Spies, Weihrauch and Zheng (2000).1.8.15 Fact. Let r ∈ R. The following equivalences hold uniformly.(i) r is ∆ 0 2 ⇔ r = lim n q n for a computable sequence (q n ) n∈N of rationals.(ii) r is left-c.e. ⇔ r = lim n q n for a non-descending computable sequence(q n ) n∈N of rationals.(iii) r is right-c.e. ⇔ r = lim n q n for a non-ascending computable sequence(q n ) n∈N of rationals.(iv) r is computable ⇔ r = lim n q n for a computable sequence (q n ) n∈N ofrationals such that abs(r − q n ) ≤ 2 −n for each n⇔ given n one can compute q ∈ Q suchthat abs(r − q) ≤ 2 −n .(v) r is difference left-c.e. ⇔ r = lim n q n for a computable sequence (q n ) n∈N ofrationals such that ∑ n abs(q n+1 − q n ) < ∞.Proof. We leave (i), (iii) and (iv) as exercises.(ii) ⇒: If W e = {q ∈ Q 2 : q < r} (via our identification), then let q n =max(W e,n ).⇐: The left cut of r is c.e. because for q ∈ Q 2 we have q

52 1 The complexity of setsnow one requires the relevant property for B. This leads to the same class if theproperty is being computable, or being ∆ 0 2. In contrast, if A is a c.e. set, then0.A is left-c.e., but not conversely. A counterexample is, for instance, Chaitin’snumber Ω; see page 108. Each left-c.e. set Z is truth-table equivalent to the c.e.set {σ : σ < L Z}.The computable real numbers form a field. More generally, for any ideal L in theTuring degrees, the real numbers with Turing degree in L form a field, because a realnumber t obtained from real numbers r, s by a field operation is computable in r⊕s. Theleft-c.e. real numbers are closed under addition but not under the operation x →−x.In contrast, Ambos-Spies, Weihrauch and Zheng (2000) proved the following surprisingfact.1.8.16 Proposition. The set D of difference left-c.e. real numbers is a subfield of R.Proof. Throughout, we use Fact 1.8.15(v). Clearly D is closed under the operationx →−x and under addition. For the closure under product, suppose r, s ∈ D. Thereare M ∈ N and effective sequences (x n) n∈N and (y n) n∈N of rationals converging to r, s,respectively, such that ∀n abs(x n) ≤ M, ∀n abs(y n) ≤ M,∑n abs(xn+1 − xn) ≤ M, and ∑ nabs(yn+1 − yn) ≤ M.Then lim n x ny n = rs and ∑ n abs(xn+1yn+1 − xnyn) ≤ ∑ nabs(xn+1yn+1∑− xnyn+1)+n abs(xnyn+1 − xnyn) ≤ 2M 2 ,sors ∈ D.It remains to show that 1/r ∈ D in case that r ≠ 0. We may assume M>abs(1/r),so there is n 0 ∈ N such that x n ≠ 0 and abs(1/x n) ≤ M for all n ≥ n 0. Then∑ 1abs( − 1 )= ∑ abs(x n − x n+1)≤ M 3 .x n+1 x n abs(xn≥n 0 n≥n nx n+1)0Note that the closure under the field operations is effective.✷Exercises.1.8.17. If r is left-c.e. then e r is left-c.e. as well.1.8.18. Suppose r i is a computable real number uniformly in i ∈ N and0 ≤ r i ≤ 2 −i for each i. Show that r = ∑ iri is computable.Effectivity notions for classes of setsThis subsection introduces two essential concepts: co-c.e. closed sets (also calledΠ 0 1 classes) and c.e. open sets (also called Σ 0 1 classes). We show how to indexsuch classes, and how to obtain effective approximations for them.By Stone duality (Facts 1.8.3 and 1.8.4), any complexity notion for sets ofstrings can be applied to closed sets and to open sets. To be in a closed set P isa property related to universal quantification: Z is in P if each initial segmentof Z lies on the representing tree. Similarly, to be in an open set is an existentialproperty. This is why the most fruitful effectiveness notion for closed sets is tobe Π 0 1, and the most fruitful one for open sets is to be Σ 0 1. We may call theseobjects either sets (when viewing them as sets of strings) or classes (when viewingthem as sets of subsets of N).

1.8.19 Definition.1.8 Cantor space 53(i) A closed set P is co-c.e. if the corresponding binary treeT P = {x: [x] ∩ P ≠ ∅} has a c.e. complement in {0, 1} ∗ . A co-c.e. closedset is usually called a Π 0 1 class.(ii) An open set R is c.e. if the corresponding set of stringsA R = {x: [x] ⊆ R} is c.e.; such an open set is also called a Σ 0 1 class.Representing Π 0 1 classesUsually we show that a class P is Π 0 1 by defining some Π 0 1 tree B such thatP = Paths(B), and using the following fact.1.8.20 Fact. Let B ⊆ {0, 1} ∗ be a Π 0 1 tree. Then P = Paths(B) is a Π 0 1 class.Proof. The subtree of B given byB ∗ = {σ : ∀n ≥ |σ| ∃ρ ∈ B [|ρ| = n & ρ ≽ σ]} (1.14)is a Π 0 1 tree since the quantifier ∃ρ is bounded. Moreover, B ∗ has no deadbranches and P = Paths(B ∗ ). Hence B ∗ = T P by the correspondence in Fact1.8.3, and P is a Π 0 1 class. ✷On the other hand, if we are willing to admit dead branches we can find arepresenting tree that is computable.1.8.21 Fact. Each Π 0 1 class is of the form Paths(B) for some computable tree B.Proof. T P is a Π 0 1 tree, so A = {0, 1} ∗ − T P is c.e., and therefore has a computableenumeration (A s ) s∈N . The tree B = {σ : ∀ρ ≼ σ [ρ ∉ A |σ| ]} is computable.Clearly T P ⊆ B and hence P ⊆ Paths(B). For the inclusion Paths(B) ⊆P ,ifZ ∉ P , then we can choose n such that Z ↾ n ∈ A s for some s. Then Z ↾ n isa dead branch of B because none of its extensions of length s are in B. HenceZ ∉ Paths(B).✷A Π 0 1 class P can be viewed as a problem. Each Z ∈ P is a solution of this problem.A Π 0 1 tree B such that P = Paths(B) is a description of the problem. By Fact 1.8.21,each problem given by a Π 0 1 class has a computable description. Fact 1.8.31 belowshows that sometimes there exists a solution but not a computable one. However, thereis always a low solution by the Low Basis Theorem 1.8.37. Later on we will encountermore examples of nonempty Π 0 1 classes without computable members, for instance thesets that are Martin-Löf-random for a constant b (see 3.2.9).If T P is computable then its leftmost path is computable. Thus, if P has no computablemember, the tree B in the proof of 1.8.21 necessarily contains dead branches.Representing c.e. open sets1.8.22 Fact. R ⊆ 2 N is c.e. open ⇔ R =[W e ] ≺ for some e.Proof. ⇒: By definition A R is c.e. Then R =[A R ] ≺ by Fact 1.8.4(i).

54 1 The complexity of sets⇐: LetŴ e = {σ : ∃ρ ∈ W e [ρ ≼ σ]}. (1.15)Then R =[Ŵe] ≺ . Since Ŵe is closed under extensions, B = {0, 1} ∗ − Ŵe is aΠ 0 1 tree. So P = Paths(B) isaΠ 0 1 class by Fact 1.8.20, and R =2 N − P .ThusA R = {0, 1} ∗ − T P . Since T P is Π 0 1, A R is c.e. So R is a c.e. open set. ✷1.8.23 Remark. We say the number e is an index for a c.e. open set R if R =[W e ] ≺ . An index for an object usually provides us with an effective approximationof the object (see page 7). In the case of c.e. open sets, the approximation forindex e is denoted by (Ŵe,s) s∈N . It is chosen in such a way that Ŵe,s is closedunder extensions within the strings of length up to s: letŴ e,s = {x: |x| ≤ s & ∃y ∈ W e,s [y ≼ x]}. (1.16)We suppress the index e for R and simply write R s for Ŵe,s. ThusR = ⋃ s [R s] ≺ .1.8.24 Example. Let Φ be a partial computable functional. For each string x,the classS x = {Z : ∀i

1.8 Cantor space 55(P s ) s∈N (1.17)of (strong indices for) clopen sets, where P s ⊇ P s+1 and P = ⋂ s P s. To obtainthis sequence let P s =[{σ : |σ| = s & σ ∉ V e,s }] ≺ . Thus P s is generated by thestrings of length s that are still on the tree B e at stage s. (This differs from thecase of an approximation (R s ) s∈Nof a c.e. open set, where R s is a set of stringsof length at most s.)1.8.28 Fact. Let G e be the Π 0 1 class given by index e. Then the set {e: G e = ∅}is Σ 0 1 .Proof. Clearly G e = ∅↔∃sG e s = ∅, and the latter condition is Σ 0 1.1.8.29 Exercise. Show that the operations ∪ and ∩ are effective on indices for c.e.open sets, as well as on indices for Π 0 1 classes.Examples of Π 0 1 classesWe are now in the position to give some interesting examples of Π 0 1 classes.The first one will be important later on, for instance in Section 4.3. Recall thatJ(e) denotes Φ e (e). A {0, 1}-valued function f is called two-valued diagonallynoncomputable, or two-valued d.n.c. for short, if it avoids being equal to J(e)whenever J(e) is defined, namely, ¬f(e) =J(e) for each e.1.8.30 Remark. A simple example of a two-valued d.n.c. function f is obtainedas follows. Since {e: J(e) =1} is c.e., there is a computable function q such thatJ(e) =1 ⇔ q(e) ∈∅ ′ . Now let f(e) =1−∅ ′ (q(e)). If J(e) = 1 then f(e) =0,otherwise f(e) = 1. Notice that f ≤ tt ∅ ′ .1.8.31 Fact. The two-valued d.n.c. functions form a nonempty Π 0 1 class Pwithout computable members. In particular, the tree T P is not computable.Proof. The set of strings B = {σ : ∀s ∀e

56 1 The complexity of setsConsider the class in (iii) when T is Peano arithmetic. Then T has no computablecompletion, so once again there is no computable solution for the problemdescribed by the Π 0 1 class of completions of T .Isolated points and perfect setsA point Z in a topological space X is called isolated if the singleton {Z} is open.For instance, if X satisfies the separation axiom T 1 (each singleton is closed)and X is finite then each point is isolated. Consider the case that X is thesubspace Paths(B) of Cantor space for a binary tree B. Then a path Z of B isisolated iff there is a number n 0 such that Z is the only path extending Z ↾ n0 .For example, consider the treeB = {0 i : i ∈ N} ∪ {0 i 10 k : i, k ∈ N}.Then Paths(B) is infinite, and every path except for 0 ∞ is isolated in X .A nonempty closed set P in a topological space is called perfect if it has noisolated points. If P ⊆ 2 N this amounts to saying that the binary tree T P has noisolated path. Thus each perfect class in Cantor space has size 2 ℵ0 .The following fact is usually applied to computable trees, and often in relativizedform, but it actually holds for Π 0 1 trees.1.8.33 Fact. Let B ⊆ {0, 1} ∗ be a Π 0 1 tree. Then each isolated path Z of B iscomputable.Proof. Let B ∗ be the Π 0 1 subtree of extendable nodes of B defined in (1.14).Then Z is an isolated path of B ∗ as well. So choose a number n 0 such that Z isthe unique path of B ∗ extending Z ↾ n0 . To compute Z(n) for n ≥ n 0 , enumerate{0, 1} ∗ − B ∗ until there remains a unique σ ≽ Z ↾ n0 such that |σ| = n + 1 andσ is on B ∗ . This must happen, for otherwise some path of B ∗ other than Z alsoextends Z ↾ n0 .Thusσ ≺ Z, so output σ(n).✷In particular, every nonempty Π 0 1 class without computable members is perfect.1.8.34 Corollary. Let B be a binary tree.(i) If Z is an isolated path of B then Z ≤ T B.(ii) If Paths(B) is finite, then Z ≤ T B for each path Z.Proof. (i). Relativize Fact 1.8.33 (the case where the binary tree is computable).(ii). It suffices to observe that each path of B is isolated.✷1.8.35 Exercise. Suppose the partial computable function ψ in 1.8.32(ii) has a coinfinitedomain. Then the class of total {0, 1}-valued extensions of ψ is perfect.The Low Basis TheoremA basis theorem (for Π 0 1 classes) states that each nonempty Π 0 1 class has amember with a particular property. We begin with an example of such a theorem,the Kreisel Basis Theorem, that the left-c.e. sets form a basis for the Π 0 1 classes.1.8.36 Fact. Every nonempty Π 0 1 class P has a left-c.e. member, namely itsleftmost path.

1.8 Cantor space 57Proof. By Fact 1.8.3 we may identify a closed P ⊆ 2 N with the tree T P ={x: [x] ∩ P ≠ ∅}, the set of strings that are on P . We show that the leftmostpath Y of P is left-c.e. Note thatτ < L Y ↔ ∀σ ≤ L τ [ |σ| = |τ| → σ is not on P ] .This is in Σ 0 1 form because the universal quantifier is bounded. Thus the set{τ : τ < L Y } is c.e. ✷The proof of a basis theorem is usually uniform: from an index for the given Π 0 1class one can compute a description of a member with the desired property. Thus, abasis theorem provides an effective choice function, picking a member with a particularproperty from the class in case the class is nonempty.To verify that the proof of Fact 1.8.36 is uniform we provide a little more detail: froman index for P (that is, an index for a Π 0 1 tree B such that Paths(B) =P ) we caneffectively obtain an index for the Π 0 1 tree B ∗ without dead branches in (1.14). SinceB ∗ = {σ : σ is on P }, wehaveτ < L Y ↔ ∀σ ≤ L τ [ |σ| = |τ| → σ ∉ B ∗] , which givesthe required c.e. index for {τ : τ < L Y }.The desired property of member Y of the given Π 0 1 class is usually a lowness property.On the other hand, Y ∈ P often means that Y is complex in some sense (computationalor descriptive). For instance, P could be the class of two-valued d.n.c. functions inFact 1.8.31. Then a basis theorem implies that a set can be computationally weak ina given sense, but complex in the sense of being in P . In particular, the Π 0 1 classes weare interested in here have no computable paths, otherwise the basis theorem is trivial.The following is due to Jockusch and Soare (1972b).1.8.37 Theorem. (Low Basis Theorem)Every nonempty Π 0 1 class has a low member.Proof. Let P be the given Π 0 1 class. We define a descending sequence of nonemptyΠ 0 1 classes (P e ) e∈N , where P 0 = P . Then, by the compactness of 2 N (in the formof Proposition 1.8.5), there is a set Y in ⋂ e P e .The class P e+1 determines whether e ∈ Y ′ : either this holds for no Y ∈ P e+1 ,or for all such Y . The halting problem ∅ ′ can decide which case applies, soY ′ ≤ T ∅ ′ . (Note that Y is the only element of ⋂ e P e , since Y ≤ m Y ′ via somefixed many-one reduction.)Construction relative to ∅ ′ of Π 0 1 classes (P e ) e∈N . At stage e we define P e . LetP 0 = P .Stage e + 1. Suppose that P e has been defined. If P e ∩ {Z : J Z (e) ↑} ̸= ∅ thenlet P e+1 be this class, otherwise let P e+1 = P e .Notice that by Example 1.8.32(i) one may effectively determine an index for theΠ 0 1 class P e ∩ {Z : J Z (e) ↑}. Thus by Fact 1.8.28, it is a Π 0 1 property of e thatthis class is nonempty. Therefore this can be decided by ∅ ′ . Clearly, e ∉ Y ′ iffthis first alternative applies at stage e + 1. Hence Y ′ ≤ T ∅ ′ .✷

58 1 The complexity of setsIn the foregoing proof we approximated a set Y not by specifying initial segmentsbut rather via the classes P e . In fact P e determines Y ′ ↾ e . We discussedapproximations by classes on page 46 at the beginning of this section.The proof of Theorem 1.8.37 actually produces a set Y ∈ P such that Y ′ isleft-c.e. Note that this property of a set Y depends on the particular way thejump operator is defined. However, the property implies that Y ′ ≡ tt ∅ ′ (namely,Y is superlow, Definition 1.5.3). To verify that Y ′ is left-c.e., we rewrite theproof, avoiding a construction relative to ∅ ′ . This also stresses its uniformity.1.8.38 Theorem. (Extends 1.8.37) Every nonempty Π 0 1 class P has a memberY such that Y ′ is left-c.e.; a c.e. index for {σ : σ < L Y ′ } (and hence areduction procedure for Y ′ ≤ tt ∅ ′ ) can be obtained effectively from an index for P .Proof idea. First consider Y ′ (0). We maintain the guess that J Y (0)↑ as longas possible, namely, till it becomes apparent that J Y (0)↓ for all Y in P (note thatthis is a Σ 0 1 event). For Y ′ (1) we do the same, but at first within the restricted Π 0 1class ˜P = P ∩ {Y : J Y (0) ↑}. Once our guess at Y ′ (0) changes to 1, we removethis restriction on ˜P . We may already have discovered that J Y (1) ↓ for all Yin ˜P . This led us to make a guess that Y ′ (1) = 1. This guess may now be revisedto Y ′ (1) = 0, in case there remains some Y in P such that J Y (1)↑. It should beclear how to continue this procedure in order to guess at Y ′ (e) for each e.Proof details. We apply the Kreisel Basis Theorem 1.8.36 to the Π 0 1 class ofstrings τ such that for some Y ∈ P , whenever e

1.8 Cantor space 59(P e ) ∈N , and let Y be an element of ⋂ e P e . However, now the class P 2e+1 is usedto determine whether e ∈ Y ′ , while P 2e+2 prevents that B = Φ Y e .Construction relative to ∅ ′ ⊕ B of Π 0 1 classes (P e ) e∈N . Let P 0 = P .Stage 2e + 1. This is similar to stage e + 1 in the proof of Theorem 1.8.37. IfP 2e ∩{X : J X (e)↑} ≠ ∅, then let P 2e+1 be this class. Otherwise, let P 2e+1 = P 2e .Stage 2e + 2. Using ∅ ′ ⊕ B as an oracle, search in parallel for the following:(a) a string σ and a number k such that σ is on P 2e+1 (i.e., [σ] ∩ P 2e+1 ≠ ∅)and B(k) ≠ Φ σ e (k). If such a pair is found let P 2e+2 = P 2e+1 ∩ [σ];(b) a number n such that P 2e+1 ∩ {X : Φ X e (n) ↑} ≠ ∅. Ifsuchanumberisfound let P 2e+2 be this class.The parallel search carried out at an even stage terminates: otherwise, Φ X eis total for each X ∈ P 2e+1 since (b) does not terminate. In that case, B iscomputable, contrary to our assumption: given k, we may compute s such thatfor all σ ∈ Ps2e+1 ,if|σ| = s then Φ σ e (k) is defined, and in addition, since (a)does not terminate, all such computations Φ σ e (k) give the same output. Thenthis common output is B(k). (Here we have used the approximation given by(1.17), which is automatically obtained from an index for the Π 0 1 class.)To see that Y ′ ≤ T B ⊕∅ ′ , as before it suffices to let B ⊕∅ ′ decide which caseapplies at each stage as this determines Y ′ . The halting problem ∅ ′ suffices forthe odd stages, and B ⊕∅ ′ can decide whether the parallel search carried out ateach single even stage terminates first in (a), or first in (b).✷Even if B is c.e., one cannot in general achieve that Y ≤ wtt ∅ ′ . A counterexample canbe derived from Exercise 8.5.23.Exercises.1.8.40. Extend 1.8.39 as follows: given incomputable sets B 1,...,B k , there is a setY ∈ P such that B 1,...,B k ≰ T Y and Y ′ ≤ T B 1 ⊕ ...⊕ B k ⊕∅ ′ .1.8.41. Suppose that P ≠ ∅ is a Π 0 1 class and B ⊆ N. Show that there is a set Y ∈ Psuch that (Y ⊕ B) ′ ≤ tt B ′ . (This is more than a mere relatization of 1.8.37.)The basis theorem for computably dominated setsMartin and Miller (1968) proved that the computably dominated sets (Definition1.5.9) form a basis for the Π 0 1 classes. Since there is a Π 0 1 class withouta computable member (Fact 1.8.31), this shows that a computably dominatedset can be incomputable. An extension of the result states that there are 2 ℵ0computably dominated sets.1.8.42 Theorem. Every nonempty Π 0 1 class P has a computably dominatedmember Y .Proof. Once again we define a descending sequence of nonempty Π 0 1 classes(P e ) e∈N and use Proposition 1.8.5 to conclude that there is Y ∈ ⋂ e P e .Webegin with P 0 = P . The class P e+1 either shows that Φ Y e is partial, or thatthere is a computable function f dominating Π Y e .

60 1 The complexity of setsConstruction relative to ∅ ′′ of Π 0 1 classes (P e ) e∈N . Let P 0 = P .Stage e + 1. Suppose P e has been determined.(a) Q x = P e ∩ {Z : Φ Z e (x) ↑} is a Π 0 1 class, uniformly in x. If there is x suchthat Q x ≠ ∅, let x be least such and let P e+1 be this class. This ensuresthat Φ Z e is partial for each set Z ∈ P e+1 .(b) Otherwise let P e+1 = P e . (We will show that there is a computable functionf dominating Φ Z e for each Z ∈ P e+1 .)Let Y ∈ ⋂ e P e .Ifg = Φ Y e is a total function then at stage e + 1, we are incase (b) since Y ∈ P e+1 . To compute a function f that dominates g, weusethe approximation (Ps e ) s∈N obtained in (1.17) on page 55. Given an input x,compute s = s(x) such that∀σ ∈ P e s [ |σ| = s → Φ σ e,s(x)↓].Such an s exists, otherwise the class Q x considered at stage e + 1 is nonempty.Now let f(x) be the maximum of all the values Φ σ e (x) where |σ| = s(x) andσ ∈ P e s(x) . Then f dominates ΦZ e for each Z ∈ P e . ✷While an incomputable computably dominated set is not ∆ 0 2 by 1.5.12, the set Yabove automatically satisfies Y ′′ ≤ tt ∅ ′′ . This fact requires some additional effort toverify. Recall from Exercise 1.4.20(iii) that Tot = {e: dom(Φ e)=N} is Π 0 2-complete.The solution of 1.4.20 actually shows that Tot Y = {e: dom(Φ Y e )=N} ≡ m N − Y ′′for each Y , and hence Partial Y := N − Tot Y ≡ m Y ′′ . The following is analogous toTheorem 1.8.38.1.8.43 Theorem. (Extends 1.8.42) Every nonempty Π 0 1 class P has a computablydominated member Y such that Y ′′ ≤ tt ∅ ′′ , and indeed {τ : τ < L Partial Y } is Σ 0 2.A truth-table reduction of Y ′′ to ∅ ′′ can be obtained effectively from an index for P .Proof. Let Y ∈ ⋂ e P e as above. We effectively obtain from P a Σ 0 2 index for the set{τ : τ < L Partial Y }. This suffices because Y ′′ ≡ m Partial Y ≡ tt {τ : τ < L Partial Y } viafixed reduction procedures.The following construction enumerates {τ : τ < L Partial Y } relative to ∅ ′ . At eachstage s>0, for each e

1.8 Cantor space 61To prove the claim we use induction on e. Clearly lim s ˜P 0 [s] =P 0 , and lim sτ s(0) = 1 iffsome x is found by S 0 such that P 0 ∩ {Z : Φ Z 0 (x)↑} ≠ ∅ iff Φ Y 0 is partial. Now supposethat the claim holds for all i ≤ e, and let s 0 be a stage by which all the limits for i ≤ ehave been reached. Then S e is not initialized at any stage s ≥ s 0,soifS e finds x ata stage s ≥ s 0, it defines ˜P e+1 [s] and τ s(e) correctly. Otherwise, ˜P e+1 [s] and τ s(e) arealready correct at stage s = s 0.✷Recall from page 56 that a nonempty closed set in a topological space is calledperfect if it has no isolated points.1.8.44 Theorem. (Extends 1.8.42) Every nonempty Π 0 1 class P without computablemembers has a perfect subclass S of computably dominated sets.Proof. We modify the proof of Theorem 1.8.42. Instead of a descending sequence(P e ) e∈N , we build a tree (P σ ) σ∈{0,1} ∗ of Π 0 1 classes.Stage 0. Let P ∅ = P .Stage e + 1. Suppose that P σ ≠ ∅ has been determined for each σ such that|σ| = e. Firstly, for every such σ let ̂P σ0 and ̂P σ1 be nonempty disjoint subclassesof P σ . They exist since there are incompatible strings τ 0 , τ 1 on P σ ,sowemaylet ̂P σi = P σ ∩ [τ i ](i ∈ {0, 1}). Now proceed as before, but with both classeŝP σi separately: if there is x such that ̂P σi ∩ {Z : Φ Z e (x)↑} ≠ ∅ then let x be leastsuch and let P σi be this class. Otherwise let P σi = ̂P σi .Verification. For each set C, there is a set Y C ∈ ⋂ e P C↾e , and, by the sameargument as before, each such Y C is computably dominated. If C ≠ D thenY C ≠ Y D . So the class S = {Y C : C ⊆ N} is as required.✷1.8.45 Corollary. There are 2 ℵ0 many computably dominated sets.Proof. Apply Theorem 1.8.44 to any Π 0 1 class without computable members,for instance the class of two-valued d.n.c. functions from Fact 1.8.31. ✷A further basis theorem is 4.3.2 below: if D computes a two-valued d.n.c. function,then each nonempty Π 0 1 class contains a set Y ≤ T D.1.8.46. ⋄ Exercise. (Kučera and Nies) Let P be a nonempty Π 0 1 class. Suppose thatB> T ∅ ′ is Σ 0 2. Then there is a computably dominated set Y ∈ P such that Y ′ ≤ T B.Hint. Combine the techniques of the Low Basis Theorem and Theorem 1.8.42 withpermitting below B relative to ∅ ′ . Fix an enumeration (B s) s∈N of B relative to ∅ ′ , anduse the function c B ≤ T B given by c B(i) =µt > i. B t ↾ i= B ↾ i for the permitting.Weakly 1-generic sets1-genericity for sets is an effective version of the notion of Cohen genericity fromset theory. We first introduce the simpler concept of weak 1-genericity. Eachweakly 1-generic set is hyperimmune. Each hyperimmune set is Turing equivalentto a weakly 1-generic set.A subset D of a topological space is called dense if D∩R ≠ ∅ for each nonemptyopen set R. A set D ⊆ 2 N is dense iff D ∩ [x] ≠ ∅ for each cylinder [x] (namely,each string is extended by a set in D). For instance, for each n, the open set

62 1 The complexity of setsE n = {Z : ∃i ≥ n [Z(i) =1]} is dense in 2 N . Note that E n is c.e. open, being aneffective union of clopen sets.In Proposition 1.8.5 we interpreted a descending sequence (P i ) i∈N of nonemptyclosed classes as desirable conditions. Compactness showed that there is a setZ ∈ ⋂ i P i . A desirable condition can also be given by a dense open set. Such acondition cannot be ruled out by any finite initial segment of a set G ⊆ N. Baire’scategory theorem for 2 N states that the intersection of a countable collection(D n ) n∈N of dense open sets contains a set G. One builds G by the method offinite extensions. Let [σ 0 ] ⊆ D 0 , and if σ n has been defined choose σ n+1 ≻ σ nsuch that [σ n+1 ] ⊆ D n+1 . Then G = ⋃ n σ n is as required. (This result holds inin any uncountable Polish space, such as Baire space N N .)The weakly 1-generic sets are the ones that meet each condition of this kindwhere the dense set is also computably enumerable.1.8.47 Definition. G ⊆ N is weakly 1-generic if G is in each dense c.e. openset D ⊆ 2 N .1.8.48 Proposition. Each weakly 1-generic set G is hyperimmune. In particular,G is not computably enumerable.Proof. G is infinite since G is in the dense set E n = {Z : ∃i ≥ n [Z(i) =1]} foreach n. Next, given a computable function f, we will show that there is n suchthat p G (n) ≥ f(n). The c.e. open setD f =[{σ0 f(|σ|) : σ ≠ ∅}] ≺is dense. Thus G ∈ D f , and we may choose σ such that σ0 f(|σ|) ≺ G. Let n = |σ|.In the worst case, σ consists only of ones, so that p G (n − 1) = n − 1 and G hasthe stringf(n){ }} {1}...1{{ }0 ... 0nas an initial segment. Even then we have p G (n) ≥ f(n).To build a weakly 1-generic set, we use the construction in the proof of Baire’scategory theorem.1.8.49 Theorem. There is a weakly 1-generic left-c.e. set G.Proof. (1) It is somewhat simpler to show that there is a weakly 1-generic∆ 0 2 set. We build an effective sequence (σ s ) s∈N of strings such that |σ s | = s.This sequence is a computable approximation of the set G, namely G(n) =lim s>n σ s (n) for each n.We use the effective listing of c.e. open sets ([Ŵe] ≺ ) e∈N , where, as in (1.15),Ŵ e = {σ : ∃ρ ∈ W e [ρ ≼ σ]}. Note that [Ŵe] ≺ =[W e ] ≺ . We meet the requirementsR e : [Ŵe] ≺ is dense ⇒ G ∈ [Ŵe] ≺ .✷

1.8 Cantor space 63Construction. Let σ 0 = ∅. Initialize all the requirements.Stage s>0. For each e, let t e,s be the maximum of e and the last stage when R ewas initialized. R e is called satisfied at stage s if σ s−1 ∈ Ŵe,s. If there is ee.We say that R e acts. Otherwise let σ s = σ s−1 0.Verification. First we show by induction that each requirement R e acts finitelyoften: when all R i , in σ s (n) exists for each n. Moreover, t e = lim s t e,s exists for each e.If [Ŵe] ≺ is dense then Ŵe contains some σ ≽ G ↾ te ,soR e is met. Thus G isweakly 1-generic.(2) To make G left-c.e., in (1.18) we only allow extensions σ such that σ s−1 < L σ.To see that R e is met, note that G is co-infinite by construction, so G↾ te 1 r 0 ≺ Gfor some r. Now R e can be satisfied permanently via some string in Ŵe extendingG↾ te 1 r+1 .✷Kurtz (1981, 1983) proved a converse of Proposition 1.8.48 for Turing degrees.1.8.50 Theorem. A is hyperimmune ⇒ A ≡ T G for some weakly 1-generic G. ✷Thus, one can characterize the hyperimmune degrees using “effective topology”. For arecent proof of Kurtz’s theorem see Downey and Hirschfeldt (20xx).1-generic setsLet X be a topological space, S ⊆ X and A ∈ X. We say that A is in the closureof S if S ∩ V ≠ ∅ for every open set V containing A. IfX is Cantor space and Sis open, this means that for each σ ≺ A there is ρ ≽ σ such that [ρ] ⊆ S. In thiscase we say that S is dense along A.1.8.51 Definition. A is 1-generic if A ∈ S for every c.e. open set S that isdense along A.1-generic sets were introduced by Jockusch (1977). If S is a dense open setthen S is dense along any A. Therefore each 1-generic set is weakly 1-generic(Definition 1.8.47). Note that A is 1-generic iff for each c.e. open set S, eitherA ∈ S or there is σ ≺ A such that [σ] ∩ S = ∅. Intuitively, if A ∉ S thenan initial segment of A already precludes A from being in S. For instance, letS = {Z : Φ Z (n) ↓} for a Turing functional Φ. If Φ A (n) ↑ then there is σ ≺ Asuch that Φ Y (n)↑ for each Y ≻ σ.Part (1) of the proof of Theorem 1.8.49 actually builds a 1-generic set. It sufficesto observe that our strategy meets the requirements

64 1 The complexity of setsR e : [Ŵe] ≺ is dense along G ⇒ G ∈ [Ŵe] ≺ .1.8.52 Theorem. There is a 1-generic set G ∈ ∆ 0 2. ✷A 1-generic set is not left-c.e., and in fact it does not even Turing bound an incomputablec.e. set. For this result of Jockusch (1977) see Odifreddi (1999, p. 662).However, the construction of a 1-generic set can be combined with permitting:Exercises.1.8.53. For each incomputable c.e. set C there is a 1-generic set A ≤ T C.1.8.54. Each 1-generic set G is in GL 1, namely, G ′ ≤ T G ⊕∅ ′ .The arithmetical hierarchy of classesSets of natural numbers are our primary objects of study. In particular, weare interested in their computational complexity, their descriptive complexity,and their randomness properties. Recall from the beginning of this chapter thatto understand these aspects of a set we study classes of sets sharing certaincomplexity or randomness properties. It will be useful to measure the descriptivecomplexity of classes as well. With the exception of Chapter 9, all classes of sets Xstudied in this book can be described by a formula in the language of arithmeticinvolving initial segments of X. Similar to the case of the arithmetical hierarchyof sets defined in 1.4.10, the descriptive complexity of a class is measured bythe complexity of its description, in terms of how many alternations of (number)quantifiers it has. The variable y n of the innermost quantifier is now used todetermine the initial segment X↾ yn of X.It is useful to extend the definitions to relations of k numbers and one set. Thecase k = 0 refers to classes.1.8.55 Definition. Let A ⊆ N k × 2 N and n ≥ 1.(i) A is Σ 0 n if〈e 1 ,...,e k ,X〉∈A ↔∃y 1 ∀y 2 ...Qy n R(e 1 ,...,e k ,y 1 ,...,y n−1 ,X↾ yn ),where R is a computable relation, and Q is “∃” ifn is odd and Q is “∀”if n is even.(ii) A is Π 0 n if the complement of A is Σ 0 n, that is,〈e 1 ,...,e k ,X〉∈A ↔∀y 1 ∃y 2 ...Qy n S(e 1 ,...,e k ,y 1 ,...,y n−1 ,X↾ yn ),where S is a computable relation, and Q is “ ∀” ifn is odd and Q is “ ∃”if n is even.A relation is arithmetical if it is Σ 0 n for some n.For instance, a Π 0 2 class is of the form {X : ∀y 1 ∃y 2 S(y 1 ,X↾ y2 )}, andaΣ 0 3 classis of the form {X : ∃y 1 ∀y 2 ∃y 3 R(y 1 ,y 2 ,X↾ y3 )}, where S and R are computablerelations. We have already introduced Π 0 1 and Σ 0 1 classes in 1.8.19. Let us verifythat these two ways of introducing Π 0 1 and Σ 0 1 classes are equivalent.

1.8.56 Fact.1.8 Cantor space 65(i) P ⊆ 2 N is Π 0 1 in the sense of 1.8.55 ⇔ P is a co-c.e. closed set, that is, Pis Π 0 1 in the sense of 1.8.19.(ii) U ⊆ 2 N is Σ 0 1 in the sense of 1.8.55 ⇔ U is a c.e. open set, that is, U isΣ 0 1 in the sense of 1.8.19.Proof. (i) ⇒: Suppose that P = {X : ∀yS(X ↾ y )} for a computable set Sand let B = {z ∈ {0, 1} ∗ : ∀z ′ ≼ zS(z ′ )}. Then B is a computable tree andP = Paths(B), so by Fact 1.8.20 P is Π 0 1 in the sense of 1.8.19.⇐: By 1.8.21, P = Paths(B) for a computable tree B. ThereforeX ∈ P ↔∀nX↾ n ∈ B, hence P is Π 0 1 in the sense of 1.8.55.The statement (ii) is now immediate by taking complements.✷Most of the classes of sets we consider will be arithmetical (the only exceptionsare the classes of Chapter 9). Often arithmetical classes are introduced by anappropriate property of sets or functions computed by a set X. This makes themarithmetical by the use principle. Sometimes they are defined by an arithmeticalgrowth condition on the initial segment complexity of their members.We verify that some classes or relations we have introduced are arithmetical:1.8.57 Proposition.(i) The relation {〈e, X〉: Φ X e is total} is Π 0 2.(ii) The class {A: A is computably dominated} is Π 0 4.Proof. (i) For each e, X we have Φ X e is total ↔ ∀x∃s Φ X↾se,s(x)↓ . The relation{〈x, e, σ〉: Φ σ e,|σ|(x)↓} is computable.(ii) This class is given by the description in Π 0 4 form∀e∃i∀y, s, w∃t[(∀y ′ ≤ y Φ A↾se,s (y ′ )↓ & Φ A↾se,s (y) =w) → w ≤ Φ i,t (y)].For, if Φ A e is total then there is i such that Φ A e (y) ≤ Φ i (y) for each y. Ontheother hand, if z is least such that Φ A e (z)↑, there are i, t such Φ A e (y) ≤ Φ i,t (y) foreach y1 and consider a Σ 0 n classA = {X : ∃y 1 ∀y 2 ...Qy n R(y 1 ,...,y n−1 ,X↾ yn )}.Then A = ⋃ y 1B y1 where B y1 = {X : ∀y 2 ...Qy n R(y 1 ,...,y n−1 ,X ↾ yn )} is aΠ 0 n−1 class uniformly in y 1 . Thus the Σ 0 n classes are the unions of effective sequencesof Π 0 n−1 classes. Similarly, the Π 0 n classes are the intersections of effectivesequences of Σ 0 n−1 classes.The notions of Π 0 n and of Σ 0 n relations can be relativized.1.8.59 Definition. For C ⊆ N and n ∈ N, we define Σ 0 n(C) relations and Π 0 n(C)relations A ⊆ N k × 2 N as in Definition 1.8.55, but for R, S ≤ T C.In the exercises we show that each Π 0 n(∅ ′ ) class is Π 0 n+1 but not conversely.Remark 1.8.58 allows us to simplify the presentation of Π 0 2 classes.

66 1 The complexity of sets1.8.60 Proposition. Let A ⊆ 2 N . Then A is Π 0 2 ⇔ there is a computable S ⊆ {0, 1} ∗such that A = {X : ∀y 1 ∃y 2 >y 1 S(X ↾ y2 )} = {X : ∃ ∞ nS(X ↾ n)}.Proof. We only have to prove the implication “⇒”. Let A = ⋂ n Gn where (Gn) n∈Nis an effective sequence of Σ 0 1 classes. By Remark 1.8.23, given t we can compute anapproximation G n,t of G n consisting of strings of length up to t. Letm(x) = max{n: x ∈ G n,|x| },and let S = {ya: y ∈ {0, 1} ∗ & a ∈ {0, 1} & m(ya) >m(y)}. Then S is computable,and for each X we have ∃ ∞ nX↾ n∈ S ↔∃ ∞ nX ∈ G n ↔ Z ∈ A. ✷The Borel classes are the subclasses of 2 N that can be obtained from the basicopen cylinders [x] via applications of the operations of complementation and countableunions. Each arithmetical class is Borel by Remark 1.8.58. The arithmetical hierarchyof classes is an effective version of the finite levels of a hierarchy of Borel classes.The description of classes in arithmetic also provides new ways of describing sets.1.8.61 Definition. Let n>1. A set Z is a Π 0 n-singleton if {Z} is a Π 0 n class.The Π 0 1-singletons coincide with the computable sets by Fact 1.8.33. The Π 0 2 singletonsalready takes us beyond the arithmetical sets. Let ∅ (ω) = ⋃ n (∅(n) ×{n}) be the effectiveunion of all the jumps ∅ (n) .1.8.62 Proposition. ∅ (ω) is a Π 0 2 singleton that is not an arithmetical set.Proof. There is a computable g such that (X [n] ) ′ = W X g(n) for each X, n, soX = ∅ (ω) ⇔ X [n] = ∅ & ∀nX [n+1] =(X [n] ) ′Thus the class {∅ (ω) } is Π 0 2.⇔∀n ∀k, s ∃t ≥ s[〈k, n +1〉∈X ↔ k ∈ W X↾ tg(n),t ].Recall the discussion of the local versus the global view of sets in Section 1.3. IfY ∈ Σ 0 n, the description showing this embodies the local view. This is even moreapparent at the lower levels Σ 0 1 and Σ 0 2 of the arithmetical hierarchy, where we stillhave a reasonable effective approximation of Y . On the contrary, a description of aset Z as a Π 0 n singleton, such as the description of ∅ (ω) above, embodies the globalview. The description only works because it involves the set as a whole.In Proposition 3.6.2 we will see that each ∆ 0 2 set is a Π 0 2 singleton. However, there isa ∆ 0 3 (even left-Σ 0 2) set that is not a Π 0 2 singleton, for instance the 2-random set Ω ∅′(page 136). Thus, there are two incomparable classes of descriptive complexity, thearithmetical sets and the Π 0 2 singletons. This is an exception to the rule that the classesof descriptive complexity we consider form a nearly linear hierarchy.Exercises. Show the following.1.8.63. The class of c.e. sets is Σ 0 3. The class of computable sets is Σ 0 3. (Also see 3.2.3.)1.8.64. The class of c.e. [computable] sets is not Π 0 2.1.8.65. Let A be a Π 0 2 class and let Ψ be a Turing functional. Then the classC = {Z : Ψ Z is total & Ψ Z ∈ A} is Π 0 2.1.8.66. For each Σ 0 3 class B the Turing upward closure G = {Z : ∃A ∈ B [A ≤ T Z]}is Σ 0 3 as well.1.8.67. (i) Each Π 0 1(∅ ′ ) class is Π 0 2. (ii) Some Π 0 2 singleton class is not Π 0 1(∅ ′ ).1.8.68. Extend 1.8.67(i): each Π 0 n(∅ ′ ) class is a Π 0 n+1 class.✷

Comparing Cantor space with Baire space1.8 Cantor space 67Let N ∗ be the set of finite sequences of numbers. König’s Lemma 1.8.2 relies onthe fact that the given tree is finitely branching. It fails in N ∗ : for instance, theinfinite tree {n0 n : n ∈ N} has no infinite path.For σ ∈ N ∗ let [σ] ={f ∈ N N : σ ≺ f}. The sets [σ] form a base of a topologyon N N , and N N equipped with this topology is called the Baire space. This space isnot compact because N N = ⋃ n[n]. Nonetheless, many notions we have discussedfor Cantor space can also be studied in the setting of Baire space. This includesnot only the notions from topology, such as open, closed, and compact setsand their representations (page 47), but also arithmetical definability of classes(1.8.55). As an example we consider closed sets. The class C ⊆ N N is closed iff C =Paths(B) for a subtree B of N ∗ , and C ⊆ N N is a Π 0 1 class of functions iff there isa computable set R ⊆ N ∗ such that C = {f : ∀n [f ↾ n ∈ R]} iff C = Paths(B) for acomputable subtree of N ∗ . The problem whether Paths(B) =∅ for a computabletree B ⊆ N ∗ is very complex, namely Π 1 1-complete (see Chapter 9, page 368). Fora computable binary tree the corresponding problem is merely Σ 0 1. In particular,the Low Basis Theorem 1.8.37 fails in Baire space.A function f ∈ N N can be encoded by a set X ∈ 2 N , its graph Γ f = {〈n, f(n)〉: n ∈ N}.For functions totality is automatic. For sets X, we have to require that X codes a function(Π 0 1) that is total (Π 0 2). In the following we will see that descriptions of functions asΠ 0 1 singletons have the same expressive power as descriptions of sets as Π 0 2 singletons.We first introduce some notation. Let D : N ∗ → N be the computable injection givenby D(n 0,...,n k−1 )= ∏ k−1i=0 pn i+1i , where p i is the i-th prime number. Let Seq denotethe range of D, the computable set of sequence numbers. Ifα = ∏ k−1i=0 pn i+1i as abovethen we write α(i) =n i.1.8.69 Proposition. Suppose that f ∈ N N and {f} is Π 0 1. Then {Γ f } is Π 0 2.Proof. Let T (X) be the Π 0 2 condition expressing that X is the graph of a function.Suppose f is the unique function satisfying the condition ∀nR(f ↾ n) where R is computable.Then Γ f is the only set X satisfying the Π 0 2 conditionT (X) &∀n ∃α ∈ Seq [|D −1 (α)| = n & R(α) &∀y

68 1 The complexity of setsExercises.1.8.71. A tree T ⊆ N ∗ is finitely branching if T ⊆ {σ ∈ N ∗ : ∀i

1.9 Measure and probability 69The property (1.19) ensures that this sum does not exceed r(∅), and that anychoice of a prefix-free set E yields the same value µ r (A).2. For a class C ⊆ 2 N we letµ r (C) = inf{µ r (A): C ⊆ A & A is open}. (1.20)It is not hard to verify that µ r is indeed an outer measure (Exercise 1.9.5).In the following we provide a useful fact about outer measures. It is a specialcase of the Lebesgue Density Theorem. First a definition.1.9.3 Definition. Suppose µ is an outer measure. If C ⊆ 2 N and σ ∈ {0, 1} ∗ issuch that µ[σ] ≠ 0, then the local outer measure of C in [σ] isµ(C |σ) =µ(C ∩ [σ])/µ[σ].The theorem states that if µC > 0 for an outer measure µ, then the local outermeasure µ(C |σ) can be arbitrarily close to 1.1.9.4 Theorem. Let µ be an outer measure, and let C ⊆ 2 ω be such that µC > 0.Then for any δ, 0 < δ < 1, there exists σ ∈ {0, 1} ∗ such that µ(C |σ) ≥ δ.Proof. Let ɛ =( 1 δ−1)µC. By the definition of an outer measure, there is an openset A ⊇ C such that µA−µC ≤ ɛ. Then µA ≤ µC +ɛ = 1 δµC, that is, δ ·µA ≤ µC.There is a prefix-free set D ⊆ {0, 1} ∗ such that A =[D] ≺ = ⋃ σ∈D[σ]. We claimthat some σ ∈ D satisfies the conclusion of the theorem. Otherwise, for eachσ ∈ D, µ[σ] ≠ 0 implies µ(C |σ) < δ. Since µC > 0, we have µ[σ] > 0 for someσ ∈ D. ThenµC = µ( ⋃ σ∈DC ∩ [σ]) since C ⊆ A≤ ∑ µ(C ∩ [σ]) by countable sub-additivity< δ ·∑σ∈Dµ[σ] since µ(C |σ) < δ for each σ ∈ Dsuch that µ[σ] > 0, and there is such a σ= δ · µA ≤ µC,contradiction.✷1.9.5 Exercise. Show that µ r is an outer measure for each measure representation r.MeasuresFor details on the following see for instance Doob (1994).1.9.6 Definition.(i) A set B ⊆ P(2 N ) is called a σ-algebra if B is nonempty, closed undercomplements, and the operations of countable union and intersection.(ii) A function µ: B → R + 0 is called a measure if µ(∅) =0andµ is countablyadditive, namely, µ( ⋃ i D i)= ∑ i µ(D i) for each countable family (D i ) i∈Nof pairwise disjoint sets. If µ(2 N ) = 1 then µ is called a probability measure.

70 1 The complexity of setsClearly countable additivity implies monotonicity. It also implies countablesubadditivity, since for each familiy (C i ) i∈N , ⋃ i C i is the disjoint union of thesets D i = C i − ⋃ k

1.9 Measure and probability 711.9.11 Proposition. Suppose a class V ⊆ 2 N contains exactly one member ofeach equivalence class of = ∗ . Then V is not measurable.Proof. Assume V is measurable. Then 2 N is the disjoint union of the measurable classesV F where F is finite. By countable additivity, 1 = λ(2 N )= ∑ FλVF [F ⊆ N is finite].This is impossible since λV F = λV for each F .✷As a consequence, there is no Borel well-order of Cantor space, for otherwise onecould take as V the Borel class of minimal elements in each equivalence class. Notethat V is a version for Cantor space of the Vitali set, where instead of 2 N and = ∗ onehas [0, 1] R with the equivalence relation {〈r, s〉: r − s ∈ Q}.For a class C and a string σ, let C | σ = {X : σX ∈ C}. The conditional outer measureλ(C |σ) of Definition 1.9.3 equals the uniform outer measure of the class C | σ.With few exceptions, classes C relevant to computability theory are Borel andclosed under finite variants, that is, X ∈ C and X = ∗ Y implies Y ∈ C. Bythe following, the uniform measure can only distinguish between small and largeclasses of this kind.1.9.12 Proposition. (Zero-one law) If a measurable class C is closed underfinite variants then λC =0or λC =1.Proof. Suppose that λC > 0. We show that λC > δ for every δ, 0≤ δ < 1. If σand ρ are strings of the same length, then by the invariance property of λ above,for each X we have σX ∈ C ↔ ρX ∈ C. Therefore,λ(C ∩ [σ]) = λ({σX : σX ∈ C})=λ({ρX : ρX ∈ C})=λ(C ∩ [ρ]).By Theorem 1.9.4 choose σ such that λ(C | σ) > δ and let n = |σ|. Thenλ(C ∩ [σ]) > δ2 −n , hence λC = ∑ |ρ|=n λ(C ∩ [ρ]) > 2n δ2 −n = δ.✷Exercises.1.9.13. If S is a prefix-free set then lim n#(S ∩ {0, 1} n )/2 n =0.1.9.14. An ultrafilter U ⊆ P(N) is called free if {n} ∉ U for each n. Show that a freeultrafilter is not measurable when viewed as a subset of 2 N .1.9.15. ⋄ Suppose that N ∈ N, ɛ > 0, and for 1 ≤ i ≤ N, the class C i is measurableand λC i ≥ ɛ. IfNɛ >kthen there is a set F ⊆ {1,...,N} such that #F = k + 1 andλ ⋂ i∈FCi > 0. For instance, if N = 5 sets of measure at least ɛ =1/2 are given, thenthree of them have an intersection that is not a null class. (This is applied in 8.5.18.)Hint. Think of integrating a suitable function g : 2 N → {0,...,N}.Uniform measure of arithmetical classesBy 1.8.58 each arithmetical class is Borel and hence measurable. The uniformmeasure of a Σ 0 n class is left-Σ 0 n. The uniform measure of a Π 0 n class is left-Π 0 n.We prove this for n = 1 and leave the general case as Exercise 1.9.22.1.9.16 Fact. If R ⊆ 2 N is c.e. open then λR is left-c.e. in a uniform way.If P ⊆ 2 N is a Π 0 1 class then λP is right-c.e. in a uniform way.Proof. According to 1.8.23, from an index for R we obtain the effective approximation(R s ) s∈N such that R = ⋃ s [R s] ≺ . Then λR = sup s λR s ,soλR is left-c.e.The second statement follows by taking complements.✷

72 1 The complexity of setsWe discuss c.e. open sets R such that λR is computable. Recall from 1.8.19 thatA R = {σ :[σ] ⊆ R}. First we note that none of the properties “λR computable”and “A R computable” implies the other.1.9.17 Example. There is a c.e. open set R such that λR = 1 and A R is notcomputable.Proof. Let R =2 N −P where P be the class of two-valued d.n.c. functions fromFact 1.8.31. Then A R = {0, 1} ∗ −T P is not computable. Since there are infinitelymany x such that J(x) =0,wehaveλP =0.✷On the other hand, the uniform measure of a c.e. open set R with computableA R can be an arbitrary left-c.e. real by Exercise 1.9.20.We provide two results needed later on when we study Schnorr randomness.1.9.18 Fact. Let R be a c.e. open set such that λR is computable. Then λ(R∩C)is computable uniformly in R, λR, and an index for the clopen set C.The same holds for Π 0 1 classes of computable measure.Proof. We use Fact 1.8.15(iv). Given n ∈ N, we can compute t ∈ N such thatλR − λ[R t ] ≺ ≤ 2 −n . Then λ(R ∩ C) − λ([R t ] ≺ ∩ C) ≤ 2 −n . Thus the rationalq n = λ([R t ] ≺ ∩ C) is within 2 −n of λ(R ∩ C).For Π 0 1 classes, one relies on the first statement and takes complements. ✷1.9.19 Lemma. Let S be a c.e. open set such that λS is computable. From arational q such that 1 ≥ q ≥ λS we may in an effective way obtain a c.e. openset ˜S such that S ⊆ ˜S and λ ˜S = q.Proof. We identify sets and real numbers according to Definition 1.8.10. Let˜S = S ∪ ⋃ {[0,x): x ∈ Q 2 & λ(S ∪ [0,x)) r.Thusf is continuous. Further, f(0) ≤ q and f(r) ≥ r for each r, so thereis a least t such that f(t) =q. Then ˜S = S ∪ [0,t), and hence λ ˜S = q.To see that ˜S is c.e., note that f(x) =x + λ(S ∩ [x, 1)) for each x ∈ Q 2 ,soby 1.9.18 f(x) is a computable real uniformly in x. Since x

Probability theory1.9 Measure and probability 73We briefly discuss binary strings and sets of natural numbers from the viewpointof probability theory. Unexplained terms in italics represent standard terminologyfound in text books such as Shiryayev (1984).A sample space is given by a set X together with a σ-algebra and a probabilitymeasure on it. For instance, a string of length n is an element of the samplespace {0, 1} n , and a set is an element of the sample space 2 N . The probabilitymeasure P is given by P ({x}) =2 −|x| in the case of strings and by the uniformmeasure λ in the case of sets.For each appropriate i we have a random variable ξ i : X → {0, 1} given byξ i (x) =x(i). We think of ξ i (x)asthei-th outcome in the sequence of experimentsdescribed by x. If X is Cantor space, this is a dynamic version of the local viewof sets (see Section 1.3): the set is revealed bit by bit. Note thatP (ξ i =0)=P (ξ i =1)=1/2.Moreover, the ξ i are independent. Such a sequence of random variables is calleda Bernoulli scheme for the probability 1/2.For n ∈ N let S n = ∑ i 0, namely that the number of ones differs by at least ɛn from the expectedvalue n/2. The Chebycheff inequality shows that this probability is atmost 1/(4nɛ 2 ). (For a numerical example, if n = 1000 and ɛ =1/10, then 1/40bounds the probability that the number of ones is at least 600 or at most 400.)In Section 2.5 we will use the improved estimate 2e −2nɛ2 for the probability ofthe event (1.21) in order to bound the number of strings of length n where thenumber of zeros and of ones is unbalanced.

2The descriptive complexity of stringsIn contrast to the remainder of the book, this chapter is on finite objects. We mayrestrict ourselves to (binary) strings, because other types of finite objects, suchas strings over alphabets other than {0, 1}, natural numbers, or finite graphs,can be encoded by binary strings in an effective way. We are interested in givinga description σ of a string x, and if possible one that is shorter than x itself. Suchdescriptions are binary strings as well. To specify how σ describes a string x, wewill introduce an optimal machine, which outputs the string x when the inputis the description σ. The descriptive complexity C(x) is the length of a shortestdescription of x. We first study the function x → C(x). Some of its drawbackscan be addressed by introducing a variant K, where the set of descriptions isprefix-free. The function K also has its drawbacks as a measure of descriptivestring complexity, but is the more fruitful one where the interaction of computabilityand randomness is concerned. The Machine Existence Theorem 2.2.17is an important tool for showing that the elements of a collection of strings haveshort descriptions in the sense of K. It will be used frequently in later chapters.A string x is b-incompressible (in the sense of C, orK) if it has no description σ(in that sense) such that |σ| ≤ |x| − b. In Section 2.5 we will see that incompressibilitycan serve as a mathematical counterpart for the informal concept ofrandomness for strings. Using some basic tools from probability theory, we showthat an incompressible string x has properties one would intuitively expect froma random string. For instance, x has only short runs of zeros.Optimal machines act as description systems for strings (analogous to the descriptionsystems for sets mentioned at the beginning of Chapter 1). However,the theory of describing strings differs from the theory of describing sets. Onlya few reasonable description systems have been introduced. Every string can bedescribed in each system. For a set, the question is whether it can be describedat all, while for strings we are interested in the length of a shortest description.For strings, we can formalize randomness by being hard to describe (that is,being incompressible). It takes more effort to formalize randomness for sets (seethe introduction to Chapter 3).The prefix-free descriptive complexity K can be used to determine the degreeof randomness of a set Z ⊆ N, because to a certain extent it is measured bythe growth rate of the function n → K(Z ↾ n ). For instance, a central notion,Martin-Löf randomness, is equivalent to the condition that for some b, eachinitial segment Z ↾ n is b-incompressible in the sense of K (Theorem 3.2.9).

2.1 The plain descriptive complexity C 75Much of the material in this chapter goes back to work of Solomonoff (1964),Kolmogorov (1965), Levin and Zvonkin (1970), Levin (1973, 1976), and Chaitin(1975). A standard textbook is Li and Vitányi (1997).Comparing the growth rate of functionsWe frequently want to measure the growth of a function g : N → R. One way isto compare g to the particularly well-behaved functions of the following type.2.0.1 Definition. An order function is a computable nondecreasing unboundedfunction f : N → N. Examples of order functions are λn. log n, λn.n 2 , and λn.2 n .In the following let f,g: N → R. We will review three ways of saying that fgrows at least as fast as g: by domination, up to an additive constant, and up toa multiplicative constant (if f and g only have nonnegative values).(1) Recall the domination preordering on functions from Definition 1.5.1: f dominatesg if ∀ ∞ n [f(n) ≥ g(n)]. An example of a growth condition on a function gsaying that g grows very slowly is to require that g is dominated by each orderfunction. We will study an unbounded function g of this type in 2.1.22.(2) To compare f and g up to additive constants, letg ≤ + f : ↔∃c ∈ N ∀n [g(n) ≤ f(n)+c].This preordering is a bit weaker than the domination preordering. We usuallyavoid explicitly mentioning the functions f,g. Instead, we write expressions definingthem, as for instance in the statement log n +16≤ + n. The preordering ≤ +gives rise to the equivalence relation g = + f : ↔ g ≤ + f ≤ + g.We often try to characterize the growth rate of a function f by determining itsequivalence class with respect to = + . For instance, the class of bounded functionscoincides with the equivalence class of the function that is constant 0.(3) Recall that abs(r) denotes the absolute value of a number r ∈ R. Letg = O(f) :↔∃c ≥ 0 ∀ ∞ n [abs(g(n)) ≤ c abs(f(n))].Moreover, g = h + O(f) means that g − h = O(f). We think of O(f) as an errorterm, namely, an unspecified function ˜f such that abs( ˜f(n)) ≤ c abs(f(n)) foralmost all n. In particular, O(1) is an unspecified function with absolute valuebounded by a constant. Thus g ≤ + f ↔ g ≤ f + O(1).For f,g: N → R + we let f ∼ g : ↔ lim n f(n)/g(n) =1.Iff ∼ g, one canreplace f by g in error terms of the form O(f).2.1 The plain descriptive complexity CMachines and descriptionsRecall that elements of {0, 1} ∗ are called strings. A partial computable functionmapping strings to strings is called a machine. (As we identify a string σwith the natural number n such that the binary representation of n + 1 is 1σ,formally speaking a machine is the same a partial computable function. It isuseful, though, to have this term for the particular context of strings.) If M is

76 2 The descriptive complexity of stringsa machine and M(σ) =x then we say σ is an M-description of x. IfM is theidentity function then x is an M-description of itself. (We will call this machinethe copying machine.) In general, an M-description of x may be shorter than x,in which case one can view σ as a compressed form of x. The machine M carriesout the decompression necessary to re-obtain x from σ.Our main interest is in measuring how well a string x can be compressed.That is, we are more interested in the length of a shortest description thanin the description itself. We introduce a special notation for this length of ashortest M-description:C M (x) = min{|σ|: M(σ) =x}. (2.1)Here min ∅ = ∞. For an example of a drastic compression, let M be the machine thattakes an input σ, views it as a natural number and outputsx =2 22σ .Then C M (x) =|σ| = + log (4) x. For instance, the string σ = 10 corresponds to thenumber 5, so M(σ) =2 232 =2 4294967296 = x. The string σ is an M-description of x,and C M (x) = 2, while x has length 2 32 .2.1.1 Definition. The machine R is called optimal if for each machine M, thereis a constant e M such that∀σ,x[M(σ) =x →∃τ (R(τ) =x & |τ| ≤ |σ| + e M )], (2.2)or, equivalently, ∀x [C R (x) ≤ C M (x)+e M ].Thus, for each M, the length of a shortest R-description of a string exceeds thelength of a shortest M-description only by a constant e M . Optimal machines areoften called universal machines.The constant e M is called the coding constant for M (with respect to R). Itbounds the amount the length of a description increases when passing from Mto R. Although we are usually only interested in the length of an R-description,we can in fact obtain an appropriate R-description τ from the M-description σ inan effective way, by trying out in parallel all possible R-descriptions τ of lengthat most |σ| + e M till one is found such that R(τ) =M(σ).An optimal machine exists since there is an effective listing of all the machines.The particular choice of an optimal machine is usually irrelevant; most of the inequalitiesin the theory of string complexity only hold up to an additive constant,and if R and S are optimal machines then ∀xC R (x) = + C S (x). Nonetheless, wewill specify a particularly well-behaved optimal machine. Recall the effective listing(Φ e ) e∈N of partial computable functions from (1.1) on page 3. We will fromnow on assume that Φ 1 is the identity.2.1.2 Definition. The standard optimal plain machine V is given by lettingfor each e>0 and ρ ∈ {0, 1} ∗ .V(0 e−1 1ρ) ≃ Φ e (ρ)

2.1 The plain descriptive complexity C 77This definition makes sense because each string σ can be written uniquely inthe form 0 e−1 1ρ. The machine V is optimal because for each machine M thereis an e>0 such that M = Φ e ,soM(σ) =x implies V(0 e−1 1σ) =x. The codingconstant with respect to V is simply an index e>0 for M. Regarding (2.2),note that a V-description τ of x is obtained in a particularly direct way from anindex e>0 for M and an M-description σ of x: by putting 0 e−1 1 in front of σ.From now on, we simply write C(x) for C V (x). For each optimal machine R wehave C R (x) ≤ + |x| since R emulates the copying machine. In particular, sincethis machine is Φ 1 , for each xC(x) ≤ |x| +1. (2.3)This upper bound cannot be improved in general by 2.1.19(ii).In Exercises 2.1.6 and 2.1.24 we study necessary conditions for a set to be the domainof an optimal machine. These conditions only depend on the number of strings in theset at each length.Exercises. Show the following.2.1.3. A shortest V-description cannot be compressed by more than a constant: thereis b ∈ N such that, if σ is a shortest V-description of a string x, then C(σ) ≥ |σ| − b.2.1.4. There is an optimal machine R such that for each x, m, the string x has at mostone R-description of length m.2.1.5. Let d ≥ 1. There is an optimal machine R such that d divides |ρ| for eachρ ∈ dom R.2.1.6. Let D = dom R for an optimal machine R. Then there is b ∈ N such that foreach n we have 2 n ≤ s n,b < 2 n+b , where s n,b =#{σ ∈ D : n ≤ |σ|

78 2 The descriptive complexity of stringsThe counting condition for a function D says that not too many strings x yieldsmall values D(x).2.1.9 Definition. A function D : {0, 1} ∗ → N ∪ {∞} satisfies the counting conditionif #{x: D(x)

2.1 The plain descriptive complexity C 79Proof. We may assume that at most one request is enumerated into W at eachstage. We define the machine M by giving a computable enumeration (M s ) s∈Nof its graph (see Exercise 1.1.18).Let M 0 = ∅. Fors>0, if a request 〈n, x〉 is in W s − W s−1 then put 〈σ,x〉into M s , i.e., let M s (σ) =x, for the leftmost string σ of length n that is notin dom(M s−1 ). By our hypothesis such a string can be found. Clearly M is amachine as required.✷A characterization of C2.1.15 Proposition. Suppose that D : {0, 1} ∗ → N ∪ {∞} is computably approximablefrom above and satisfies the counting condition. Then there is a machineM such that ∀xC M (x) =D(x)+1 (where ∞ +1=∞).Proof. Suppose the function λx, s.D s (x) is a computable approximation of Dfrom above. The c.e. set of requests W is given as follows: whenever s>0 andn = D s (x)

80 2 The descriptive complexity of stringsInvarianceA machine N on input x cannot increase C(x) by more than a constant:2.1.20 Fact. For each machine N and each x such that N(x)↓ we haveC(N(x)) ≤ + C(x). IfN is a one-one function then C(N(x)) = + C(x).Proof. Let M be a machine such that M(σ) ≃ N(V(σ)) for each σ. ThenC M (N(x)) ≤ C(x), so that C(N(x)) ≤ C(x) +e M where e M is the codingconstant for M. IfN is one-one then the same argument applies to its inverseN −1 ,soC(x) ≤ + C(N(x)) as well.✷In particular, if π is a computable permutation of N, then C(x) = + C(π(x)) foreach x. Also, C(|x|) ≤ + C(x).Continuity properties of CRecall that abs(z) denotes the absolute value of z ∈ Z.2.1.21 Proposition.(i) abs(C(x) − C(y)) ≤ + 2 log abs(x − y) for each pair of strings x, y.(ii) If the string y is obtained from the string x by changing the bit in oneposition, then abs(C(x) − C(y)) ≤ + 2 log |x|.Proof. (i) follows from the stronger Proposition 2.4.4 below, so we postpone theproof till then. We leave (ii) as an exercise.✷The growth of CThe function C is unbounded but slowly growing in the sense that large argumentscan have small values. To make this more precise, we consider the fastestgrowing non-decreasing function that bounds C from below: letC(x) = min{C(y): y ≥ x}.Fact 2.1.10 states that C satisfies the counting condition, that is, for each kthere are fewer than 2 k strings x such that C(x) r n foralmost all n. Since h is non-decreasing, this implies that h dominates C. ✷

2.1 The plain descriptive complexity C 81Exercises.2.1.23. (Stephan) Let z m be the largest number such that C(z m)

82 2 The descriptive complexity of stringsTo show that B is weak truth-table complete we define a c.e. set of requests W .If n ∈∅ ′ s −∅ ′ s−1 then we put the request 〈n, x〉 into W , where x is the leftmoststring of length 2n such that C s (x) ≥ 2n. Note that x exists by Fact 2.1.13.Let M be the machine obtained from W via Proposition 2.1.14, and let d bethe coding constant for M. We provide a weak truth-table reduction of a finitevariant of ∅ ′ to B:on input n>d, using B as an oracle, compute a stage s such thatB s (x) =B(x) for all strings x of length 2n and output ∅ ′ s(n).If n ∈∅ ′ t −∅ ′ t−1 for some t>s, then for some string x of length 2n such thatC s (x) ≥ 2n this causes C M (x) ≤ n and hence C(x) ≤ n + dd.The use of this reduction procedure is computably bounded.✷2.1.29 Corollary. The function C is not computable. ✷Kummer (1996) proved that the set {x: C(x) < |x|} is in fact truth-table complete.Exercises.2.1.30. Show that the set A = {〈x, n〉: C(x) ≤ n} is c.e. and wtt-complete.2.1.31. Prove that ∃ ∞ x [C(x) < C g (x)] for each computable function g such that∀ng(n) ≥ n. Show that this cannot be improved to ∀ ∞ x [C(x)

Drawbacks of C2.2 The prefix-free complexity K 83Complexity dipsRecall from 2.1.12 that a string w is d-incompressible C if C(w) > |w| − d. Weproved in Fact 2.1.13 that the number of d-incompressible C strings of length nis at least 2 n −2 n−d+1 . One may ask whether there is a string w of length n suchthat each prefix of w is d-incompressible C when n is large compared to d. Theanswer is negative, because a machine N can “cheat” in the decompression, byencoding some extra information into the length of a description.2.2.1 Proposition. There is a constant c with the following property. For eachd ∈ N and each string w of length at least 2 d+1 + d there is an x ≼ w such thatC(x) ≤ |x| − d + c.Proof. We use the notation introduced in (1.5) on page 13. The machine Nis given by N(σ) =string(|σ|)σ. It is sufficient to obtain a prefix x of w withan N-description of length |x| − d. Let k =number(w ↾ d ) (so that k

84 2 The descriptive complexity of stringsthe machine. Since M is prefix-free the following sum converges and expressesthe probability that M halts in this experiment:Ω M = ∑ σ2 −|σ| [[M(σ)↓]]. (2.4)Thus Ω M = λ[dom M] ≺ is the measure of the open set generated by the domainof M. Since this open set is c.e., Ω M is a left-c.e. real number (see 1.9.16). LetΩ M,s = λ[dom(M s )] ≺ , then (Ω M,s ) s∈N is a nondecreasing computable approximationof Ω M .2.2.5 Example. Suppose M is a machine which halts on input σ iff σ is of theform 0 i 1 for even i. Then Ω M =0.101010 ...=2/3.Optimal prefix-free machines2.2.6 Definition. We say that a machine R is an optimal prefix-free machineif R is prefix-free, and for each prefix-free machine M there is a constant d Msuch that ∀xK R (x) ≤ K M (x) +d M . As before, the constant d M is called thecoding constant of M (with respect to R).2.2.7 Proposition. An optimal prefix-free machine R exists.Proof. We first provide an effective listing (M d ) d∈N of all the prefix-free machines.M d is a modification of the partial computable function Φ d in (1.1) onpage 3. When the computation Φ d (σ) =x converges at stage s, then declareM d,s (σ) =x unless M d,t (τ) has already been defined at a stage t0. By the PaddingLemma 1.1.3 we can leave out M 0 ,soR is optimal.✷For the moment we will fix an arbitrary optimal prefix-free machine R andwrite K(x) for C R (x) = min{|σ|: R(σ) =x}. Like any other machine, R isemulated by V. ThusC(x) ≤ + K(x).The string complexity K amends the drawbacks of C discussed on page 83.Firstly, K(〈x, y〉) ≤ + K(x)+K(y), because given an R-description σ of x andan R-description τ of y, a new prefix-free machine can decompress the descriptionρ = στ. It recovers σ and τ, then it runs R on both to compute x and y, andfinally it outputs 〈x, y〉. (In Theorem 2.3.6 we give an explicit expression forK(〈x, y〉) using a conditional version of K.)Secondly, by Proposition 2.5.4 there are arbitrarily long strings x such thateach prefix of x is d-incompressible in the sense of K, for some fixed d.Upper bounds for KBy (2.3) we have C(x) ≤ |x|+1. A drawback of K is the absence of a computableupper bound, in terms of the length, that is tight for almost all lengths. For C,the upper bound was based on the copying machine Φ 1 , which is not prefixfree.In fact this upper bound fails for K (see Remark 2.5.3 below). To obtain

2.2 The prefix-free complexity K 85a crude computable upper bound (to be improved subsequently) consider themachine M given by M(0 |x| 1x) =x for each string x. This machine is prefix-freesince 0 |x| 1x ≼ 0 |y| 1y first implies |x| = |y|, and then x = y. ThusK(x) ≤ + 2|x|. (2.5)The idea in the foregoing argument was to precede x by an encoding of its lengthn = |x| in such a way that the strings encoding lengths form a prefix-free set.Instead of 0 n 1, we can take any other such encoding. So, why not take a shortestpossible prefix-free encoding of n, using R itself for the decompression? Thenwe get as close to the copying machine as we possibly can within the prefix-freesetting. This yields the following improved upper bounds.2.2.8 Proposition. K(x) ≤ + K(|x|)+|x| ≤ + 2 log |x| + |x| for each string x.Proof. The second inequality follows by applying (2.5) to |x|. For the firstinequality, define a prefix-free machine N as follows:on input τ search for a decomposition τ = σ x such thatn = R(σ)↓ and |x| = n. If one is found output x.Clearly N is prefix-free. Given x, let σ be a shortest R-description of |x|. ThenN(σx) =x. This shows that K(x) ≤ + K(|x|)+|x|.✷Incorporating N into the machine R obtained in the proof of Proposition 2.2.7,one may achieve that the constant in the inequality K(x) ≤ + K(|x|)+|x| is 1. Theidea is to emulate N with a loss in compression of only 1. Since N is already basedon the optimal machine to be defined, we have to use the Recursion Theorem.2.2.9 Theorem. (Extends 2.2.7) There is an optimal prefix-free machine Usuch that K(x) ≤ K(|x|)+|x| +1 for each x, where K(x) =K U (x).Proof. Let (M d ) d∈N be the effective listing of all the prefix-free machines fromthe proof of 2.2.7. Given e, we define a prefix-free machine N e by N e (σx) =x iffM e (σ) =|x|. By the Parameter Theorem 1.1.2 there is a computable function qsuch that{Φ q(e) (0 d−1 N e (ρ) if d =11ρ) ≃M d (ρ) if d>1.By the Recursion Theorem 1.1.5 (and 1.1.7) there is an i>1 such that Φ q(i) =Φ i . Since Φ q(e) is prefix-free for each e, Φ i is prefix-free and hence Φ i = M i .Bythe Padding Lemma 1.1.3 it does not matter to leave out M 0 and M 1 in theemulation, so U := Φ i is an optimal prefix-free machine.Given x, let σ be a shortest U-description of |x|, then U(1σx) =Φ q(i) (1σx) =N i (σx) =x. Therefore K(x) ≤ K(|x|)+|x| +1.✷From now on, we will simply write K(x) for K U (x) = min{|σ|: U(σ) =x}. Wealso let

86 2 The descriptive complexity of stringsK s (x) = min{|σ|: U s (σ) =x} (2.6)(where min ∅ = ∞). Note that C(x) ≤ + K(x) since U is emulated by V.Exercises.2.2.10. Redo the Exercises 2.1.3–2.1.5, 2.1.17 and 2.1.23 for prefix-free machines and K.2.2.11. Show that K(x) ≤ + 2 log log |x| + log |x| + |x|.2.2.12. (Calude, Nies, Staiger and Stephan, 20xx) Let U be an optimal prefix-freemachine, and let C = {S ⊆ {0, 1} ∗ : S is prefix-free & dom U ⊆ S}.Show that (i) C is a Π 0 1 class, and therefore contains a low set; (ii) no Π 0 1 set S is in C.The Machine Existence Theorem and a characterization of KWe proceed as in the characterization of C on page 79: firstly, we characterize,up to = + , the class of functions of the form K M for a prefix-free machine M, by(1) being computably approximable from above, and(2) satisfying the weight condition defined in 2.2.13.Secondly, K is the least function in this class with respect to ≤ + .2.2.13 Definition. A function D : N → N ∪ {∞} satisfies the weight conditionif ∑ x 2−D(x) ≤ 1. (Here 2 −∞ := 0.)The weight condition implies #{x: D(x) =i} ≤ 2 i for each i, which in turn impliesthe counting condition in 2.1.9. The weight condition holds for any functionof the type K M because ∑ x 2−K M (x) ≤ ∑ σ 2−|σ| [[M(σ)↓] = Ω M ≤ 1.To proceed with the characterization of the functions K M we need an analogof Proposition 2.1.14, which we will call the Machine Existence Theorem (it isalso refered to in the literature as the Kraft-Chaitin Theorem). Unlike 2.1.14,it is nontrivial. It will be an important tool in future constructions. First wediscuss Kraft’s Theorem. Then we introduce our tool which is an effectivization ofKraft’s Theorem. It appeared in Chaitin (1975). A similar result had already beenobtained by Levin (1973); see Exercise 2.2.23. We prefer the Machine ExistenceTheorem to the setting of Levin because its notation is more convenient in ourapplications.Let N ∈ N ∪ {∞} and suppose we want to encode strings x i , i < N,bystrings σ i in such a way that the set of strings σ i is prefix-free. In this case wecall the (finite or infinite) list 〈σ 0 ,x 0 〉, 〈σ 1 ,x 1 〉,...a prefix-free code. For instance,if we let σ i =0 |xi| 1x i we obtain a prefix-free code. (For practical applications,such a code is useful when each x i represents a letter in an alphabet. Then atext is a concatenation of the x i ’s, and the text is encoded by the correspondingconcatenation τ of the strings σ i .Nowτ can be decoded in a unique way byscanning from left to right, splitting off one string σ i at a time.)Recall from 2.1.14 that a request is a pair 〈r, x〉 from N × {0, 1} ∗ , meaning thatwe want a description of x with length r. The following is due to Kraft (1949).2.2.14 Proposition. Suppose (〈r i ,x i 〉) i

2.2 The prefix-free complexity K 87Proof. The necessity of the weight condition is clear since the basic cylinders [σ i ]are pairwise disjoint, and therefore ∑ i

88 2 The descriptive complexity of strings2.2.17 Theorem. (Machine Existence Theorem) For each bounded requestset W , one can effectively obtain a prefix-free machine M = M d , d>1, suchthat∀r, y [〈r, y〉 ∈WMoreover, Ω M equals the total weight of W .↔ ∃w (|w| = r & M(w) =y)].We say that M d is a prefix-free machine for W . Recall that d is the codingconstant for M d (with respect to the standard optimal prefix-free machine). Inorder to avoid explicit mention of M d , we will also refer to d as the coding constantfor W .Bounded request sets are useful because they are easier to handle than prefix-freemachines. The machine existence theorem provides the coding constant d for thecorresponding machine. So we know that, if we enumerate the request 〈r, y〉, thenK(y) ≤ r + d (usually this is all we care about).Proof of Theorem 2.2.17. Let 〈r n ,y n 〉 n

2.2 The prefix-free complexity K 89(3) To obtain R n , first remove z n from R n−1 .Ifw n ≠ z n then also add thestrings z n 0 i 1, 0 ≤ i ∑ z 2−|z| [[z ∈ R n−1 ] by (b) for n − 1. Then ∑ ni=0> 1 2−risince ∑ z 2−|z| [[z ∈ R n−1 ]] + ∑ n−1i=0= 1 by (c) for n − 1. This contradicts the2−riassumption that W is a bounded request set.Clearly (b) for n holds if w n = z n .Ifw n ≠ z n then |z n | < |w n | but also |w n | isless than the length of the shortest string in R n−1 that is longer that z n ,so(b)holds by the definition of R n . Finally, (c) is satisfied by the definition of R n .The machine M given by M(w n )=y n is prefix-free by (c). As M was determinedeffectively from W we may obtain an index d>1 for M. For each requestρ = 〈r, y〉 ∈W we put a new description w of length r into dom(M), so Ω Mequals the total weight of W .✷To give an example how to apply Theorem 2.2.17, we reprove the inequalityK(x) ≤ + |x| + 2 log |x| in 2.2.8. If c is chosen sufficiently large then theset∑W = {〈|x| + 2 log |x| + c, x〉: x ∈ {0, 1} ∗ } is a bounded request set, sinceρ[[ρ ∈ W ]] ≤ 2 −c ( ∑ 2−(ρ)0 n>0 2n 2 −n−2 log n ) ≤ 1. Let M be the prefix-freemachine for W . Then K M (x) ≤ |x|+2 log |x|+c and thus K(x) ≤ + |x|+2 log |x|.Theorem 2.2.17 can be used to characterize the functions of the type K M fora prefix-free machine M. This is analogous to Proposition 2.1.15.2.2.18 Proposition. Suppose that D : N → N ∪ {∞} is computably approximablefrom above and satisfies the weight condition ∑ x 2−D(x) ≤ 1. Then thereis a prefix-free machine M such that ∀xK M (x) =D(x)+1.Proof. Suppose that λx, s. D s (x) is a computable approximation of D fromabove. The c.e. set W is defined exactly as in the proof of 2.1.15: wheneverr = D s (x) u 2−m =2 −u since m>ufor such a request. All the requests with second component x together contributeless than 2 −D(x) . Since D satisfies the weight condition, W is a bounded requestset. Applying the Machine Existence Theorem 2.2.17 to W yields a machine Mas required.✷

90 2 The descriptive complexity of stringsWe now obtain a machine-independent characterization of K (and in fact ofany function K R where R is an optimal prefix-free machine as defined in 2.2.6).2.2.19 Theorem. K can be characterized as follows up to = + : it is the leastwith respect to ≤ + among the functions D that are computably approximablefrom above and satisfy the weight condition ∑ x 2−D(x) ≤ 1.✷The following application of 2.2.19 will be needed for Theorem 8.1.9. It can beseen as an effective version of Exercise 1.9.13.2.2.20 Proposition. Let B be a c.e. prefix-free set and let b m =#(B ∩{0, 1} m )for m ∈ N. Then ∀m [K(m) ≤ + m − log b m ].Proof. Clearly the function D given by D(m) = m − log b m is computablyapproximable from above. For each m we have λ[B ∩ {0, 1} m ] ≺ =2 −m b m . Since2 log bm ≤ b m ,wehave ∑ m 2−m+log bm ≤ λ[B] ≺ ≤ 1. Therefore D satisfies theweight condition.✷2.2.21 Remark. (Coding constants given in advance.) In building a boundedrequest set L, by the Recursion Theorem we often assume that a coding constantd for a machine M d is given, and we think of M d as a prefix machine for L.This is useful because d represents the “loss” that occurs when putting a request〈r, x〉 into L; its enumeration merely ensures that K(x) ≤ r + d. As usual inapplications of the Recursion Theorem, somewhat paradoxically, we can assumethat d is given even if we are building L. Here is why.(1) From an index for a c.e. set G ⊆ N × 2 1) be the machine effectively obtained from ˜G via the MachineExistence Theorem.(3) Our construction yields a bounded request set L uniformly in d.We have to show that L is a bounded request set for each d. IfG = L, which willhappen for some effectively given G by the Recursion Theorem, then we mayconclude that G is a bounded request set and M d is a machine for L. (We needto show that L is always a bounded request set, for otherwise we might end upwith a fixed point where G = L but ˜G ≠ G, which would be of no interest tous.) See Proposition 5.2.13 for our first application of this method.Exercises.2.2.22. Show that ∑ x f(K(x))2−K(x) = ∞ for each order function f.2.2.23. (Levin) A discrete c.e. semimeasure on {0, 1} ∗ is a function m: {0, 1} ∗ → R + 0such that m(x) is a left-c.e. real number uniformly in x and ∑ xm(x) ≤ 1. Notethat S → m(S) = ∑ x∈S m(x) defines a measure on all subsets S of {0, 1}∗ . Forinstance, the function m(x) =2 −x−1 is a discrete c.e. semimeasure; a further exampleis m(x) =2 −K(x) . Verify that a function D : N → N∪{∞} is computably approximablefrom above and satisfies the weight condition iff m(x) = 2 −D(x) is a discrete c.e.

2.2 The prefix-free complexity K 91semimeasure. Next, show that there is a close correspondence between discrete c.e.semimeasures and bounded request sets:(a) For each bounded request set W , the function m(x) =wgt W ({x}) is a discrete c.e.semimeasure on {0, 1} ∗ such that m(S) =wgt W (S).(b) If m is a discrete c.e. semimeasure, there is a bounded request set W such thatm(x) =wgt W ({x}) for each x. Also, L = {〈k +1,x〉: 2 −k ≤ m(x)} is a boundedrequest set such that 4 · wgt L ({x}) ≥ m(x) ≥ wgt L ({x}) for each x.For details see Li and Vitányi (1997).2.2.24. (Gács) (i) Let p, δ ∈ [0, 1) R , δ > 0, p + δ ≤ 1. Show that there is a string wsuch that δ < 2 −|w|+2 and I(w) ⊆ [p, p + δ).(ii) Use (i) to prove the slightly weaker version of 2.2.17 where the conclusion is∀r, y [〈r, y〉 ∈ W ↔ ∃w (|w| ≤ r +2 & M(w) = y)] (that is, the M-descriptioncorresponding to the request 〈r, y〉 may be chosen by 2 longer than r).The Coding TheoremThe probability that a prefix-free machine M outputs a string x isThus Ω M = ∑ x P M (x).P M (x) = ∑ σ 2−|σ| [[M(σ) =x]] = λ[{σ : M(σ) =x}] ≺ .2.2.25 Theorem. (Coding Theorem) From a prefix-free machine M we mayeffectively obtain a constant c such that ∀x 2 c 2 −K(x) >P M (x).Proof. We show that the function D(x) =⌈− log 2 P M (x)⌉ is computably approximablefrom above and satisfies the weight condition ∑ x 2−D(x) ≤ 1. Then weapply Theorem 2.2.19.Let P M,s (x) = λ[{σ : M s (σ) = x}] ≺ , then D s (x) = ⌈− log 2 P M,s (x)⌉ is acomputable approximation of D from above.By definition D(x) ≥−log 2 P M (x) >D(x) − 1, so2 −D(x) ≤ P M (x) < 2 −D(x)+1 .This implies the weight condition for D, since ∑ x 2−D(x) ≤ ∑ x P M (x) ≤ 1.By 2.2.19 we have K ≤ + D, so there is a constant c such that ∀xK(x) − c ≤D(x) − 1. Then for each x we have 2 c 2 −K(x) ≥ 2 −D(x)+1 >P M (x). ✷Note that 2 −K M (x)≤ P M (x) because a shortest M-description of x contributes2 −K M (x) to P M (x). If M = U then by the Coding Theorem we also have 2 c 2 −K(x) >P U (x) for each x, so2 −K(x) and P U (x) are proportional. By another application of theCoding Theorem, for each prefix-free machine M we have P M (x) =O(P U (x)).In Fact 2.1.13 we proved that if d ∈ N then for each n ∈ N the number of d-compressible C strings of length n is less than 2 n−d+1 . We apply the CodingTheorem to obtain an analog of this for K. By Theorem 2.2.9, K(x) ≤ |x| +K(|x|)+1. We will calculate an upper bound for the number of strings of length nsuch that K(x) ≤ |x| + K(|x|) − d.2.2.26 Theorem. There is a constant c ∈ N such that the following hold.(i) ∀d ∈ N ∀n #{x: |x| = n & K(x) ≤ n + K(n) − d} < 2 c 2 n−d(ii) ∀b ∈ N ∀n #{x: |x| = n & K(x) ≤ K(n)+b} < 2 c 2 b .

92 2 The descriptive complexity of stringsProof. (i). Let M be the prefix-free free machine given by M(σ) =|U(σ)|. Bythe Coding Theorem, let c be the constant such that 2 c 2 −K(n) >P M (n) foreach n. If|x| = n and K(x) ≤ n + K(n) − d, then a shortest U-description of x,being an M-description of n, contributes at least 2 −n−K(n)+d to P M (n). If thereare at least 2 n+c−d such x, then P M (n) ≥ 2 n+c−d 2 −n−K(n)+d =2 c 2 −K(n) ,acontradiction.(ii) follows from (i) letting d = n − b. (Note that (i) is vacuously true for negatived.)✷The estimate in (ii) will be applied in Section 5.2 to show that each K-trivial setis ∆ 0 2. Another application of the Coding Theorem is Theorem 2.3.6 where we give anexpression for the descriptive complexity of an ordered pair K(〈x, y〉).2.2.27 Exercise. Use Exercise 2.2.23 and the Coding Theorem to show that m(x) =2 −K(x) is an optimal discrete c.e. semimeasure on {0, 1} ∗ , namely, for each discrete c.e.semimeasure v we have ∀x [e · m(x) ≥ v(x)] for an appropriate constant e>0.2.3 Conditional descriptive complexityWe study the conditional descriptive complexity of a string x. An auxiliarystring y is available to help with the decompression. We consider conditionalversions of both C and K.BasicsThe conditional descriptive complexity C(x | y) is the length of a shortest descriptionof x using the string y as auxiliary information. For a simple example,if y is the first half of a string x of even length, we expect C(x | y) to be quitea bit smaller than C(x). Also C(f(x) | x) should be bounded by a constant foreach computable function f.To develop the formal theory of conditional C-complexity, one simply allowsall the machines involved to have two inputs, the second being the auxiliaryinformation y. Such a machine is called binary machine. The standard optimalbinary machine is given byV 2 (0 e−1 1σ,y) ≃ Φ 2 e(σ,y)where e>0 and σ ∈ {0, 1} ∗ . Now defineC(x | y) = min{|σ|: V 2 (σ,y)=x}.Next we introduce a conditional version of the prefix-free string complexity K.A binary machine M is called prefix-free when fixing the second component if foreach string y the set {σ : M(σ,y) ↓} is prefix-free. There is an effective listing(Md 2) d∈N of all such machines. Now we may adapt the proof of Theorem 2.2.9,using the Recursion Theorem with Parameters 1.1.6, in order to obtain an optimalbinary machine U 2 that is prefix-free when fixing the second componentand such that U 2 (0 d−1 1σ,y)=Md 2 (σ,y) for d>1. We letK(x | y) = min{|σ|: U 2 (σ,y)=x}.

2.3 Conditional descriptive complexity 93Then K(x | y) ≤ K(n | y)+n + 1 where n = |x|.The empty string is of no use as an auxiliary information. Hence C(x | ∅) = +C(x) and K(x | ∅) = + K(x). (We could modify the definitions of V and of U inorder to achieve an equality here.)2.3.1 Fact. Let N be a machine.(i) For each z and each y such that N(y)↓ we have C(N(y) | z) ≤ + C(y | z).(ii) For each y and each z such that N(z)↓ we have C(y | z) ≤ + C(y | N(z)).The same holds with K in place of C, and still for an arbitrary machine N.Proof. (i) is a version of Fact 2.1.20 for conditional complexity and proved inthe same way; (ii) is left as an exercise. The case of K is similar.✷In the following we will need a notation for a particular shortest U-descriptionof a string x such that the decompression takes the least amount of time.2.3.2 Definition. For a string x, ift = µs. K s (x) =K(x), let x ∗ be the leftmoststring σ such that |σ| = K(x) and U t (σ) =x.The reason for this particular choice of a shortest description becomes apparentin the proof of the following fact.2.3.3 Fact. K(y | x ∗ )= + K(y | 〈x, K(x)〉).Proof. Given x ∗ , a machine N can compute the pair 〈x, K(x)〉 = 〈U(x ∗ ), |x ∗ |〉.Conversely, from the pair 〈x, K(x)〉 a machine N ′ can first compute t =µs. K s (x) =K(x) and then x ∗ . Now we apply Fact 2.3.1(ii) for K. ✷Exercises. Show the following.2.3.4. C(x) = + C(x, C(x)) and K(x) = + K(x, K(x)).2.3.5. For all x, y, z we have K(x | z) ≤ + K(x | y)+K(y | z).An expression for K(x, y) ⋆Using conditional K-complexity we will find an expression for the prefix-free complexityof an ordered pair 〈x, y〉. We write K(x, y) as a shorthand for K(〈x, y〉).(1) Clearly K(x, y) ≤ + K(x)+K(y) via the prefix-free machine which on input ρtries to find a decomposition ρ = στ such that U(σ)↓= x and U(τ)↓= y, and inthat case outputs 〈x, y〉.(2) This can be improved to K(x, y) ≤ + K(x)+K(y | x) because the machinealready has x when it decompresses τ to obtain y.(3) A further improvement is the inequality K(x, y) ≤ + K(x)+K(y | x ∗ ): themachine looks for a decomposition ρ = στ such that U(σ)↓= x and U 2 (τ, σ)↓= y.Once it finds this, it outputs 〈x, y〉. This machine is prefix-free because U 2 isprefix-free when fixing the second component. The last upper bound for K(x, y)is our final one by the following result of Levin in Gács (1974).2.3.6 Theorem. K(x, y) = + K(x)+K(y | 〈x, K(x)〉) = + K(x)+K(y | x ∗ ).

94 2 The descriptive complexity of stringsProof. By the preceding discussion and Fact 2.3.1, it remains to show thatK(x)+K(y | x ∗ ) ≤ + K(x, y). We will apply the Coding Theorem 2.2.25 to themachine G(σ) ≃ (U(σ)) 0 which outputs the first component of U(σ) viewed as anordered pair in case U(σ)↓. Let c be a constant such that ∀x 2 c 2 −K(x) >P G (x).Construction of a uniformly c.e. sequence of bounded request sets (L σ ).Fix σ. Suppose U(σ) ↓= x at stage s. IfU(ρ) =〈x, y〉 at a stage t>s, put therequest 〈|ρ|−|σ|+c, y〉 into L σ , but only as long as the total weight of L σ definedafter (2.8) does not exceed 1. (Putting the request into L σ causes an increase ofthis weight by 2 −|ρ|+|σ|−c .)Let M f(σ) be the prefix-free machine effectively obtained from σ via the MachineExistence Theorem 2.2.17. The binary machine N on inputs ρ, σ first waitsfor U(σ) ↓. If so, it simulates M f(σ) (ρ). If σ = x ∗ then 2 c 2 −|σ| > P G (x) =∑ρ 2−|ρ| [[∃z U(ρ) =〈x, z〉]. Therefore the total weight put into L σ is at mostP G (x)2 |σ|−c ≤ 1, hence we never threaten to exceed the weight 1 in the constructionof L σ . Thus N(ρ, σ) =y for some ρ such that |ρ| = K(x, y) − |σ| + c,which implies that K(y | x ∗ ) ≤ + |ρ| = + K(x, y) − K(x), as required. ✷2.4 Relating C and KBasic interactionsProposition 2.2.8 states that K(x) ≤ + K(|x|)+|x|. In the proof we constructed aprefix-free machine N that tries to split the input τ in the form τ = σρ, where σis a prefix-free description. The point is that such a decomposition is unique ifit exists. We will obtain some facts relating C and K by varying this idea. Forinstance, a modification of N, where we attempt to compute V(ρ) instead ofcopying ρ, yields the following.2.4.1 Proposition. K(x) ≤ + K(C(x)) + C(x).Proof. On input τ, the machine Ñ first searches for a decomposition τ = σρsuch that n = U(σ) ↓ and |ρ| = n. Once it is found Ñ simulates V(ρ). (That is,Ñ behaves exactly like V on input ρ, including its output.)Clearly Ñ is prefix-free. Given x, let ρ be a shortest V-description of x andlet σ be a shortest U-description of |ρ|, then Ñ(σρ) =x. SoK(x) ≤+ |σ| + |ρ| =K(C(x)) + C(x).✷In the following we use this fact to show that K(x) exceeds C(x) by at most2 log(C(x)) ≤ + 2 log(|x|), up to a small additive constant. Hence C ∼ K, so inseveral results C can be replaced by K. An example is Proposition 2.1.22.2.4.2 Corollary. C(x) ≤ + K(x) ≤ + C(x) + 2 log(C(x)) ≤ + C(x) + 2 log(|x|).Proof. We have already observed that C(x) ≤ + K(x). For the second inequality,let m = C(x). Then K(m) ≤ + 2 log m by (2.5) and since |m| = log(m+1) when mis viewed as a string (see page 13). Now apply 2.4.1.✷

2.4 Relating C and K 95The term 2 log(C(x)) in the middle could even be decreased to log(C(x))+2 log log(C(x))by 2.2.8. If x has length 10000, say, then K(x) − C(x) is bounded by about 20.One more time, we apply the idea to split off a prefix-free description.2.4.3 Proposition. For all strings x and y we have C(xy) ≤ + K(x)+C(y).Proof. On input τ, the machine M first searches for a decomposition τ = σρsuch that U(σ) ↓= x. Once found, M simulates the computation of V on inputρ. IfV(ρ) =y then M outputs xy. Clearly, C M (xy) ≤ K(x)+C(y). ✷Next, we prove a continuity property of C that strengthens Proposition 2.1.21.2.4.4 Proposition. abs(C(x) − C(y)) ≤ + K(abs(x − y)) ≤ + 2 log abs(x − y).Proof. First suppose that C(x) ≥ C(y). Note that x = y ± abs(x − y). We maydescribe x based on (1) a U-description σ of z = abs(x − y), (2) a V-description ρof y, and (3) a further bit b telling us whether to add or to subtract z. If we put thisinformation into the form τ = σbρ, then a machine S on input τ can first find σ ≺ τsuch that U(σ) =z, then read b, and then apply V to the rest ρ to obtain y. Now it hasall the information needed in order to calculate x. ThusC(x) ≤ + K(abs(x−y))+C(y).Note that y = x ± abs(x − y), so if C(x) 1 is an index for M in the effective listing of binary machines that areprefix-free when fixing the second component.“≤ + :” Let N be the machine that, on input τ, searches for a decomposition τ = σρsuch that n = U(σ) ↓ and then simulates U 2 (ρ,n+ |ρ|). If C(x)

96 2 The descriptive complexity of stringsProof. For notation involving error terms see page 75. Also recall from Definition 2.3.2that x ∗ is a shortest U-description of x. To save on brackets we write KC(x) insteadof K(C(x)), etc.We begin with a brief outline of the proof. We first prove Equation (i) and thenobtain Equation (ii) from (i). To prove (i), by Proposition 2.4.5 C(x) = + K(x | C(x)).Note that K(x | C(x) ∗ ) ≤ + K(x | C(x)) by Fact 2.3.1(ii). In Claim 2.4.7 we show thatit is by at most K (2) C(x) smaller. This makes it possible to work with K(x | C(x) ∗ )instead of K(x | C(x)). Then, applying Levin’s Theorem 2.3.6 twice, we obtain thatwithin the small error of K (2) C(x), C(x) can be replaced byK(x | C(x) ∗ )= + K(x, C(x)) − KC(x) = + K(x)+K(C(x) | x ∗ ) − KC(x).Based on this we will be able to reach Equation (i).2.4.7 Claim. C(x) ≤ + K(x | C(x) ∗ )+K (2) C(x).Let σ = C(x) ∗ . By Exercise 2.3.5,K(x | C(x)) ≤ + K(x | σ)+K(σ | C(x)).We will estimate the second term on the right hand side. For any y we haveK(y ∗ | y) = + K(K(y) | y), since, given y, we can effectively produce a prefix-freedescription of y ∗ from one of K(y) (wait till the first U-description of y that has lengthK(y) comes up), and conversely (take the length). We apply this to y = C(x) andobtainK(σ | C(x)) = K(σ | y) = + K(K(y) | y) ≤ + K (2) (y) =K (2) C(x).This establishes Claim 2.4.7. Next we provide the main technical result.2.4.8 Claim. C(x) =K(x) − KC(x)+O(K (2) C(x)).Let Q(x) =C(x)+KC(x) − K(x). To establish the claim, we will show that∀ ∞ x [abs(Q(x)) ≤ 2K (2) C(x)]. (2.9)By 2.4.1 Q(x) is bounded from below by a constant. So it suffices to show that∀ ∞ x [Q(x) ≤ 2K (2) C(x)]. We apply Levin’s Theorem 2.3.6 twice:K(x | C(x) ∗ )= + K(x, C(x)) − KC(x) = + K(x)+K(C(x) | x ∗ ) − KC(x).Substituting this into Claim 2.4.7 we obtain an estimate for C(x):C(x) ≤ + K(x)+K(C(x) | x ∗ ) − KC(x)+K (2) C(x). (2.10)Our next task is to find a suitable upper bound for the term K(C(x) | x ∗ ):K(C(x) | x ∗ ) ≤ + K(C(x) | K(x))≤ + K(abs(C(x) − K(x)))≤ + K(abs(C(x)+KC(x) − K(x))) + K (2) C(x).To obtain the last line, we attached a U-description of KC(x) (and a further bit b)at the beginning of a U-description of abs(C(x) − K(x)) as in the proof of 2.4.4. Thiscauses the extra term K (2) C(x). After rearranging, the estimate (2.10) now turns into

2.5 Incompressibility and randomness for strings 97Q(x) ≤ + K(abs(Q(x))) + K (2) C(x).Recall that we already have a constant lower bound for Q(x), so to obtain the upperbound we may assume Q(x) ≥ 0. Then, since K(m) ≤ + 2 log m for each m ∈ N, wehaveQ(x) ≤ + 2 log Q(x) +K (2) C(x), which implies Q(x) ≤ + 2K (2) C(x). This establishes(2.9) and hence Claim 2.4.8.To proceed we need two facts.2.4.9 Fact. K(n + m) =K(n)+O(K(m)).For K(n + m) ≤ + K(n)+K(m) and also K(n) ≤ + K(n + m)+K(m). It follows thatabs(K(n + m) − K(n)) ≤ + K(m).2.4.10 Fact. Let f,g: N → N. Ifg(x) =f(x)+O(K(f(x))), then K(g(x)) ∼ K(f(x)).To see this, we apply K to both sides of the hypothesis. By Fact 2.4.9 we obtain thatK(g(x)) = K(f(x)) + O(log K(f(x))), so K(g(x)) ∼ K(f(x)).To establish Equation (i) of Theorem 2.4.6 we want to replace the C’s by K’s on theright hand side of Claim 2.4.8. We apply Claim 2.4.8 itself for this! In view of Fact 2.4.9,after applying K to both sides of the claim we obtainKC(x) =K (2) (x)+O(K (2) C(x)) (2.11)(an error term O(K (3) C(x)) has been absorbed into the larger O(K (2) C(x))). This isequivalent to K (2) (x) =KC(x)+O(K (2) C(x)), so by Fact 2.4.10, where f(x) =KC(x),we obtainK (3) (x) ∼ K (2) C(x)). (2.12)Substituting the last two equations into Claim 2.4.8 yields Equation (i).Next we prove Equation (ii). By Equation (i) with C(x) instead of x,C (2) (x) =KC(x)+O(K (2) C(x)), (2.13)where the error term O(K (3) C(x)) has been absorbed into O(K (2) C(x)). Using (2.11)and then (2.12), this turns intoC (2) (x) =K (2) (x)+O(K (3) (x)).Applying the last equation to C(x) instead of x and using Fact 2.4.10 yieldsC (3) (x) ∼ K (2) C(x). (2.14)Rearranging Claim 2.4.8 yields K(x) =C(x) +KC(x) +O(K (2) C(x)). Now we substitutefirst (2.13) and then (2.14) in order to obtain Equation (ii). This completes theproof of Theorem 2.4.6.✷2.5 Incompressibility and randomness for stringsAn object is random if it is disorganized, has no patterns, no regularities. Patternsoccuring in a string x yield a description of the string that is shorter than itstrivial description (the string itself). For instance, no one would regard x asrandom if all the bits of x in the even positions are 0. One can use this patternto show that C(x) ≤ + |x|/2 (see Exercise 2.1.7). On page 99 we will gather

98 2 The descriptive complexity of stringsfurther evidence for the thesis that, for strings, the intuitive notion of randomnesscan be formalized by being hard to describe, that is, incompressibility as definedin 2.1.12. We provide two variants of the concept of incompressibility with respectto C, the first weaker and the second stronger. They yield alternative formalrandomness notions for strings.2.5.1 Definition. The string x is d-incompressible in the sense of K, ordincompressibleK for short, if K(x) > |x| − d. Else x is called d-compressible K .An incompressibility notion for strings can also be used to study the randomnessaspect of an infinite sequence of zeros and ones (i.e., a subset of N) via its finiteinitial segments. Incompressibility in the sense of K turns out to be the mostuseful tool; see Section 3.2.2.5.2 Definition. We say that a string x is strongly d-incompressible K if K(x) >|x| + K(|x|) − d, namely, K(x) differs by at most d from its upper bound|x| + K(|x|) + 1 given by Theorem 2.2.9.While the intuitive concept of randomness for strings does not involve anynumber parameter, each of the incompressibility notions for a string is definedin terms of a constant d. It is somewhat arbitrary how large a constant wewant to accept when formalizing randomness by incompressibility. Generally, theconstant should be small compared to the length of the string. For each fixed d,randomness is an asymptotic concept when interpreted as incompressibility withparameter d. The arbitrariness in the choice of the constant only disappearscompletely when we consider subsets of N.It is harder to obtain a formal notion of computational complexity for strings, becausethe computational complexity of a set Z is given by what an algorithm can do with Zas an oracle, and such an algorithm only makes sense when considered over the wholerange of inputs. (A possible measure for the computational complexity of a string y isC(x) − C(x | y), taken over a variety of strings x.)Comparing incompressibility notionsThe implications between the incompressibility notions arestrongly incompressible K(1)⇒ incompressible C(2)⇒ incompressible K .More precisely, for each p ∈ N there is a q ∈ N such that each strongly p-incompressible K string is q-incompressible C , and similarly for the second implication.2.5.3 Remark. For each length, most strings are incompressible in the strongestsense. For it is immediate from Theorem 2.2.26(i) that the number of strongly d-incompressible K strings of length n is at least 2 n −2 c+n−d where c is the constantof the theorem. We can make the proportion of strings that are not stronglyincompressible K as small as we wish (independently of n) by choosing d sufficientlylarge.

2.5 Incompressibility and randomness for strings 99The implication (2) holds for a minor change of constants since C(x) ≤ + K(x).Actually, if x is incompressible C then all the prefixes of x are incompressible K :2.5.4 Proposition. Fix b ∈ N such that C(yz) ≤ K(y) +|z| + b for each y, z(by 2.4.3). Then for each d ≥ b, each prefix of a (d-b)-incompressible C string xis d-incompressible K . That is, C(x) > |x| − (d − b) →∀y ≼ x [K(y) > |y| − d].Proof. Suppose that y ≼ x. Let x = yz. IfK(y) ≤ |y| − d then by the choiceof b we have C(x) ≤ K(y)+|x| − |y| + b ≤ |x| + b − d.✷In particular, there are arbitrarily long strings x such that each prefix of xis d-incompressible K . There is no analog of this for C in place of K because foreach c, ifx has length about 2 c then the C-complexity of some prefix w of xdips below |w| − c (Proposition 2.2.1). This implies that for each c there is a d-incompressible K string w such that C(w) ≤ |w| − c. Thus the converse of theimplication (2) fails. Next, we address the implication (1).2.5.5 Proposition. There is c ∈ N such that the following holds. For each d,K(x) > |x| + K(|x|) − (d − c) → C(x) > |x| − 2d.Proof. We define a bounded request set W to ensure thatC(x) ≤ |x| − 2d → K(x) ≤ |x| + K(|x|) − (d − c), (2.15)where c − 2 is the coding constant of W given by Theorem 2.2.17.Construction of W . At stage s, for each d, each n and each x of length n, ifK s (n)

100 2 The descriptive complexity of strings(b) zeros and ones occur balancedly in incompressible strings.A property G of strings pointing towards organization should be rare in thatthere are not too many strings of each length with that property. Also, G shouldbe not too hard to recognize: we will mostly consider rare properties G thatare also decidable. Having long runs of zeros is such a property, having a largeimbalance between zeros and ones is another.Given a set of strings G, for each n let G n = G ∩ {0, 1} n . The following lemmashows that for any c.e. set of strings G that is rare in the sense that #G n /2 nis bounded by O(n −3 ), the strings in G n can be compressed to n − log n in thesense of K. (Recall from page 13 that we use the notation log n = max{k ∈N: 2 k ≤ n}.) This lemma will be applied to prove (a) and (b).2.5.6 Lemma. Let G be a c.e. set of strings such that #G n = O(2 n−3 log n ) foreach n. Then ∀x ∈ G n [K(x) ≤ + n − log n].Proof. The idea is that for each string x ∈ G the length n and its position inthe computable enumeration of G n together provide a short description of x.Instead of giving that description explicitly, we rely on the Machine ExistenceTheorem 2.2.17.Since n − 3 log n>0 for n ≥ 10, there is k ∈ N such that for all n ≥ 10 wehave #G n ≤ 2 k 2 n−3 log n . Define a bounded request set W as follows: when x isenumerated into G and |x| = n ≥ 10, put the request〈n − log n + k +1,x〉into W . By the hypothesis, the weight contributed by G n for n ≥ 10 is at∑most 2 −k−1 2 −n+log n 2 k 2 n−3 log n ≤ (n − 1) −2 /2 (since (n − 1) −1 ≥ 2 − log n ). Asj>0 1/j2 = π 2 /6 < 2 this implies that W is a bounded request set. By 2.2.17,x ∈ G and |x| = n imply K(x) ≤ + n − log n.✷Our first application of the foregoing lemma is a proof of (a) above: a run ofzeros in an incompressible K string of length n is bounded by 4 log n + O(1). Weneed to show that the relevant property is rare.2.5.7 Proposition. Let n ∈ N and let x be a string of length n. If there is arun of 4 log n zeros in x then K(x) ≤ + n − log n.Proof. Define the computable set G by lettingG n = {x: |x| = n & x has a run of 4 log n zeros}.Such a string x is determined by the least position i

2.5 Incompressibility and randomness for strings 101if the sample space is {0, 1} n , the random variable S n (x) denotes the number ofoccurrences of ones in a string x of length n. Ford ∈ N − {0}, letA n,d = {x ∈ {0, 1} n : abs(S n (x)/n − 1/2) ≥ 1/d}. (2.16)Thus, A n,d is the event that the number of ones differs by at least n/d from theexpectation n/2. The Chernoff bounds (Shiryayev, 1984, Eqn. (42) on pg. 69)yieldP (A n,d ) ≤ 2e −2n/d2 . (2.17)This considerably improves the estimate P (A n,d ) ≤ d 2 /(4n) after (1.21), page 73,obtained through the Chebycheff inequality. For instance, for P (A 1000,10 ) theupper bound is 4.2 · 10 −9 rather than 1/40.2.5.8 Theorem. There is a constant c and a computable function λd.n d suchthat ∀d ∈ N − {0} ∀n ≥ n d ∀x ∈ A n,d [K(x) ≤ n − log n + c].Proof. For each d ∈ N − {0} let n d be the least number such that∀n ≥ n d [e 2n/d2 /2 ≥ n 3 ].Define the computable set G by letting G n = ∅ for n2 b+c and n ≥ n d then the requiredinequality holds, else x ∈ A n,d and thus K(x) ≤ n − log n + c ≤ n − b. ✷If incompressibility is taken in a stronger sense one also obtains stronger conclusions.For instance, the following exercise shows that for almost all n, if a string x of length nis incompressible in the sense of C then abs(S n(x) − n/2) is bounded by √ n ln n.Incompressibility in the sense of K merely yields a bound in o(n).2.5.10 Exercise. For almost all n, if|x| = n and abs(S n(x) − n/2) ≥ √ n ln n thenC(x) ≤ + n − log n.Hint. Modify the proof of 2.5.8, using Proposition 2.1.14 rather than the MachineExistence Theorem.

3Martin-Löf randomness and its variantsSets computable by a Turing machine are the mathematical equivalent of setsdecidable by an algorithm. In this chapter we introduce a mathematical notioncorresponding to our intuitive concept of randomness for a set. The intuition isvaguer here than in the case of computability. The central randomness notionwill be the one of Martin-Löf (1966). In order to address criticisms to the claimthat this notion is the appropriate one, we will consider weaker and strongerrandomness notions for sets as alternatives. In this way a (mostly linear) hierarchyof randomness notions emerges. Martin-Löf randomness stands out becausewe arrive at it from two different directions (Theorem 3.2.9), and also becauseit interacts best with computability theoretic concepts.Section 3.1 contains background on randomness and tests. Section 3.2 providesthe basics on Martin-Löf randomness, and some interesting further results. InSections 3.3 and 3.4 we study its interaction with computability theoretic notionsvia reducibilities and relativization, respectively. In Section 3.5 we introduceweaker variants of Martin-Löf randomness, and in 3.6 stronger variants.3.1 A mathematical definition of randomness for setsOur intuitive concept of randomness for a set Z has two related aspects:(a) Z satisfies no exceptional properties, and(b) Z is hard to describe.Towards a mathematical definition of randomness for a set, at first we will considereach of the two aspects separately.(a) Z satisfies no exceptional properties. Think of the set Z as the overall outcomeof an idealized physical process that proceeds in time. It produces infinitelymany bits. The bits are independent. Zero and one have the same probability.An example is the repeated tossing of a coin. The probability that a string x isan initial segment of Z is 2 −|x| . Exceptional properties are represented by nullclasses (with respect to the uniform measure λ on Cantor space, Definition 1.9.8).We give examples of exceptional properties P and Q. The first states that allthe bits in even positions are zero:P(Y ) ↔∀iY(2i) =0. (3.1)The second states there are at least twice as many zeros as ones in the limit:Q(Y ) ↔ lim inf #{i

3.1 A mathematical definition of randomness for sets 103The corresponding classes are null. Hence, according to our intuition, they shouldnot contain a random set. In order to obtain a mathematical definition of randomness,we impose extra conditions on the null classes a set must be avoid.Otherwise no set Z would be random at all because the singleton {Z} itself is anull class! Some effectivity, or definability, conditions are required for the class(or sometimes a superclass). For instance, one can require that the null class isΠ 0 2 (see Definition 1.8.55). The classes given by the properties above are of thistype; {Y : P(Y )} is actually Π 0 1.(b) Z is hard to describe. A random object has no patterns, is disorganized. Onpage 99 we provided evidence that in the setting of finite objects, the intuitiveconcept of randomness corresponds to being hard to describe. We relied on thefact that there are description systems, called optimal machines, that emulateevery other description system of the same type, and in particular describe everypossible string. Being hard to describe can be formalized by incompressibilitywith respect to an optimal machine. Incompressible strings have the propertiesthat one intuitively expects from a random string.For sets, as for strings our intuition is that some degree of organization makesthe object easier to describe. However, we cannot formalize being hard to describein such a simple way as we did for strings, because each description system onlydescribes countably many sets. To make more precise what we mean by beinghard to describe for sets, we recall close descriptions from page 46. The null Π 0 1classes represent a type of close description; so do the null Π 0 2 classes. A set ishard to describe in a particular sense (say, Π 0 2 classes) if it does not admit a closedescription in this sense (for instance, it is not in any Π 0 2 null class).If we want to introduce a mathematical notion of randomness, to incorporateaspect (a) we need a formal condition restricting null classes, and for aspect (b)a formal notion of close description. Both are given by specifying a test concept.This determines a mathematical randomness notion: Z is random in that specificsense if it passes all the tests of the given type. Tests are themselves objects thatcan be described in a particular way; thus only countably many null classes aregiven by such tests. If (A n ) n∈N is a list of all null classes of that kind, then thecorresponding class of random sets is 2 N − ⋃ n A n, which is conull. Remark 2.5.3is the analog of this for strings. It states that most strings of each length areincompressible.In the following we give an overview of some test notions. They determineimportant randomness concepts. A test can be a conceptually simple objectsuch as a Π 0 2 null class. It can also be a more elaborate object such as a sequenceof c.e. open sets or a computable betting strategy. The power of tests is directlyrelated to their position in the hierarchy of descriptive complexity for sets andclasses (pages 21 and 64). The formal details are provided in later sections ofthis chapter, and in Chapter 7.

104 3 Martin-Löf randomness and its variantsMartin-Löf tests and their variantsRecall from Fact 1.9.9 that a class A ⊆ 2 N is null iff there is a sequence (G m ) m∈Nof open sets such that lim m λG m = 0 and A ⊆ ⋂ m G m. Motivated by testconcepts from statistics, Martin-Löf (1966) introduced an effective version of thischaracterization of null classes. He required that the sequence (G m ) m∈N of opensets be uniformly c.e., and the convergence be effective, namely, for each positiverational δ one can compute m such that λG m ≤ δ. Taking a suitable effectivesubsequence, one might as well require that λG m ≤ 2 −m for each m. A sequenceof open sets with these properties will be called a Martin-Löf test, and a set Zis Martin-Löf random if Z ∉ ⋂ m G m for all Martin-Löf tests (G m ) m∈N .Forinstance, to see that a Martin-Löf random set does not satisfy the property Pin (3.1), let G m = {Z : ∀i

3.1 A mathematical definition of randomness for sets 105The analog of a Martin-Löf test is a uniformly Π 1 1 sequence (G m ) m∈N of opensets such that λG m ≤ 2 −m . For an even stronger notion based on higher computability,one can take as tests the null Π 1 1 classes C ⊆ 2 N (9.1.1). Sets Z enterthe class at stages that are ordinals computable relative to Z.The choice of the “superlevel” in the hierarchy of descriptions, namely whetherone takes feasible computability, computability, or higher computability, dependson the applications one has in mind. Interestingly, similar patterns emerge at thethree superlevels.Schnorr’s Theorem and universal Martin-Löf testsIf the same notion arises from investigations of different areas, it deserves specialattention. Among randomness notions, this is the case for Martin-Löf randomness.One obtains the same class when requiring that all the finite initial segmentsbe incompressible in the sense of the prefix-free algorithmic complexity K definedin Section 2.2: a theorem of Schnorr (1973) states that Z is Martin-Löfrandom if and only if there is b ∈ N such that ∀nK(Z ↾ n ) >n− b. ThusZ isMartin-Löf random if and only if all its initial segments are random as strings!To some extent, the “randomness = incompressibility” paradigm used for finitestrings carries over to sets.Schnorr’s Theorem only holds when we formalize randomness of a string x bybeing b-incompressible in the sense of K (for some b that is small compared tothe length of x). We cannot use the plain descriptive complexity C because ofits dips at initial segments of a sufficiently long string; see Proposition 2.2.1.However, a similar theorem using a monotonic version Km of string complexitywas announced by Levin (1973).A Martin-Löf test (U b ) b∈N is called universal if ⋂ b U b contains ⋂ m G m for everyMartin-Löf test (G m ) m∈N . Schnorr’s Theorem shows that (R b ) b∈N is universal,where R b is the open set generated by {x ∈ {0, 1} ∗ : K(x) ≤ |x| − b}, the b-compressible K strings. The existence of a universal test is a further criterion forbeing natural for a randomness notion. For instance, the argument of Schnorrcan be adapted to the Π 1 1 version of Martin-Löf randomness, so a universal testexists as well. Also, there is a largest Π 1 1 null class Q ⊆ 2 N . To be random in thisstrong sense means to be not in Q (Theorem 9.3.6).The initial segment approachSchnorr’s Theorem is our first application of the following:Characterize C ⊆ 2 N via the initial segment complexity of its members.This is very useful because one can determine whether Z ∈ C by looking at itsinitial segments, rather than at Z as a whole. We will later provide several otherexamples of such characterizations. As mentioned above, in Schnorr’s Theoremwe use K to measure the descriptive complexity of the initial segment Z ↾ n .Forother classes one may use different measures, such as the plain complexity C orits conditional variant λz.C(z | n) where n = |z|.

106 3 Martin-Löf randomness and its variantsOften membership in C is determined by a growth condition on the initialsegment complexity; 2-randomness (Theorem 3.6.10 below) is somewhat differentas one requires that the plain complexity is infinitely often maximal (up to aconstant).3.2 Martin-Löf randomnessWe provide the formal details for the discussion in the previous section.The test conceptIn 1.8.22 we introduced the effective listing ([W e ] ≺ ) e∈N of all the c.e. open sets(here W e is regarded as a subset of {0, 1} ∗ ). A uniformly c.e. sequence (G m ) m∈Nof open sets is given by a computable function f such that G m =[W f(m) ] ≺for each m. Martin-Löf (1966) effectivized the description of null classes fromFact 1.9.9.3.2.1 Definition.(i) A Martin-Löf test, orML-test for short, is a uniformly c.e. sequence(G m ) m∈N of open sets such that ∀m ∈ N λG m ≤ 2 −m .(ii) A set Z ⊆ N fails the test if Z ∈ ⋂ m G m, otherwise Z passes the test.(iii) Z is ML-random if Z passes each ML-test. Let MLR denote the class ofML-random sets. Let non-MLR denote its complement in 2 N .Note that Z ∈ ⋂ m G m ↔ ∀m ∃k, s [Z ↾ k ] ⊆ G m,s , so ⋂ m G m is a particularkind of Π 0 2 null class. Admitting all the Π 0 2 null classes as tests is equivalent toreplacing the condition ∀m ∈ N λG m ≤ 2 −m by the weaker one that ⋂ m G m bea null class. In this case, we have more tests and therefore a stronger randomnessnotion. This notion is called weak 2-randomness, introduced on page 134.3.2.2 Example. A simple example of a ML-test (G m ) m∈N is the test showingthat a computable set Z is not ML-random: let G m =[Z ↾ m ]. Note that if Z iscomputable then {Z} is a null Π 0 1 class. More generally, for any null Π 0 1 class Pthere is a ML-test (G m ) m∈N such that P = ⋂ m G m: let (P s ) s∈N be the effectiveapproximation of P by clopen sets from (1.17) on page 55, then P = ⋂ s P s.Let f be an increasing computable function such that λP f(m) ≤ 2 −m for each m,and let G m = P f(m) .We do not require in 3.2.1 that a ML test (G m ) m∈N satisfy G m ⊇ G m+1 foreach m. It would make little difference if we did, since we can replace each G mby ⋂ i≤m G i without changing the described null class ⋂ m∈N G m. The tests inthe examples above have this monotonicity property anyway.3.2.3 Remark. While each computable set is given by a ML-test, there is no singleML-test that corresponds to the property of being computable, because the class ofcomputable sets is not Π 0 2 by Exercise 1.8.64.

A universal Martin-Löf test3.2 Martin-Löf randomness 107We say that a Martin-Löf test (U b ) b∈N is universal if ⋂ b U b contains ⋂ m G m forany Martin-Löf test (G m ) m∈N . In other words, ⋂ b U b =2 N − MLR. Martin-Löf(1966) proved that a universal ML-test exists. Fix a listing (G e k ) k∈N (e ∈ N) ofall the ML-tests in such a way that G e kis c.e. uniformly in e, k.3.2.4 Fact. Let U b = ⋃ e∈N Ge b+e+1 . Then (U b) b∈N is a universal ML-test.Proof. The sequence (U b ) b∈N is a ML-test since it is uniformly c.e. andλU b ≤ ∑ e 2−(b+e+1) =2 −b . For the universality, suppose that Z is not MLrandom.Then there is e such that Z ∈ ⋂ k Ge k ,soZ ∈ U b for each b. ✷In 1.8.55 we introduced the arithmetical hierarchy for classes of sets. Since⋂m G m is Π 0 2 for any ML-test (G m ) m∈N , the existence of a universal ML-testimplies the following.3.2.5 Proposition. MLR is a Σ 0 2 class. ✷This shows that Martin-Löf randomness is among the simplest randomness notionsas far as the descriptive complexity is concerned; other notions are typicallyΣ 0 3 or even more complex.Characterization of MLR via the initial segment complexityRecall from 2.2.4 that K M (x) = min{|σ|: M(σ) = x} for a prefix-free machineM. The open sets generated by the strings that are b-compressible in thesense of K M form a monotonic test (R M b ) b∈N. IfM is an optimal prefix-freemachine this test turns out to be universal.3.2.6 Definition. For b ∈ N let R M b=[{x ∈ {0, 1} ∗ : K M (x) ≤ |x| − b}] ≺ .3.2.7 Proposition. (R M b ) b∈N is a ML-test.Proof. The condition K M (x) ≤ |x| − b is equivalent to∃σ ∃s [M s (σ) =x & |σ| ≤ |x| − b],which is a Σ 0 1-property of x and b. Hence the sequence of open sets (RbM ) b∈N isuniformly c.e.To show λRbM ≤ 2 −b , let VbM be the set of strings in {x: K M (x) ≤ |x|−b} thatare minimal under the prefix ordering. Then ∑ x 2−|x| [[x ∈ Vb M ]] = λRb M . Notethat 2 −|x∗ M | ≥ 2 b 2 −|x| for each x ∈ VbM , where x ∗ M is a shortest M-descriptionof x similar to 2.3.2. Then 1 ≥ ∑ x 2−|x∗ M | [[x ∈ VbM ]] ≥ 2 b ∑ x 2−|x| [[x ∈ Vb M ]],and therefore λRb M ≤ 2 −b . ✷Recall U is the optimal prefix-free machine defined in 2.2.9, and K(x) =K U (x).3.2.8 Definition. For b ∈ N, let R b = R U b =[{x ∈ {0, 1}∗ : K(x) ≤ |x| − b}] ≺ .Thus, R b is the open set generated by the b-compressible K strings as definedin 2.5.1. The theorem of Schnorr (1973) states that Z is ML-random iff there

108 3 Martin-Löf randomness and its variantsis b such that each initial segment of Z is b-incompressible in the sense of K.A similar theorem using monotonic string complexity is in Levin (1973).3.2.9 Theorem. The following are equivalent for a set Z.(i) Z is Martin-Löf random.(ii) ∃b ∀n [K(Z ↾ n ) >n− b], that is, ∃b Z∉ R b .Theorem 3.2.9 simply states that (R b ) b∈N is universal. It actually holds for(Rb M ) b∈N, where M is an optimal prefix-free machine, by the same proof.Proof. The implication (i)⇒(ii) holds because (R b ) b∈N is a ML-test by 3.2.7.For (ii)⇒(i), suppose that Z is not ML-random, that is, Z ∈ ⋂ m G m for someML-test (G m ) m∈N . Since we can replace G m by G 2m we may assume that λG m ≤2 −2m for each m.We define a bounded request set L (see 2.2.15). By the representation ofc.e. open sets in 1.8.26 we may uniformly in m obtain an effective antichain(x m i ) i

3.2 Martin-Löf randomness 109Ω M,s := λ[dom M s ] ≺ . We identify co-infinite subsets of N with real numbers in[0, 1) according to Definition 1.8.10. In particular, for a real number r, weletr ↾ n denote the first n bits of the binary expansion of r.3.2.11 Theorem. (Chaitin) The halting probability Ω R is ML-random for eachoptimal prefix-free machine R.Proof. We will write Ω for Ω R , and we let Ω s = Ω R,s . Let N be the (plain)machine that works as follows on an input x of length n.(1) Wait for t such that 0.x ≤ Ω t < 0.x +2 −n .(2) Output the least string y not in the range of R t .If x = Ω↾ n then such a t exists. By stage t all R-descriptions of length ≤ n haveappeared, for otherwise Ω ≥ Ω t +2 −n . Thus K(y) >nwhere y = N(x). Thisimplies ∀n [K(Ω↾ n )+c ≥ K(N(Ω↾ n )) >n] for an appropriate constant c. ✷From now on we will write Ω for Ω U . We let Ω s := λ[dom U s ] ≺ .Facts about ML-random setsFacts derived from Schnorr’s Theorem. In the following we think of a set Z as asequence of experiments with outcomes zero or one, like tossing a coin. The lawof large numbers for a set Z says that the number of occurrences of zeros andones is asymptotically balanced:3.2.12 Definition. A set Z satisfies the law of large numbers iflim n (#{i

110 3 Martin-Löf randomness and its variantsMartin-Löf random, then for each c there is an x ≺ Z such that K(x) ≤ |x| − c,so K(Z ↾ m ) ≤ + m − c for m ∈ R as above, contrary to the hypothesis. ✷A tail Y of a set Z is what one obtains after taking off an initial segment x.Thus, if |x| = n, the tail is the set Y given by Y (i) =Z(i + n). In other words,Y = f −1 (Z) where f is the one-one function λi. i + n.3.2.15 Proposition. Suppose Y and Z are sets such that Y = ∗ Z,orY is atail of Z. Then Z is ML-random ⇔ Y is ML-random.Proof. Under either hypothesis we have ∀nK(Y ↾ n )= + K(Z ↾ n ). Now we applySchnorr’s Theorem 3.2.9.✷The fact that tails of a ML-random set are also ML-random can be strengthened: thepre-image of a ML-random set under a one-one computable function is ML-random.Also, recall from Section 1.3 that, with very few exceptions, classes introduced in computabilitytheory are closed under computable permutations. By the next result this isthe case for ML-randomness.3.2.16 Proposition. Suppose the one-one function f : N → N is computable. If Z isML-random then so is Y = f −1 (Z).Proof. For a class C ⊆ 2 N let C ∗ = {Z : f −1 (Z) ∈ C}. Since f is one-one, we haveλ[x] ∗ =2 −|x| for each string x. IfG ⊆ 2 N is open, let (x i) i0and z 0,...,z i−1 have been defined. The tail Y of Z obtained by taking off z 0 ...z i−1 isnot ML-random by Proposition 3.2.15. Thus there is z i ≺ Y such that K(z i) ≤ |z i| − 1.(ii)⇒(iii) is trivial. For (iii)⇒(i), fix n and consider the prefix-free machine N whichon input σ first searches for an initial segment ρ ≼ σ such that U(ρ) =n and thenlooks for ν 0,...,ν n−1 ∈ dom(M) such that ρν 0 ...ν n−1 = σ. If the search is successfulit prints M(ν 0) ...M(ν n−1).Given a string z 0 ...z n−1, let σ be a concatenation of a shortest U-description of nfollowed by shortest M-descriptions of z 0,...,z n−1. Then N(σ) =z 0 ...z n−1 and henceK(z 0 ...z n−1) ≤ + K(n) + ∑ i

3.2 Martin-Löf randomness 111if Z ∈ ⋂ m G m. To define Solovay tests we will broaden the test definition andrelax the failure condition. Although Solovay tests are more general than Martin-Löf tests and therefore easier to build, they still determine the same notion ofrandomness.3.2.18 Definition. A Solovay test is a sequence (S i ) i∈N of uniformly c.e. opensets such that ∑ i λS i < ∞. Z fails the test if Z ∈ S i for infinitely many i,otherwise Z passes the test.3.2.19 Proposition. Z is ML-random ⇔ Z passes each Solovay test.Proof. ⇐: Suppose Z fails the ML-test (G m ) m∈N . Each Martin-Löf test is aSolovay test, and since Z fails (G m ) m∈N as a ML-test (namely, ∀mZ ∈ G m ),Z fails it as a Solovay test (namely, ∃ ∞ mZ ∈ G m ).⇒: Suppose Z fails the Solovay test (S i ) i∈N , that is, Z ∈ S i for infinitely many i.We may omit finitely many of the S i , and hence assume that ∑ i λS i ≤ 1. LetG m =[{σ :[σ] ⊆ S i for at least 2 m many i}] ≺ ,then (G m ) m∈N is a u.c.e. sequence of open sets. Given m, let (σ k ) k∈N be a listingof the minimal strings σ under the prefix ordering such that [σ] ⊆ G m . Then1 ≥ ∑ i λS i ≥ ∑ ∑i k λ(S i ∩ [σ k ]) ≥ 2 m ∑ k 2−|σk| =2 m λG m .Thus λG m ≤ 2 −m , and (G m ) m∈N is a Martin-Löf test. Since Z ∈ ⋂ m G m, Z isnot Martin-Löf random.✷We give two applications of Solovay tests. The first is to show that ML-randomsets only have short runs of zeros. By Proposition 2.5.7 the length of a run of zerosin an incompressible K string of length m is bounded by 4 log m+O(1). If the runstarts at the n-th bit position of a Martin-Löf random set, then its length is atmost K(n) (for almost all n). This is rather short since K(n) ≤ + log n+2 log log nby Proposition 2.2.8.3.2.20 Proposition. Let Z be ML-random. Then for each r, for almost all n,a run of zeros starting at position n has length at most K(n) − r.Proof. Recall from (2.6) that K t (n) is a nonincreasing computable approximationof K(n) at stages t. Forn>0, letS n,t =[{σ0 max(Kt(n)−r,0) : |σ| = n}] ≺ ,and let S n = ⋃ t S n,t. Then (S n ) n∈N is a uniformly c.e. sequence of open sets.Moreover, λS n = 2 −K(n)+r , so that ∑ n λS n ≤ Ω2 r . The ML-random set Zpasses this test, that is, for almost all n we have Z ∉ S n . This implies that theruns of zeros in Z satisfy the required bound on the length.✷3.2.21 Proposition. Z is ML-random ⇔ lim n K(Z ↾ n ) − n = ∞.Informally, for every set Z the function λn. K(Z ↾ n ) avoids being close to n:either for each d there is n such that it dips below n − d (when Z is not MLrandom),or for each b eventually it exceeds n + b (when Z is ML-random).

112 3 Martin-Löf randomness and its variantsProposition 3.2.21 will be improved in Theorem 7.2.8: Z is ML-random ⇔∑n 2−K(Z↾n)+n < ∞.Proof. ⇐: This is immediate by Theorem 3.2.9.⇒: Suppose that there is b such that ∃ ∞ nK(Z ↾ n ) − n = b. Let S n =[{x: |x| =n & K(x) ≤ n + b}] ≺ , then Z is in infinitely many sets S n , so we can concludethat Z is not ML-random once we have shown that (S n ) n∈N is a Solovay test.Clearly (S n ) n∈N is uniformly c.e. Applying Theorem 2.2.26(i) where d = K(n)−b,the number of strings x of length n such that K(x) ≤ n + b is at most 2 c 2 n−d =2 c 2 n−K(n)+b . Hence λS n ≤ 2 c 2 −K(n)+b , and ∑ n λS n < 2 c+b Ω < ∞. ✷3.2.22 Remark. Let (S i ) i∈N be a Solovay test. By Fact 1.8.26, each open set S iis generated by a uniformly computable prefix-free set of strings. We might aswell put the basic open cylinders corresponding to the strings in all those prefixfreesets together, because failing the test is equivalent to being in infinitely manyintervals. In this way we obtain an equivalent Solovay test where each open setis just a basic open cylinder [x]. Thus, we may alternatively represent a Solovaytest G by an effective listing of strings x 0 ,x 1 ,...(possibly with repetitions) suchthat ∑ i 2−|xi| < ∞. Usually we will not distinguish between the two types ofrepresentation; if we want to stress this difference we will call G an intervalSolovay test. For instance, in the foregoing proof, the interval Solovay test is aneffective listing of the set {x: K(x) ≤ |x| + b}.3.2.23 Example. Let B be a c.e. prefix-free set such that λ[B] ≺ < 1. Foreach n let S n be the open set generated by the n-th power B n of B, namelyS n =[{x 1 ...x n : ∀i x i ∈ B}] ≺ . Clearly, λS n =(λS 1 ) n . Since λS 1 < 1, we mayconclude that (S n ) n∈N is a Solovay test. If λS 1 ≤ 1/2 then (S n ) n∈N is in fact aML-test.Tails of sets were defined after Prop. 3.2.13. The following is due to Kučera.3.2.24 Proposition. Suppose P is a Π 0 1 class such that λP > 0. Then eachML-random set Z has a tail in P .Proof. Let B ⊆ {0, 1} ∗ be a c.e. prefix-free set such that [B] ≺ =2 N − P . Definethe Solovay test (S n ) n∈N as in 3.2.23. Since Z is ML-random, there is a leastn ∈ N such that Z ∉ S n .Ifn>0, let x ≺ Z be shortest such that x ∈ B n−1 ,otherwise let x = ∅. Then the tail Y of Z obtained by taking x off is not in[B] ≺ , and hence Y ∈ P .✷In particular, if P ⊆ MLR (say, P =2 N − R b for some b), then Z is ML-randomiff some tail of Z is in P . For, if some tail of Z is in P , then Z is ML-random byProposition 3.2.15.Exercises.3.2.25. Give an alternative proof of Proposition 3.2.20 using 3.2.21.3.2.26. Use Solovay tests to show that each ML-random set satisfies the law of largenumbers.

3.2 Martin-Löf randomness 113Left-c.e. ML-random reals and Solovay reducibilityWe use the term “real” for real number. Via the identifications in Definition 1.8.10we can apply definitions about sets of natural numbers to reals.Solovay reducibility ≤ S is used to compare the randomness content of left-c.e.reals. We obtain two characterizations of the left-c.e. ML-random reals:(a) They are the reals of the form Ω R for a optimal prefix-free machine R. Thus,Theorem 3.2.11 accounts for all the left-c.e. ML-random reals in [0, 1).(b) They are the Solovay complete left-c.e. reals. Thus, they play a role similarto the creative sets for many-one reducibility on the c.e. sets.We identify the Boolean algebra of clopen sets in Cantor space with the Booleanalgebra Intalg [0, 1) R as in 1.8.11. We let α, β, γ denote left-c.e. reals. If γ is leftc.e.then there is a computable sequence of binary rationals (γ s ) s∈N such thatγ s ≤ γ s+1 for each s and γ = sup s γ s (Fact 1.8.15). We say that (γ s ) s∈N is anon-decreasing computable approximation of γ.First we show that adding a left-c.e. real to a left-c.e. ML-random real keepsit ML-random.3.2.27 Proposition. Suppose α, β are left-c.e. reals such that γ = α + β < 1.If α or β is ML-random, then γ is ML-random.Proof. Suppose that (G n ) n∈N is a ML-test such that γ ∈ ⋂ n G n. We showthat α is not ML-random. We may assume that λG n ≤ 2 −n−1 , and that β isnot a binary rational. Let (α s ) s∈N and (β s ) s∈N be nondecreasing computableapproximations of α, β, and let γ s = α s + β s . We enumerate a ML-test (H n ) n∈Nsuch that α ∈ ⋂ n H n.At stage s, ifγ s ∈ I where I =[x, y) is a maximal subinterval ofG n,s , then put the interval J =[x − β s − (y − x),y− β s )intoH n .If in fact γ ∈ I, then since β s < β < β s +(y − x) and x ≤ γ we have y − β s >γ − β s > α = γ − β; also x − β s − (y − x) ≤ x − β ≤ α. Thusα ∈ J. The lengthof an interval J is twice the length of an interval I, so that λH n ≤ 2 −n . ✷The converse of Proposition 3.2.27 was proved by Downey, Hirschfeldt and Nies(2002): if γ is a ML-random left-c.e. real and γ = α+β for left-c.e. reals α, β, thenα or β is ML-random. They also introduced the following algebraic definition of≤ S . It is equivalent to the original definition of Solovay (1975) by Exercise 3.2.33.3.2.28 Definition. Let α, β ∈ [0, 1) be left-c.e. reals. We write β ≤ S α if∃d ∈ N ∃γ left-c.e. [ 2 −d β + γ = α ] .Thus, Solovay completeness in (b) above yields an algebraic characterization ofbeing ML-random within the left-c.e. reals. By Exercise 3.2.33 ≤ S is transitiveand implies ≤ T .We proceed to the main result. The implications (iii)⇒(ii)⇒(i) are due toCalude, Hertling, Khoussainov and Wang (2001), and (i)⇒(iii) to Kučera andSlaman (2001).

114 3 Martin-Löf randomness and its variants3.2.29 Theorem. The following are equivalent for a real α ∈ [0, 1).(i) α is left-c.e. and ML-random.(ii) There is an optimal prefix-free machine R such that Ω R = α.(iii) α is Solovay complete, that is, β ≤ S α for each left-c.e. real β.Proof. (iii)⇒(ii). Choose d ∈ N and a left-c.e. real γ such that 2 −d Ω + γ = α.We define a bounded request set L of total weight α such that the associatedprefix-free machine R is optimal. Note that Ω R = α.Choose a nondecreasing computable approximation (γ s ) s∈N of γ. We may assumethat γ s+1 − γ s is either 0 or of the form 2 −n for some n.Construction of L. Let L 0 = ∅.Stage s>0. If U s−1 (σ) ↑ and U s (σ) =y, put the request 〈|σ| + d, y〉 into L sunless it is already in L s−1 ; in this case, to record the increase of Ω in L, put〈|σ| + d, i〉 into L s for a number i>snot mentioned so far. If γ s+1 − γ s =2 −nthen put 〈n, j〉 into L s for a number j>snot mentioned so far.It is clear that L is a bounded request set as required.(ii)⇒(i). Ω M is left-c.e. for each prefix-free machine M. For an optimal prefix-freemachine R, the real Ω R is ML-random by Theorem 3.2.11.(i)⇒(iii). Suppose the left-c.e. real β ∈ [0, 1) is given. Let (β s ) s∈N and (α s ) s∈Nbe nondecreasing computable approximations of β and α, respectively, whereα s , β s ∈ Q 2 . For each parameter d ∈ N we build a left-c.e. real γ d uniformly in d,attempting to ensure that 2 −d β + γ d = α. In the end we will define a ML-test(G d ) d∈N .Ifd is a number such that α ∉ G d then we succeed with γ d .The construction with parameter d runs only at active stages; 0 is active. If sis active, and t is the greatest active stage less than s, then let ɛ s =2 −d (β s −β t ).We wish that α increase by an amount of at least ɛ s in order to record theincrease of β. So we put the interval [α s , α s + ɛ s )intoG d .Ifα ∉ G d , then α willincrease eventually, and in that case we have reached the next active stage. If αhas increased too much the excess is added to γ. For the formal construction, itis easier to first update γ, and then define the next interval.Construction for the parameter d.Stage 0 is declared active. Letq 0 =0and γ d,0 =0.Stage s>0. Let t 1 then stop.γ d,s − γ d,t 2 −d (β s − β t )q tα sq s

3.2 Martin-Löf randomness 115Verification. Let G d = ⋃ {[α s ,q s ): s active}. Then G d is (identified with) a c.e.open set in Cantor space uniformly in d, and λG d ≤ 2 −d β.Thus(G d ) d∈N isa ML-test. Choose d such that α ∉ G d , then the construction for parameter dhas infinitely many active stages. Inductively, between (1) and (2) of each activestage s>0wehave2 −d β t + γ d,s = α s , where t is the preceding active stage.Hence 2 −d β + γ d = α, where γ d = sup s is active γ d,s .✷The following fact of Calude and Nies (1997) can alternatively be proved usingTheorem 4.1.11 below.3.2.30 Proposition. ∅ ′ ≡ wtt Ω R for each optimal prefix-free machine R.Proof. Clearly Ω R ≤ wtt ∅ ′ . To show that ∅ ′ ≤ wtt Ω R , define a prefix-free machineM by M(0 n 1) = s if n ∈∅ ′ at s. Let d be the coding constant for M withrespect to R. The reduction procedure is as follows: on input n, using the oracleΩ R , compute t such that Ω R ↾ n+d+1 = Ω R,t ↾ n+d+1 . Output ∅ ′ t(n).If n enters ∅ ′ at a stage s>tthen Ω R increases by at least 2 −(n+d+1) , contraryto the choice of t.✷3.2.31 Corollary. Every ML-random left-c.e. set is wtt-complete. ✷We cannot hope to characterize the ML-random ω-c.e. sets as easily: such a setcan be superlow by 3.2.10(ii), but it can also be weak truth-table complete.As an immediate consequence of Theorem 3.2.29, Z is right-c.e. and ML-random iffthere is an optimal prefix-free machine R such that 1 − Ω R =0.Z. How about MLrandomreals that are difference left-c.e.? (See (iv) of Definition 1.8.14.) Rettinger(unpublished) has shown that one does not obtain anything new.3.2.32 Proposition. Let r ∈ [0, 1) R be difference left-c.e. and ML-random. Then r iseither left-c.e. or right-c.e.Proof. Assume r is neither. By Fact 1.8.15, r = lim iq i for an effective sequence(q i) i∈N of dyadic rationals such that ∑ iabs(qi+1 − qi) < ∞. For each m ∈ N wehave sup i≥m q i >r, otherwise r = lim i≥m max{q k : m ≤ k ≤ i}, sor is left-c.e. Similarly,inf i≥m q i

116 3 Martin-Löf randomness and its variants3.2.34 Remark. Fix a base b ∈ N such that b ≥ 2, and write b N for the set of functionsN → {0,...,b−1}. Product topology and uniform measure can be defined on the set b N .One can think of the elements of b N as the overall result of a sequence of experimentswith b outcomes, each one occurring with probability 1/b. For instance if b = 6 theexperiment could be rolling a dice. Similar to (1.13), define a mapF b : {Z ∈ b N : Z is not eventually periodic} → [0, 1) R − Qby F b (Z) = ∑ i Z(i)b−i−1 . This map preserves topology and measure in both directions.One can also adapt the definition of ML-randomness in 3.2.1 to base b. It is easy tocheck that F b preserves ML-randomness in both directions. Thus, if Z ∈ b N is noteventually periodic, then Z is ML-random ↔ F b (Z) is ML-random (in the sense ofreals) ↔ the set F −12 (F b (Z)) is ML-random in the sense of Definition 3.2.1. Thus,ML-randomness is a base-independent concept.Giving preference to base 2 is not necessary but convenient, because in computabilitytheory one studies subsets of N rather than functions in b N for b>2.A nonempty Π 0 1 subclass of MLR has ML-random measure ⋆We consider the uniform measure of nonempty Π 0 1 subclasses of MLR, suchas2 N − R 1.This yields examples of right-c.e. ML-random reals other than 1 − Ω.3.2.35 Theorem. If P ⊆ MLR is a nonempty Π 0 1 class then λP is ML-random.Proof. Note that λP > 0, since otherwise P ∩ MLR = ∅ by 3.2.2. Suppose that λPis not ML-random, and let Z be the co-infinite set identified with the real number λP(namely, λP =0.Z).Firstly we show that for each b ∈ N, there is y on P such that K(y) ≤ |y| − b. Weuse the fact that an appropriate initial segment x of Z is sufficiently compressible inthe sense of K to obtain a b-compressible K string y that is guaranteed to be on P :ifyfalls off P then the measure decreases so much that the approximation 0.x is wrong.Let (P t) t∈N be the effective approximation of P by clopen sets from (1.17) on page 55.The prefix-free machine M works as follows on an input σ.(1) Wait for s such that U s(σ)↓= x. Let n = |x|. Let c = ⌊(n − |σ|)/2⌋.(2) Wait for t ≥ s such that 0.x ≤ λP t < 0.x +2 −n .(3) If there is a string y of length n − c such that λ(P t ∩ [y]) ≥ 2 −n then output theleftmost such y.If x ≺ Z and M outputs y, then y is on P , for otherwise [y] ∩ P = ∅ and henceλ(P t − P ) ≥ 2 −n where n = |x|. Let d be a coding constant for M. Given b ∈ N,since we are assuming that λP is not ML-random, there is x ≺ Z that is sufficientlycompressible in the sense of K, namely, b + d ≤ (n − K(x))/2, where n = |x|; wemayalso require that 2 −c ≤ λP , where c = ⌊(n − K(x))/2⌋. Thus there is some y of lengthn − c such that λ(P ∩ [y]) ≥ 2 −(n−c) 2 −c =2 −n .If σ is a shortest U-description of x, then M on input σ outputs a string y on P suchthat |y| = n − c. ThusK M (y) ≤ |y| − c ≤ |y| − (b + d) and hence K(y) ≤ |y| − b.We now iterate the foregoing argument and obtain a sequence of strings y 0 ≺ y 1 ≺ ...such that each y i is on P and K(y i) ≤ |y|−i. Then the set Y = ⋃ y i is not ML-randomand Y ∈ P .For S ⊆ 2 N let S c =2 N − S. Let y 0 = ∅. Suppose i>0 and y i−1 has been defined.Then λ(P ∩ [y i−1]) is not ML-random by Proposition 3.2.27: the left-c.e. real number

3.3 Martin-Löf randomness and reduction procedures 117λP c is not ML-random, and for each length m, wehaveλP c = ∑ |y|=m λ(P c ∩ [y]), sothat λ(P c ∩ [y]) is not ML-random for any y. So we may apply the argument above tothe Π 0 1 class P ∩ [y i−1] in order to obtain y i ≻ y i−1 on P such that K(y i) ≤ |y i| − i.✷3.3 Martin-Löf randomness and reduction proceduresWe are mostly interested in the interactions between the degree of randomnessand the absolute computational complexity of sets; a summary of such interactionswill be given in Section 8.6. However, here we address the interaction ofrandomness with the relative computational complexity of sets. Firstly, we considerreducibilities, and then, in the next section, we look at ML-randomnessrelative to an oracle.Each set is weak truth-table reducible to a ML-random setAn arbitrarily complex set A can be encoded into an appropriate Martin-Löfrandom set: there is a ML-random set Z such that A ≤ wtt Z. This result wasobtained independently by Kučera (1985) and by Gács (1986). For instance, thereis a ML-random set Z weak truth-table above ∅ ′′ . Thus, ML-random sets can haveproperties that fail to match our intuition on randomness, a view also supportedby the existence of left-c.e. ML-random sets. These particular properties arealready incompatible with the somewhat stronger notion of weak 2-randomnessintroduced in 3.6.1 below. Ultimately, the reason why the coding is possible forML-random sets, but not for sets satisfying a stronger randomness property, isthat MLR contains a Π 0 1 class of positive measure, for instance 2 N − R 1 . We willprovide a mechanism for encoding a set A into members of such a Π 0 1 class. Itrelies on a simple measure theoretic lemma. Recall from 1.9.3 that for measurableS ⊆ 2 N , λ(S|z) is the local measure 2 |z| λ(S ∩ [z]). For each n, λS is the average,over all strings z of length n, of the local measures λ(S|z).3.3.1 Lemma. Suppose that S ⊆ 2 N is measurable and λ(S|x) ≥ 2 −(r+1) wherer ∈ N. Then there are distinct strings y 0 ,y 1 ≽ x, |y i | = |x| + r +2, such thatλ(S|y i ) > 2 −(r+2) for i =0, 1.Proof. We may assume that x = ∅. Let y 0 be a string of length r + 2 such thatλ(S|y 0 ) is greatest among those strings. Assume that λ(S|y) ≤ 2 −(r+2) for eachy ≠ y 0 of length r + 2. ThenλS =∑|y|=r+2λ(S ∩ [y]) = λ(S ∩ [y 0 ]) +∑y≠y 0&|y|=r+2λ(S ∩ [y])≤ 2 −(r+2) +(2 r+2 − 1)2 −(r+2) 2 −(r+2) < 2 −(r+1) . ✸3.3.2 Theorem. Let Q be a nonempty Π 0 1 class of ML-random sets. Then foreach set A there is Z ∈ Q such that A ≤ wtt Z ≤ wtt A ⊕∅ ′ . The wtt-reductionsare independent A.

118 3 Martin-Löf randomness and its variantsProof. Since λQ >0, by Theorem 1.9.4 there is a string σ such that λ(Q | σ) ≥1/2, so we may as well assume that λQ ≥ 1/2. Let f be the function given byf(0) = 0 and f(r +1)=f(r)+r + 2 (namely, f(r) =r(r +3)/2). Let ̂Q be theΠ 0 1 class of paths through the Π 0 1 treeT = {y : ∀r [f(r) ≤ |y| → λ(Q|(y ↾ f(r) )) ≥ 2 −(r+1) ]}. (3.4)Note that ̂Q ⊆ Q. Since λQ ≥ 1/2, by Lemma 3.3.1 ̂Q is nonempty.The set Z will be a member of ̂Q. Suppose that so far we have coded A ↾rinto the initial segment x of Z of length f(r). The idea is to code A(r) intothe initial segment of length k = f(r + 1), as follows: if A(r) =0,Z takes theleftmost length k extension of x which is on ̂Q, otherwise it takes the rightmostone. By the lemma above, these two extensions are distinct. Knowing Z ↾ k ,wemay enumerate the complement of ̂Q till we see which case applies. In this waywe determine A(r).The details are as follows. We define strings (x τ ) τ∈{0,1} ∗ on ̂Q such that |x τ | =f(|τ|). Let x ∅ = ∅. Ifx τ has been defined, let x τ0 be the leftmost y on ̂Qsuch that x τ ≺ y and |y| = f(|τ| + 1), and let x τ1 be the rightmost such y. ByLemma 3.3.1, x τ0 and x τ1 exist and are distinct.For each A, the ML-random set Z coding A simply is the path ⋃ τ≺A x τ of Tdetermined by A.Firstly, we describe a reduction procedure for A ≤ wtt Z, where f(r +1) boundsthe use for input r. To determine A(r) let x = Z ↾ f(r) and y = Z ↾ f(r+1) . Find ssuch that̂Qs ∩ [{v ≽ x: |v| = |y| & v< L y}] ≺ = ∅, or̂Q s ∩ [{v ≽ x: |v| = |y| & v> L y}] ≺ = ∅.In the first case, output 0. In the second case, output 1.Next we check that Z ≤ wtt A ⊕∅ ′ . Suppose that x = Z ↾ f(r) has been determined.If A(r) = 0, with ∅ ′ as an oracle find the leftmost extension y of x on̂Q such that |y| = f(r + 1). Otherwise, with ∅ ′ as an oracle find the rightmostsuch extension y of x. Then y = Z ↾ f(r+1) . Clearly the use on ∅ ′ is bounded bya computable function. Also, the wtt-reductions employed do not depend on theparticular set A.✷For oracles other than Z, the search in the reduction procedure for A ≤ wtt Z may notterminate. Recall from 1.2.20 that a Turing reduction is called a truth-table reductionif it is total for all oracles. In Theorem 4.3.9 below we show that ∅ ′ ≰ tt Z for eachML-random Z. Sothewtt-reduction obtained above must be partial for some oracles.In the exercises we indicate a proof of Theorem 3.3.2 for Q =2 N − R 1 closer toKučera’s original proof, avoiding the class ̂Q. Kučera actually did not use K for hiscoding. He used an idea taken from the proof of Gödel’s incompleteness theorem. Asimilar coding works for the Π 0 1 class of two-valued d.n.c. functions from Fact 1.8.31.The proof of Gács (1986) used martingales. We will apply his method in Lemma 7.5.6.Exercises.3.3.3. ⋄ Show that given an effective listing (P e ) e∈N of Π 0 1 classes, one may effectivelyobtain a constant c ∈ N such that λ(P e ∩ Q) ≤ 2 −K(e)−c → P e ∩ Q = ∅.

3.3 Martin-Löf randomness and reduction procedures 119Hint. Use Theorem 2.2.17 and Remark 2.2.21.3.3.4. Show that there is c ∈ N such that if x is on Q then λ(Q ∩ [x]) ≥ 2 −K(x)−c .Now let h(n) = 2 log(n) +n + c K, where by Proposition 2.2.8 the constant c K ischosen so that ∀x [K(x) 2 −h(|x|) for every x on Q. Hencethere are distinct strings y 0,y 1 ≻ x on Q such that |y 0| = |y 1| = h(|x|). Use this todefine (x τ ) τ∈{0,1} ∗ on Q as before, where |x τ | = h (|τ|) (0).Autoreducibility and indifferent sets ⋆A is called autoreducible (Trahtenbrot, 1970) if there is a Turing functional Φsuch that∀x [A(x) =Φ(A − {x}; x)]. (3.5)Thus one can determine A(x) via queries to A itself, but distinct from x. Intuitively,A is redundant. For example, each set Y ⊕ Y is autoreducible via thefunctional Φ defined by Φ(A;2n + a) =A(2n +1− a) (n ∈ N,a∈ {0, 1}). Thuseach many-one degree contains an autoreducible set.Autoreducibility is a bit like the following hat game. Players sit around a table. Eachone has to determine the color of his hat. A player cannot see his own hat, only thehats of the other people. He has to derive his hat color from this information and someknown assumptions about the distribution of hat colors.3.3.5 Proposition. Some low c.e. set A is not autoreducible.Proof sketch.The P e -strategy is as follows.To ensure that A is not autoreducible, we meet the requirementsP e : ∃x ¬A(x) =Φ e (A − {x}; x).(1) Choose a large number x.(2) When Φ e (A − {x}; x) = 0 enumerate x into A and initialize the strategiesfor weaker priority requirements P i , i>e. The initialization is an attempt topreserve the computation Φ e (A − {x}; x) =0.To make A low we satisfy the usual lowness requirements (1.10) on page 32.This merely needs some extra initialization of the P e strategies.✷In the following, for a string σ, i

120 3 Martin-Löf randomness and its variants{σ : ∃x 0

3.4 Martin-Löf randomness relative to an oracle 121to an oracle. In this section we study these concepts in relativized forms. Forinstance, we interpret the definition of ML-randomness in 3.2.1 relative to anoracle A:(i) A ML-test relative to A is a sequence (G A m) m∈N of uniformly c.e. relative to Aopen sets such that ∀m ∈ N λG A m ≤ 2 −m .(ii) Z ⊆ N fails the test if Z ∈ ⋂ m GA m.(iii) Z is ML-random relative to A, orML-random in A, ifZ passes each ML-testrelative to A. MLR A denotes the class of sets that are ML-random relative to A.Note that MLR A is conull and MLR A ⊆ MLR for each A. More generally, foreach A, B we have B ≤ T A → MLR B ⊇ MLR A : the stronger the oracle iscomputationally, the stronger are the tests, and therefore the harder for a set toescape them.In this section we study relative randomness as a binary relation between sets Zand A. An important fact is the symmetry of relative randomness, Theorem 3.4.6.Moreover, in Theorem 5.1.22 of Section 5.1 we consider the situation that someset Z is ML-random in A and also Z ≥ T A. We show that this is a strong lownessproperty of a set A.In Section 3.6 we fix A; mostly A will be ∅ (n−1) for some n>0:3.4.1 Definition. Let n>0. A set Z is called n-random if Z is ML-randomrelative to ∅ (n−1) .Thus, 1-randomness is the same as ML-randomness. Recall that 1-random setsmay fail to match our intuition of randomness because of facts like the Kučera-Gács Theorem 3.3.2, or the existence of a left-c.e. ML-random set. In Section 3.6we will see that both concerns are no longer valid for 2-randomness.We can also fix a ML-random set Z and consider the class of oracles A suchthat Z is ML-random in A. ForZ = Ω this yields the lowness property of beinglow for Ω, defined in 3.6.17 and studied in Section 8.1.Relativizing C and KA Turing functional viewed as a partial map M :2 N × {0, 1} ∗ → {0, 1} ∗ is calledan oracle machine. Thus, M is an oracle machine if there is e ∈ N such thatM(A, σ) ≃ Φ A e (σ) for each oracle A and string σ. We extend Definition 2.1.2,namely, we let V A (0 e−1 1ρ) ≃ Φ A e (ρ), for each set A, each e>0, and ρ ∈ {0, 1} ∗ .We write C A (x) for the length of a shortest string σ such that V A (σ) =x.3.4.2 Definition. (i) A prefix-free oracle machine is an oracle machine M suchthat, for each set A, the domain of M A is prefix-free. We let Ω A M = λ(dom M A ).K M A(x) denotes the length of a shortest string σ such that M A (σ) =x.(ii) A prefix-free oracle machine R is optimal if for each prefix-free oracle machineM there is a constant e M such that ∀A ∀x [K R A(x) ≤ K M A(x) +e M ].Thus, the coding constant is independent of the oracle.To show that an optimal prefix-free oracle machine exists, we view the proofof Proposition 2.2.7 relative to an oracle A. We may in fact relativize Theorem2.2.9, using the Recursion Theorem in its version 1.2.10. Thus, there is an

122 3 Martin-Löf randomness and its variantsoptimal prefix-free oracle machine U such that, where K A (x) =K U A(x), we have∀x[K A (x) ≤ K A (|x|)+|x| + 1]. We letΩ A = λ[dom U A ] ≺ . (3.6)The notation Ω A is short for Ω A U . By the definition of U as an oracle machine,for each prefix-free oracle machine M there is d>1 such that∀X ∀ρ [M X (ρ) ≃ U X (0 d−1 1ρ)]. (3.7)We say that d is a coding constant for M (with respect to U).Exercises. For a string α, C α (x) denotes the length of a shortest string σ such thatV α (σ) =x, and K α (x) denotes the length of a shortest string σ such that U α (σ) =x.3.4.3. (i) Show that C(x | α) ≤ + C α (x). (ii) Show that for each n there is α of length nsuch that C α (n) ≥ + C(n) (while C(n | α) =O(1)).3.4.4. Show that for each A, B such that A ≤ T B we have ∃d ∀yK B (y) ≤ K A (y)+d.Basics of relative ML-randomnessThe relativized form of Theorem 3.2.9 is:Z is ML-random relative to A ⇔ ∃b ∀n K A (Z ↾ n ) >n− b.In other words, (R A b ) b∈N is a universal ML-test relative to A, where R A b=[{x ∈{0, 1} ∗ : K A (x) ≤ |x| − b}] ≺ . We obtain examples of sets that are ML-randomrelative to A by relativizing the results on page 108. In particular, Ω A definedin (3.6) is ML-random in A.Fact 3.2.2 that no computable set is ML-random can be relativized:3.4.5 Fact. If Z ≤ T A then Z is not ML-random relative to A.Proof. Suppose Z = Φ A for a Turing reduction Φ. Let G A m =[Φ A ↾ m ], then(G A m) m∈N is a ML-test relative to A and Z ∈ ⋂ m GA m. ✷Symmetry of relative Martin-Löf randomnessPerhaps the most important fact on relative ML-randomness as a relation onsets is the theorem of van Lambalgen (1987) that A ⊕ B is ML-random ⇔ B isML-random and A is ML-random relative to B. By Proposition 3.2.16 A ⊕ Bis ML-random ⇔ B ⊕ A is ML-random. So, we also have that A ⊕ B is MLrandom⇔ A is ML-random and B is ML-random relative to A. Thus, relativeML-randomness is a symmetric relationship between sets A and B that are bothML-random: A is ML-random in B ⇔ B is ML-random in A. This is surprisingbecause on the left side we consider the randomness aspect of A and on the rightside its computational power.3.4.6 Theorem. Let A, B ⊆ N. Then A⊕B is ML-random ⇔ B is ML-randomand A is ML-random relative to B.

3.4 Martin-Löf randomness relative to an oracle 123Proof. ⇒: For a string β we let R β b =[{x ∈ {0, 1}∗ : K β (x) ≤ |x| − b}] ≺ .Asinthe proof of Proposition 3.2.7 we have λR β b ≤ 2−b for each string β. (We couldalso use some other universal oracle ML-test here.)If A⊕B is ML-random then B is ML-random by Proposition 3.2.16, where thecomputable one-one function f is given by f(n) =2n +1.Suppose that A is not ML-random in B. Then A ∈ ⋂ b RB b. We show thatA ⊕ B ∈ ⋂ b G b for some ML-test (G b ) b∈N . For each b, n ∈ N letG b (n) =[{u ⊕ β : |u| = |β| = n & ∃x ≼ uK β (x) ≤ |x| − b}] ≺ .Then G b (n) is a c.e. open set uniformly in b and n, andλG b (n) ≤ ∑ β 2−n λR β b [[|β| = n]] ≤ 2−b .Clearly G b (n) ⊆ G b (n + 1) for each n. Let G b = ⋃ n G b(n), then (G b ) b∈N is aML-test. If A ∈ ⋂ b RB b then for each b there is x ≺ A such that KB (x) ≤ |x| − b,and thus K B↾n (x) ≤ |x| − b for some n ≥ |x|. Then A ⊕ B ∈ G b (n).⇐: Suppose A⊕B is not ML-random, then A⊕B ∈ ⋂ d V d for a ML-test (V d ) d∈Nsuch that λV d ≤ 2 −2d for each d. Firstly, we build a Solovay test (S d ) d∈N in anattempt to show that B is not ML-random. For a string x let [∅⊕x] denote theclopen set {Y 0 ⊕ Y 1 : x ≺ Y 1 }. LetS d = ⋃ {[x] :λ(V d ∩ [∅⊕x]) ≥ 2 −d−|x| }.By Fact 1.9.16, S d is a c.e. open set uniformly in d. We claim that λS d ≤ 2 −d .Let (x i ) be a listing of the minimal strings x (under the prefix relation) suchthat λ(V d ∩ [∅ ⊕x]) ≥ 2 −d−|x| . Then S d = ⋃ i [x i]. Since the sets V d ∩ [∅ ⊕x i ]are pairwise disjoint and λV d ≤ 2 −2d , we see that ∑ i 2−d−|xi| ≤ 2 −2d and henceλS d = ∑ i 2−|xi| ≤ 2 −d .If B ∈ S d for infinitely many d then B fails the Solovay test, and hence is notML-random. Now suppose there is d 0 such that B ∉ S d for all d ≥ d 0 . LetH d (n) =[{w : |w| = n &[w ⊕ B ↾ n ] ⊆ V d }] ≺ .Then λH d (n) ≤ 2 −d for each d ≥ d 0 since B ∉ S d . Moreover H d (n) ⊆ H d (n +1)for each n. Let H d = ⋃ n H d(n), then λH d ≤ 2 −d , and H d is a c.e. open setrelative to B uniformly in d. Since A ∈ H d for each d ≥ d 0 , A is not ML-randomrelative to B.✷Since MLR A is conull for each A, the symmetry of relative ML-randomnessshows that a ML-random set is ML-random relative to almost every set:3.4.7 Corollary. For ML-random A, λ{B : A is ML-random in B} =1. ✷The following is an immediate consequence of the implication “⇒” in Theorem3.4.6 and Fact 3.4.5. It yields an alternative proof of the Kleene–Post Theorem1.6.1, and shows that no ML-random set is of minimal Turing degree.3.4.8 Corollary. If A ⊕ B is ML-random then A | T B. ✷The van Lambalgen Theorem holds for k-randomness (k ≥ 1), as well as for thenotions of Π 1 1-ML-randomness and Π 1 1-randomness introduced in Chapter 9. However,

124 3 Martin-Löf randomness and its variantsit also fails for some important randomness notions, such as Schnorr randomness (3.5.8).See Remark 3.5.22.3.4.9. ⋄ Exercise. Let k ≥ 1. Then A ⊕ B is k-random ⇔ B is k-random and A isML-random relative to B (k−1) (that is, A is k-random relative to B).Computational complexity, and relative randomnessWe prove two results of independent interest that will also be applied later. Theyare due to Nies, Stephan and Terwijn (2005).In 3.2.10 we built a (super)low ML-random set by applying the Low BasisTheorem in the version 1.8.38 to the Π 0 1 class 2 N − R 1 . The proof of 1.8.38 isan explicit construction of the low set. We may also obtain a low ML-randomset A by a direct definition: take as A the bits in the odd positions of anyML-random ∆ 0 2 set, say Ω. Such a set is low by the following more generalfact which also applies to sets A not in ∆ 0 2. Recall from Definition 1.5.4 thatGL 1 = {A: A ′ ≡ T A ⊕∅ ′ }.3.4.10 Proposition. If some ∆ 0 2 set Z is ML-random in A then A is in GL 1 .Proof. Fix a computable approximation (Z t ) t∈N of Z (see Definition 1.4.1). Letf : N → N be the function given by f(r) =µs ∀t ≥ s [Z t ↾ r = Z s ↾ r]. Recallthat J A (e) ≃ Φ A e (e). Let Ĝe be the open set, uniformly c.e. in A, given by{[Z se ↾ e+1 ] if s e is the stage at which J A (e) convergesĜ e =∅ if J A (e)↑ .Let G n = ⋃ e≥n Ĝe, then (G n ) n∈N is a ML-test relative to A. Since Z/∈ ⋂ n G n,only finitely many of the Ĝe contain Z. Thus f(e) ≥ s e for almost all e such thatJ A (e)↓. Hence, for almost all e, wehaveJ A (e)↓↔ Jf(e) A (e)↓. Since f ≤ T ∅ ′ , thisimplies that A ′ ≤ T A ⊕∅ ′ . (Also see Exercise 4.1.14.)✷3.4.11 Corollary. Suppose the ∆ 0 2 set A = A 0 ⊕ A 1 is ML-random.Then A 0 and A 1 are low.Proof. The ∆ 0 2 set A 1 is ML-random in A 0 by Theorem 3.4.6. Thus A 0 is in GL 1 ,and hence A 0 is low. The same argument applies to A 1 .✷In 3.4.17 we proved that Ω ∅′ is high. So a high set can be in GL 1 :3.4.12 Corollary. Ω ∅′ is in GL 1 .Proof. Ω ∅′ is ML-random relative to ∅ ′ ≡ T Ω, so by Theorem 3.4.6 Ω is MLrandomrelative to Ω ∅′ .ThusΩ ∅′ is in GL 1 by Proposition 3.4.10. ✷Our second result will be applied, for instance, in the proof of Theorem 5.1.19.3.4.13 Proposition. Let B be c.e., Z be ML-random, and suppose that∅ ′ ≰ T B ⊕ Z. Then Z is ML-random relative to B.

3.4 Martin-Löf randomness relative to an oracle 125Proof idea. Suppose Z is not ML-random relative to B. ThusZ ∈ ⋂ d∈N RB d .Since B ⊕ Z ≱ T ∅ ′ , infinitely many numbers x enter ∅ ′ after a stage where Zenters R B x with B correct on the use. This allows us to convert (R B d ) d∈N into anunrelativized ML-test (S d ) d∈N such that Z fails this test.Proof details. Let R B d[s] =[{x: KBs s (x) ≤ |x| − d}] ≺ be the approximation ofR B dat stage s. Notice that λRB d [s] ≤ 2−d for each s. An enumeration of Z intoR Y dat a stage s is due to a computation UY (σ) =z where z ≺ Z convergingat s, and hence has an associated use on the oracle Y . The following function iscomputable in B ⊕ Z:f(x) =µs. Z ∈ R B x [s] with use u & B s ↾ u = B ↾ u ].Because B s ↾ u = B ↾ u we have Z ∈ R B [t] for all t ≥ s (here we need that Bis c.e., not merely ∆ 0 2). Let m(x) ≃ µs. x ∈∅ ′ s. Then ∃ ∞ x ∈∅ ′ [m(x) ≥ f(x)],otherwise one could compute ∅ ′ from B ⊕ Z because, for almost all x, x ∈∅ ′ ↔x ∈∅ ′ f(x) . Let S d = ⋃ x>d RB x [m(x)]. The sequence (S d ) d∈N is uniformly c.e., andµS d ≤ 2 −d . Also, Z ∈ ⋂ d S d because m(x) ≥ f(x) for infinitely many x. Thiscontradicts the assumption that Z is ML-random.✷3.4.14 Exercise. In 3.4.10, if in addition Z is ω-c.e. and A is c.e., then A is superlow.3.4.15. ⋄ Problem. Determine whether Ω 0 can be superlow for some optimal machine.The halting probability Ω relative to an oracle ⋆Recall from (3.6) that Ω A = λ[dom U A ] ≺ . We study the operator 2 N → R givenby A ↦→ Ω A . The results are from Downey, Hirschfeldt, Miller and Nies (2005).We begin with some observations on the computational complexity of Ω A .3.4.16 Fact. A ′ ≡ T A ⊕ Ω A for each set A.Proof. By Proposition 3.2.30 ∅ ′ ≡ T Ω, so A ′ ≤ T A ⊕ Ω A by relativization. Onthe other hand A ′ ≥ T A ⊕ Ω A since Ω A is left-c.e. relative to A.✷3.4.17 Proposition. If a ∆ 0 2 set A is high then Ω A is high.Proof. By the foregoing fact, ∅ ′′ ≤ T A ′ ≡ T A ⊕ Ω A ≤ T (Ω A ) ′ .Recall from 3.4.2 that Ω A M = λ(dom M A ) for a prefix-free oracle machine M.For a string σ we let Ω σ M = λ(dom M σ ). Some facts below hold for all theoperators Ω M , while at other times it is crucial that the prefix-free machine beoptimal.3.4.18 Proposition. For each prefix-free oracle machine M, the real numbersr 0 = inf{Ω X M : X ∈ 2N } and r 1 = sup{Ω X M : X ∈ 2N } are left-c.e.Proof. For any rational q ∈ [0, 1] we have q

126 3 Martin-Löf randomness and its variantsproved in their Corollary 9.5 that for M = U the supremum is assumed by aleft-Σ 0 2 set as well. In general, operators Ω M do not assume their supremum; seeExercise 3.4.21.An operator F : 2 N → R is called lower semicontinuous at A if for all ɛ > 0there is n ∈ N such that ∀X ≻ A↾ n [F (X) >F(A)−ɛ]. Dually, F is called uppersemicontinuous at A if for all ɛ > 0 there is n ∈ N such that ∀X ≻ A↾ n [F (X) 0 ∃k ∈ N [Ω A M − Ω A↾ kM< ɛ]. (3.8)Hence Ω A M − ɛ < ΩX M for every X ≻ A↾ k.✷In the following we characterize the class of sets A such that the operatorX ↦→ Ω X is continuous at A as the 1-generic sets introduced in 1.8.51.3.4.20 Theorem. A is 1-generic ⇔ the operator X ↦→ Ω X is continuous at A.Proof. ⇒: This implication actually holds for Ω M where M is any prefix-freeoracle machine. Since Ω M is lower semicontinuous at every set, it suffices to showthat Ω M is upper semicontinuous at A. Suppose this fails for the rational ɛ. Letr ≤ Ω A M be a rational such that ΩA M − r ΩA M ,contradiction.⇐: We assume that A is not 1-generic and show that there is ɛ > 0 such that∀n ∃X ≻ A↾ n [Ω X ≥ Ω A + ɛ]. Let the c.e. open S be dense along A but A ∉ S.Let N X be the prefix-free oracle machine such that N X (∅)↓ at the first stage tsuch that [X ↾ t ] ⊆ S t . Let d>1 be the coding constant for N with respect to Uaccording to (3.7), and let ɛ =2 −d−1 . Choose k as in (3.8) for ɛ and M = U.Since N A is nowhere defined, U A↾k (0 d−1 1ρ) ↑ for each string ρ. On the otherhand, since S is dense along A, for each n ≥ k there is a set X ≻ A↾ n such thatX ∈ S. Then N X (∅)↓, soΩ X ≥ Ω A↾k +2ɛ ≥ Ω A + ɛ.✷Since the implication from left to right holds for every prefix-free oracle machine M,we have also proved that A is 1-generic ⇔ for any prefix-free oracle machine M, Ω X M iscontinuous at A. We call a prefix-free oracle machine R uniformly optimal if for eachprefix-free oracle machine N there is a fixed α such that ∀X ∀ρ [N X (ρ) ≃ R X (αρ)].The implication from right to left in 3.4.20 used that U is uniformly optimal.Theorems 5.5.14 and 8.1.2 provide further results about the operators Ω M .

3.5 Notions weaker than ML-randomness 127Exercises.3.4.21. Give an example of a prefix-free oracle machine M such thatr 1 = sup{Ω X M : X ∈ 2 N } is not assumed.3.4.22. Show that for a prefix-free oracle machine M, ifΩ A M = sup{Ω X M : X ∈ 2 N },then the operator Ω M is continuous at A.3.4.23. Show that the ML-random real 1 − Ω is not of the form Ω A for any set A.3.5 Notions weaker than ML-randomnessOn page 104 we discussed two criticisms of the notion of ML-randomness.1. Schnorr (1971) maintained that Martin-Löf tests are too powerful to beconsidered algorithmic. He suggested to study randomness notions weakerthan ML-randomness.2. ML-random sets can be left-c.e., and each set Y ≥ T ∅ ′ is Turing equivalentto a ML-random set. This is not consistent with our intuition on randomness.Thus, from an opposite point of view it also makes sense to considerrandomness notions stronger than ML-randomness.In this and the next section we vary the notion of a ML-test in order to introducerandomness notions that address these criticisms. Table 3.1 summarizesthe three main variants of the concept of a ML-test, and names the correspondingrandomness notions. Each time, the tests are u.c.e. sequences of open sets(G m ) m∈N such that ⋂ m G m is a null class, possibly with some extra conditionson the effectivity of a presentation, and how fast λG m converges to 0.Table 3.1. Variants of the concept of ML-test. Throughout (G m ) m∈N is a uniformlyc.e. sequence of open sets.Test notion Definition Randomness notionKurtz test (G m ) m∈N is an effective sequence of (weakly random)clopen sets such that λG m ≤ 2 −mSchnorr test λG m ≤ 2 −m is a computable real Schnorr randomuniformly in mMartin-Löf ∀m λG m ≤ 2 −m Martin-Löf randomtest⋂Generalizedm G m is a null classML-testweakly 2-randomAs before, we say that a set Z fails the test if Z ∈ ⋂ m G m. Otherwise Zpasses the test. Schnorr tests are the Martin-Löf tests where λG m is computableuniformly in m. The corresponding randomness notion is called Schnorr randomness.We begin with the most restricted test concept, Kurtz tests, where thesets G m are clopen sets given by a strong index for a finite set. The correspondingproperty will be called weak randomness.The implications between the notions are

128 3 Martin-Löf randomness and its variantsML-random ⇒ Schnorr random ⇒ weakly random.The converse implications fail.Weak randomnessClopen sets are given by strong indices for finite sets, as explained before 1.8.6.3.5.1 Definition. A Kurtz test is an effective sequence (G m ) m∈N of clopen setssuch that ∀m λG m ≤ 2 −m . Z is weakly random if it passes each Kurtz test.Kurtz tests are equivalent to null Π 0 1 classes in a uniform way. Thus, a set isweakly random if and only if it avoids all null Π 0 1 classes.3.5.2 Fact.(i) If (G m ) m∈N is a Kurtz test then P = ⋂ m G m is a null Π 0 1 class.(ii) If P is a null Π 0 1 class then P = ⋂ m G m for some Kurtz test (G m ) m∈N .Proof. (i) The tree {x: ∀m [x] ∩ G m ≠ ∅} has a c.e. complement in {0, 1} ∗ .Thus P is a Π 0 1 class (see Definition 1.8.19).(ii) The test obtained in 3.2.2 is a Kurtz test as required.✷A weakly random set Z is incomputable, otherwise Z would be a member of thenull Π 0 1 class given in 3.2.2.3.5.3 Remark. Weak randomness behaves differently depending on whetherthe set is of hyperimmune degree or computably dominated.(i) Each hyperimmune degree contains a weakly 1-generic set by 1.8.50. Thelaw of large numbers (see 3.2.12) may fail for such a set. Weak 1-genericityimplies weak randomness. Thus, each hyperimmune degree contains a weaklyrandom set that is far from being random in the intuitive sense.(ii) If a weakly random set is computably dominated then this set is MLrandom,and in fact weakly 2-random.We discuss (i) here, and postpone (ii) to Proposition 3.6.4.3.5.4 Fact. Each weakly 1-generic set is weakly random.Proof. Each conull open set D is dense, for otherwise D ∩ [x] =∅ for somestring x, whence λD ≤ 1 − 2 −|x| < 1. By the definition, a set G is weakly 1-generic iff G is in every dense c.e. open set. So no weakly 1-generic set is in anull Π 0 1 class.✷3.5.5 Proposition. The law of large numbers fails for every weakly 1-genericset Z. In fact, lim inf n (#{i 0, consider the function f(m) =(k − 1)m. As in the proofof Proposition 1.8.48 let D f be the dense c.e. open set [{σ0 f(|σ|) : σ ≠ ∅}] ≺ .There is a string σ of length m such that σ0 (k−1)m ≺ Z, soforn = km we have#{i

3.5 Notions weaker than ML-randomness 129A similar argument shows that lim sup n (#{i

130 3 Martin-Löf randomness and its variants3.5.10 Fact. If ∃ ∞ i [Z ∈ G i ] for some Schnorr test (G i ) i∈N then Z is notSchnorr random.Proof. Let Ĝm = ⋃ i>m G i. We show that (Ĝm) m∈N is a Schnorr test. ClearlyλĜm ≤ 2 −m . To see that λĜm is uniformly computable, we apply Fact 1.8.15(iii):⋃given m, r ∈ N, we compute a rational that is within 2 −r+1 of λĜm. Let Ĝm,s =i>m G i,s. For each i we may compute s i such that λG i − λG i,si ≤ 2 −i−r . Lett = max{s i : i ≤ m + r}, thenλĜm − λĜm,t ≤ λ ⋃(G i − G i,t )≤i>mm+r∑i=m+12 −i−r +∞∑i=m+r+1Then Z is not Schnorr random since Z ∈ ⋂ m Ĝm.2 −i ≤ 2 −r+1 .On the other hand, by the uniformity of Lemma 1.9.19 we may turn a Schnorrtest into one where the measure of the m-th open set is not only uniformlycomputable, but in fact it equals 2 −m .3.5.11 Fact. For each Schnorr test (G m ) m∈N , one may effectively find a Schnorrtest ( ˜G m ) m∈N such that ∀mG m ⊆ ˜G m and ∀m λ ˜G m =2 −m .✷The analog of Proposition 3.2.16 holds by the same proof.3.5.12 Proposition. Suppose f is a computable one-one function. If Z isSchnorr random then so is f −1 (Z).High degrees and Schnorr randomness. We postpone the proof that there is aSchnorr random, but not ML-random set to Section 7.3, where we actually separatecomputable randomness from ML-randomness. Computable randomness is anotion defined in terms of computable betting strategies (martingales) which liesproperly in between Schnorr and ML-randomness. We will prove in Section 7.3that each high Turing degree contains a computably random set, and that eachc.e. high degree contains a left-c.e. computably random set. If this c.e. degree isTuring incomplete then the set cannot be ML-random by Theorem 4.1.11 below.On the other hand, if the Turing degree of a set is not high, then Schnorrrandomness is equivalent to ML-randomness. This is an instance of the heuristicprinciple that, as we lower the computational complexity of sets, randomnessnotions tend to coincide. Such an interaction from the computational complexityof sets towards randomness already occurred in Remark 3.5.3(ii), that weakrandomness coincides with weak 2-randomness for computably dominated sets.3.5.13 Proposition. If Z is Schnorr random and not high, then Z is alreadyMartin-Löf random.✷✷

3.5 Notions weaker than ML-randomness 131Proof. Suppose that (G m ) m∈N is a ML-test such that Z ∈ ⋂ m G m. Then thefunction f given by f(m) ≃ µs. Z ∈ G m,s is total. Since f ≤ T Z and Z is nothigh, by Theorem 1.5.19 there is a computable function h not dominated by f,namely, ∃ ∞ mh(m) >f(m). Let S m = G m,h(m) , then (S m ) m∈N is a Schnorr(even Kurtz) test such that Z ∈ S m for infinitely many m. Then Z is not Schnorrrandom by Fact 3.5.10.✷Computable measure machinesRecall from (2.4) on page 84 that Ω M = λ[dom M] ≺ = ∑ σ 2−|σ| [[M(σ)↓]] isthe halting probability of a prefix-free machine M. Downey and Griffiths (2004)introduced a restricted type of prefix-free machines.3.5.14 Definition. A prefix-free machine M is called a computable measuremachine if Ω M is computable.(They used the term “computable machines”, but we prefer the present termbecause a machine is the same as a partial computable function.) Similar toTheorem 3.2.9, they characterized Schnorr randomness by a growth conditionon the initial segment complexity. In the case of Schnorr randomness one needsan infinite collection of descriptive complexity measures for strings, namely allthe functions K M for a computable measure machine M.Recall that we approximate K M (x) byK M,s (x) :=min{|σ|: M s (σ) =x}. Fora computable g such that ∀ng(n) ≥ n we have the time bounded version of K Mgiven by K g M (x) =K M,g(|x|)(x) (similar to the definition of C g on page 81).3.5.15 Proposition. Suppose that M is a computable measure machine withrange {0, 1} ∗ . Then the function K M is computable. In fact, there is a computablefunction g such that K M = K g M .Proof. Since Ω M is computable, one can determine K M (x) as follows: computethe least stage s = s x such that Ω M − Ω M,s ≤ 2 −n , where n = K M,s (x) < ∞.If a description of length less than n appears after stage s, then this causes anincrease of Ω M by at least 2 −n+1 .ThusK M,s (x) =K M (x).If we let g(m) = max({m} ∪ {s x : |x| = m}) then K M = K g M .✷Our principal tool for building prefix-free machines is the Machine ExistenceTheorem 2.2.17: from a given bounded request set W one obtains a prefix-freemachine M for W such that Ω M equals wgt W ({0, 1} ∗ ), the total weight of W .Ifthe total weight of W is computable then M is a computable measure machine.Exercises.3.5.16. From a computable measure machine M and Ω M , one may effectively obtaina computable measure machine N such that Ω N = 1 and ∀xK N (x) ≤ K M (x).3.5.17. Let M be a computable measure machine. Show that from r ∈ N one cancompute a strong index for the finite set {y : K M (y) ≤ r}.

132 3 Martin-Löf randomness and its variantsSchnorr tests can be emulated by computable measure machinesDowney and Griffiths (2004) proved the analog of Schnorr’s Theorem 3.2.9: aset Z is Schnorr random ⇔ for each computable measure machine M there is bsuch that each initial segment of Z is b-incompressible in the sense of K M . Firstwe need a lemma saying that the tests (RbM ) b∈N from Definition 3.2.6, whereM is a computable measure machine, are Schnorr tests that can emulate anyother Schnorr test.3.5.18 Lemma.(i) Let M be a computable measure machine. Then (RbM ) b∈N is a Schnorr test.(ii) Let (G m ) m∈N be a Schnorr test. Then we may effectively obtain a computablemeasure machine M such that ⋂ m G m ⊆ ⋂ b RM b .Proof. (i) We extend the proof of Proposition 3.2.7 that (RbM ) b∈N is a ML-test.It suffices to show that λRbM is computable uniformly in b. As before, let VbM bethe set of strings in RbM which are minimal under the prefix ordering. Let Vb,s M bethe set of minimal strings in Rb,s M . Then Ω M − Ω M,s ≥ 2 b (λ[VbM ] ≺ − λ[Vb,s M ]≺ ) foreach s. Since Ω M is computable, by Fact 1.8.15(iv) this shows that λ[Vb M ] ≺ =λRb M is computable uniformly in b.(ii) We extend the proof of (ii)⇒(i) in Theorem 3.2.9. Let (G m ) m∈N be a Schnorrtest. We may assume that λG m ≤ 2 −2m for each m. Let L be the bounded requestset defined as in the proof of Theorem 3.2.9, and let M be the machine obtainedfrom L via the Machine Existence Theorem 2.2.17. By the construction of L, foreach Z ∈ G m there is x ≺ Z such that K M (x) ≤ |x| − m + 1. Therefore it sufficesto show that the total weight α of L is computable and hence M is a computablemeasure machine. Recall that the contribution of G m to the total weight of L isat most 2 −m−1 . In the notation of the proof of 3.2.9, letL t = {〈|x m i | − m +1,xm i 〉 : m ∈ N & i

3.6 Notions stronger than ML-randomness 133in the lemma is computable. In that case, the proof produces a bounded requestset W with a computable total weight α. For, we showed that the weight contributedby each G n , n ≥ 10, is at most (n − 1) −2 /2. Now ∫ ∞x −2 dx =1/r andrhence ∑ n>r+1 (n − 1)−2 ≤ 1/r. Forr ≥ 10 let the rational α r be the weightcontributed by all the G n ,10≤ n ≤ r + 1. Since G is computable, α r can becomputed from r, and α − α r ≤ 1/(2r).The sets G in the proofs of Proposition 2.5.7 and of Corollary 2.5.9 are computable.So we may sharpen these results in the sense that the descriptive complexitycan be taken relative to computable measure machines:3.5.20 Proposition. There are computable measure machines M and N withthe following properties.(i) If x is a string of length n with a run of 4 log n zeros thenK M (x) ≤ + n − log n.(ii) Fix b. For each d, for almost every n, each string x of length n such thatK N (x) >n− b satisfies abs(S n (x)/n − 1/2) < 1/d.✷This leads to the desired statistical properties of Schnorr random sets.3.5.21 Theorem. Let Z be Schnorr random.(i) For each n, any run of zeros in Z ↾ n has length ≤ + 4 log n.(ii) Z satisfies the law of large numbers in 3.2.12.Proof. Let M,N be the computable measure machines from the previous proposition.By Theorem 3.5.19, there is b ∈ N such that Z ∉ RbM and Z ∉ Rb N.Now(i) and (ii) follow.✷3.5.22 Remark. Stronger results along these lines have been derived. For instance,the law of iterated logarithm holds for Schnorr random sets. This suggests that therandomness notion of Schnorr can be seen as a sufficient formalization of the intuitiveconcept of randomness as far as statistical properties are concerned. In contrast, Schnorrrandom sets can have computability theoretic properties that are not consistent withour intuitive notion of randomness. For instance, as Kjos-Hanssen has pointed out,there is a Schnorr random set A = A 0 ⊕ A 1 such that A 0 ≡ T A 1. To see this, let a bea high minimal Turing degree, which exists by the Cooper Jump Inversion Theorem(see Lerman 1983, pg. 207). By Theorem 7.5.9 let A ∈ a be Schnorr random. LetA = A 0 ⊕ A 1, then A 0,A 1 are Schnorr random by 3.5.12, and hence incomputable.Thus A 0 ≡ T A 1. (Since A 0 is not Schnorr random relative to A 1, this example alsoshows that van Lambalgen’s Theorem 3.4.6 fails for Schnorr randomness.)3.6 Notions stronger than ML-randomnessIn this section we mainly study weak 2-randomness and 2-randomness. The implicationsare2-random ⇒ weakly 2-random ⇒ ML-random.The converse implications fail. The new notions address the second concern outlinedat the beginning of Section 3.5, that for instance left-c.e. sets should notbe considered random.

134 3 Martin-Löf randomness and its variantsWhen one is interested in applying concepts related to randomness in computabilitytheory, these notions are less relevant (see the beginning of Chapter 4).Examples like Ω are important for the interaction in this direction. It is at itsstrongest when one considers the ∆ 0 2 sets, and a weakly 2-random set alreadyforms a minimal pair with ∅ ′ .In the converse direction, computational complexity is essential for our understandingof weak 2-randomness and 2-randomness. Within the ML-random sets,both notions are characterized by lowness properties: forming a minimal pairwith ∅ ′ in the former case, and being low for Ω in the latter.An interesting alternative to deal with the second concern at the beginningof Section 3.5 is Demuth randomness, a further notion between 2-randomnessand ML-randomness. It turns out to be incomparable with weak 2-randomness.In Theorem 3.6.25 we show that there is a Demuth random ∆ 0 2 set, and inTheorem 3.6.26 that all Demuth random sets are in GL 1 .Weak 2-randomnessThe concept of a Martin-Löf test was obtained by effectivizing Fact 1.9.9, thatthe null classes are the classes contained in ⋂ m G m for some sequence of opensets (G m ) m∈N such that lim m λG m = 0. To be a Martin-Löf-test, the sequence(G m ) m∈N has to be uniformly c.e. and λG m has to converge to 0 effectively. Ifone drops the effectivity in the second condition, a strictly stronger randomnessnotion is obtained.3.6.1 Definition. A generalized ML-test is a sequence (G m ) m∈N of uniformlyc.e. open sets such that ⋂ m G m is a null class. Z fails the test if Z ∈ ⋂ m G m,otherwise Z passes the test. Z is weakly 2-random if Z passes each generalizedML-test. Let W2R denote the class of weakly 2-random sets.By Remark 1.8.58, the null Π 0 2 classes coincide with the intersections of generalizedML-tests. To be weakly 2-random means to be in no null Π 0 2 class. This is oneof the conceptually simplest randomness notions we will encounter. Note that, byProposition 1.8.60, these tests are determined by computable sets S ⊆ {0, 1} ∗ ,since each Π 0 2 class is of the form {X : ∃ ∞ nS(X ↾ n )} for such an S.By the following, the ML-random set Ω is not weakly 2-random.3.6.2 Proposition. If Z is ∆ 0 2 then {Z} is a null Π 0 2 class. In particular, Z isnot weakly 2-random.Proof. We will turn a computable approximation (Z s ) s∈N of Z into a generalizedML-test (V m ) m∈N such that {Z} = ⋂ m V m. We may assume that Z is infiniteand Z 0 (m) = 0 for each m. To enumerate V m , for each s>m,ifx is least suchthat Z s (x) ≠ Z s−1 (x), put [Z s ↾ x] intoV m,s . To see that {Z} ⊇ ⋂ m V m, notethat, if Z ↾ k is stable from stage t on, then for m>t, all the strings enumeratedinto V m extend Z ↾ k . Next we show that Z ∈ ⋂ m V m. Given m, there is a least xsuch that Z s (x) ≠ Z s−1 (x) for some s>m. Let s be the greatest such stage forthis x. Then Z ↾ x = Z s ↾ x is put into V m at stage s.✷

3.6 Notions stronger than ML-randomness 1353.6.3 Corollary. There is no universal generalized ML-test.Proof. Let (G m ) m∈N be a generalized ML-test, and choose m such that theΠ 0 1 class P =2 N − G m is nonempty. By the Kreisel Basis Theorem 1.8.36, theleftmost path Z of P is left-c.e. Since Z is not weakly 2-random but passes thetest, the test is not universal.✷How does one obtain a weakly 2-random set? One way is to take a weaklyrandom computably dominated set, which exists by Theorem 1.8.42. (See Remark3.5.3 for more details.) Each 2-random set has hyperimmune degree byCorollary 3.6.15 below, so the example we obtain is not 2-random. The result issimilar to Proposition 3.5.13.3.6.4 Proposition. If a set Z is computably dominated and weakly randomthen Z already is weakly 2-random.Proof. If Z is not weakly 2-random then Z ∈ ⋂ m G m for a generalized MLtest(G m ) m∈N . As in the proof of Proposition 3.5.13, the function f given byf(m) =µs. Z ∈ G m,s is total, and f ≤ T Z. There is a computable functiong dominating f. Let H m = G m,g(m) , then (H m ) m∈N is a Kurtz test such thatZ ∈ ⋂ m H m.ThusZ is not weakly random.✷Characterizing weak 2-randomness within the ML-random sets. We will laterstrengthen Proposition 3.6.2. By Exercise 1.8.65, for each ∆ 0 2 set A and eachTuring functional Φ the class {Z : Φ Z = A} is Π 0 2. In Lemma 5.1.13 below weshow that this class is null for incomputable A. Thus, if Z is weakly 2-randomand A ≤ T Z, ∅ ′ then A is computable. In Theorem 5.3.16 we will prove that thisproperty characterizes the weakly 2-random sets within the ML-random sets:Z is weakly 2-random ⇔ Z is ML-random and Z, ∅ ′ form a minimal pair.As a consequence, within the ML-random sets, the weakly 2-random sets aredownward closed under Turing reducibility. The proof is postponed because itrelies on the cost function method of Section 5.3.One cannot replace the condition that Z and ∅ ′ form a minimal pair by Z | T ∅ ′ :3.6.5 Fact. (J. Miller) Some ML-random set Z | T ∅ ′ is not weakly 2-random.Proof. Let Ω 0 be the bits of Ω in the even positions, then Ω 0 is low by thecomment after Corollary 3.4.11. Let V be a 2-random set such that Ω 0 ⊕V ≱ T ∅ ′ ,which exists since {X : Ω 0 ⊕ X ≱ T ∅ ′ } is conull. Let Z = Ω 0 ⊕ V , then Z | T ∅ ′ ,Z is ML-random by 3.4.6, and Z does not form a minimal pair with ∅ ′ . ✷On the other hand, we also cannot expect a stronger condition to characterizeweak 2-randomness within MLR. For instance, each Σ 0 2 set B> T ∅ ′ bounds aML-random computably dominated set (and hence a weakly 2-random set) byExercise 1.8.46.Exercises. Firstly, compare weak 2-randomness with weak randomness relative to ∅ ′ .Recall Definition 1.8.59. Each Π 0 1(∅ ′ ) class is a Π 0 2 class by 1.8.67. Thus, each weakly2-random set is weakly random relative to ∅ ′ , which yields an alternative proof ofProposition 3.6.2. The converse fails.

136 3 Martin-Löf randomness and its variants3.6.6. A set that is weakly random relative to ∅ ′ can fail the law of large numbers.Secondly, show that no Σ 0 2 set is weakly random relative to ∅ ′ .3.6.7. If B ⊆ N is an infinite Σ 0 2 set, then {Z : B ⊆ Z} is a Π 0 1(∅ ′ ) null class.3.6.8. ⋄ (Barmpalias, Miller and Nies, 20xx) Show that some ML-random set is weaklyrandom in ∅ ′ but not weakly 2-random.3.6.9. ⋄ Problem. To what extent does Theorem 3.4.6 hold for weak 2-randomness?2-randomness and initial segment complexityIn Section 2.5 we formalized randomness for strings by incompressibility. Theorem3.2.9 shows that Z is ML-random iff for some b, each x ≺ Z is b-incompressiblein the sense of K. In this subsection we study 2-randomness, that is,ML-randomness relative to ∅ ′ (Definition 3.4.1). The main goal is a characterizationbased on the incompressibility of initial segments. Relativizing Schnorr’sTheorem to ∅ ′ , a set Z is 2-random iff for some b, each x ≺ Z is b-incompressiblein the sense of K ∅′ . This is not very satisfying, though: we would rather like acharacterization based on the incompressibility of initial segments x with respectto an unrelativized complexity measure such as C. One cannot require that forsome b, all x ≺ Z are b-incompressible in the sense of C. Such sets do not existbecause of the complexity dips of C in Proposition 2.2.1. Surprisingly, the weakercondition that(⋆) for some b, infinitely many x ≺ Z are b-incompressible Cprecisely characterizes the 2-random sets. This property was studied by Li andVitányi (1997). Each set satisfying (⋆) is ML-random, because each prefix ofan incompressible C string is incompressible K (see 2.5.4 for the detailed form ofthis statement). On the other hand, Ding, Downey and Yu (2004) proved thateach 3-random set satisfies (⋆). The equivalence of 2-randomness and (⋆) isnowobtained by extending the arguments in both proofs in such a way that theywork with 2-randomness.Two variants of (⋆) will be considered using different notions of incompressibility.The first appears weaker, the second stronger.1. Recall from page 81 that for a computable function g we definedC g (x) = min{|σ|: V(σ) =x in g(|x|) steps}. (3.9)The first variant of (⋆) is to require that for some b, infinitely many x ≺ Zare b-incompressible with respect to C g , for an appropriate fixed g (which underan additional assumption on V can be chosen in O(n 3 )). This is actuallyequivalent to 2-randomness of Z. Using this fact and that C g is computable, wegive a short proof of the result by Kurtz (1981) that no 2-random set is computablydominated. This shows that 2-randomness is indeed stronger than weak2-randomness, because a weakly 2-random set can be computably dominated byProposition 3.6.4. (By Exercise 3.6.22, there is also a weakly 2-random set ofhyperimmune Turing degree that is not 2-random.)

3.6 Notions stronger than ML-randomness 1372. For the second variant, recall from 2.5.2 that x is strongly d-incompressible Kif K(x) is within d of its maximum possible value |x| + K(|x|) + 1, and thatbeing strongly incompressible K implies being incompressible C . The second variantof (⋆) is to require that for some b, infinitely many x ≺ Z are strongly b-incompressible K . J. Miller proved that, once again, this seemingly stronger conditionis actually equivalent to 2-randomness of Z. See Theorem 8.1.14.We now prove that Z is 2-random ⇔ Z it satisfies (⋆) above. This is due toNies, Stephan and Terwijn (2005); the implication “⇐” was also independentlyand slightly earlier obtained by Miller (2004).3.6.10 Theorem. Z is 2-random ⇔∃b ∃ ∞ n [ C(Z ↾ n ) >n− b ] .Proof. ⇒: Firstly, we will sketch a proof of the easier result of Ding, Downeyand Yu (2004) that any 3-random set Z satisfies (⋆). If (⋆) fails then Z ∈ ⋂ b V b,where V b = ⋃ t P b,t, and P b,t = { X : ∀n ≥ t [ C(X ↾ n ) ≤ n − b ]} . Note thatλP b,t ≤ 2 −b+1 for each t. Hence λV b ≤ 2 −b+1 ,asP b,t ⊆ P b,t+1 for each t. Theclass V b is Σ 0 2. It is not open, but can be enlarged in an effective way to an openclass Ṽb that is Σ 0 1 relative to C ′ (the function C is identified with its graph) andhas at most twice the measure of V b . Then Z fails (Ṽb+2) b∈N which is a ML-testrelative to C ′ ≡ T ∅ ′′ .In order to obtain a ML-test relative to ∅ ′ we introduce a concept of independentinterest. A function F : {0, 1} ∗ → {0, 1} ∗ is called compression functionif F is one-one and ∀x [ ̂F (x) ≤ C(x)], where ̂F (x) =|F (x)|. For instance, if, foreach x, F (x) is a shortest V-description of x, then F is a compression functionsuch that ̂F (x) =C(x). The idea is to replace in the argument sketched above Cby ̂F for a compression function F such that F ′ ≡ T ∅ ′ . In this way we obtain aML-test relative to ∅ ′ (and not ∅ ′′ ). For the duration of this proof we say that aset Z is complex for ̂F if there is b ∈ N such that ̂F (Z ↾ n ) >n− b for infinitelymany n. IfZ is complex for ̂F then (⋆) holds. We now extend the argumentabove to an arbitrary compression function.3.6.11 Lemma. Let F be a compression function. Suppose that Z is not complexfor F , namely,∀b ∃t ∀n ≥ t [ ̂F (Z ↾n ) >n− b ] . (3.10)Then Z is not ML-random relative to F ′ .Subproof. We identify F with its graph {〈n, m〉: F (n) =m}. LetP b,t = {X : ∀n ≥ t [ ̂F (X ↾ n ) ≤ n − b ]},then P b,t is a Π 0 1 class relative to F uniformly in b, t. By (3.10), Z ∈ ⋂ b V b whereV b = ⋃ t P b,t. Note that λP b,t ≤ 2 −b+1 for each t, because as F is one-one, thereare fewer than 2 t−b+1 strings x of length t such that ̂F (x) ≤ t−b.AsP b,t ⊆ P b,t+1for each t this implies λV b ≤ 2 −b+1 . For each b, t, k, using F as an oracle, we cancompute the clopen setR b,t,k = {X : ∀n [ t ≤ n ≤ k → ̂F (X ↾ n ) ≤ n − b ]}.

138 3 Martin-Löf randomness and its variantsSince P b,t = ⋂ k R b,t,k, using F ′ as an oracle, uniformly in b, on input t we cancompute k(t) such thatλ(R b,t,k(t) − P b,t ) ≤ 2 −(b+t) .Let T b = ⋃ t R b,t,k(t). Then the T b are open sets and the corresponding set ofstrings {x :[x] ⊆ T b } is Σ 0 2(F ) uniformly in b. Moreover, V b = ⋃ t P b,t ⊆ T b andλ(T b − V b ) ≤ ∑ t 2−(b+t) =2 −b+1 ,soλT b ≤ 4 · 2 −b . Hence Z fails (T b+2 ) b∈N ,which is a ML-test relative to F ′ .✸3.6.12 Lemma. There is a compression function F such that F ′ ≤ T ∅ ′ .Subproof. (This version is due to L. Bienvenue) A Π 0 1 class P ⊆ N N is calledbounded if there is a computable function g such thatP ⊆ Paths({σ ∈ N ∗ : ∀i

3.6 Notions stronger than ML-randomness 139On input τ, let t = |τ| and search for a decomposition τ = σy such thatU ∅′ (σ)[t]↓= x. If such a decomposition is found, then output xy.Let d be the coding constant for M with respect to the optimal machine V.Suppose that σ is a shortest U ∅′ -description of x, and let t 0 > |σ| be least suchthat the computation U ∅′ (σ)[t 0 ] is stable. If |y| ≥ t 0 and τ = σy, then σy is theonly decomposition of τ that M can find, because the domain of U ∅′ t is prefix-freefor each t. ThusM(σy) =xy, whence C(xy) ≤ K ∅′ (x)+|y| + d.✸We may now argue as before. Given b, let x ≺ Z be such that K ∅′ (x) ≤ |x|−b−d.By the foregoing Lemma, for almost all y we have C(xy) ≤ |x| + |y| − b. Inparticular, C(Z ↾ n ) ≤ n − b for almost all n. Thus(⋆) fails.✷We will improve Theorem 3.6.10. As already mentioned, using the time boundedversion C g in (3.9) for an appropriate computable g, actually Z is 2-random iff(⋆) g ∃b ∃ ∞ n [C g (Z ↾ n ) >n− b].Clearly (⋆) g is implied by (⋆) because C g (z) ≥ C(z) for each z. Under a reasonableimplementation of U by a Turing program we may ensure that g(n) =O(n 3 ).3.6.14 Lemma. (Extends Lemma 3.6.13) There is a computable functiong such that, for some d ∈ N,∀x ∀ ∞ y [ C g (xy) ≤ K ∅′ (x)+|y| + d ] . (3.12)Subproof. We determine a bound on the running time of the machine M inthe proof of Lemma 3.6.13. On input τ, if n = |τ|, in the worst case, for eachσ ≼ τ , M has to evaluate U ∅′ (σ)[n] (which may be undefined). This takes O(n 2 )steps. After that, M needs O(n) steps for printing xy. We assume that if no σis found, M halts with the empty string as an output. So M runs O(n 2 ) stepson any input of length n.The copying machine maps each string z to itself. Let g(n) be the maximumof the number of steps it takes V to simulate both M and the copying machine,for any input of length n. (Namely, g(n) bounds the maximum number of stepsit takes V(0 e−1 1σ) to halt, where e is an index of either machine and |σ| = n.)If K ∅′ (x) ≥ |x| then (3.12) is satisfied via the copying machine. Otherwise, let σbe a shortest U ∅′ -description of x. Ify is sufficiently long, we have M(σy) =xy;since |σx| ≤ |xy| we obtain (3.12) via M.✸The proof that each Z satisfying (⋆) g is 2-random can be completed as before,using C g in place of C.✷The following result was originally obtained in a different way by Kurtz (1981).3.6.15 Corollary. No 2-random set Z is computably dominated.Proof. Choose b so that (⋆) g holds for the computable function g obtainedabove. Since C g is computable, the function f defined byG(m) =µr. ∃G ⊆ {0,...,r} [ #G = m & ∀n ∈ G[C g (Z ↾ n ) >n− b] ]

140 3 Martin-Löf randomness and its variantsis total and computable in Z. Assume for a contradiction that a computablefunction h dominates f. Then Z is a path of the computable tree{x: ∀m |x|≥h(m) ∃G ⊆ {0,...,|x|} [ #G = m & ∀n ∈ G[C g (x↾ n ) >n− b] ] }.Each path of this tree satisfies (⋆) g , and is therefore 2-random. Since its leftmostpath is left-c.e. this is a contradiction.✷In fact, Kurtz (1981) proved that each 2-random set Z is c.e. relative to some setA< T Z. ThusA< T Z ≤ T A ′ , whence Z is of hyperimmune degree by Exercise 1.5.13.For a recent proof of this stronger result see Downey and Hirschfeldt (20xx).3.6.16 Exercise. Show that there is a low compression function for K, namely aone-one function F : N → N such that ∀x [|F (x)| ≤ K(x)] and ∑ x 2−|F (x)| ≤ 1.2-randomness and being low for ΩThe following lowness property will be studied in more detail in Section 8.1. Hereit allows us to characterize the 2-random sets within the ML-random sets.3.6.17 Definition. A is low for Ω if Ω is ML-random relative to A. The classof sets that are low for Ω is denoted by Low(Ω).In Proposition 8.1.1 we will see that the class Low(Ω) does not depend on theparticular choice of the optimal prefix-free machine. We make two basic observations.The second observation follows from Proposition 3.4.10.3.6.18 Fact. (i) Low(Ω) is closed downward under Turing reducibility.(ii) Low(Ω) ⊆ GL 1 .The proof of Corollary 3.4.12 actually shows that the 2-random set Ω ∅′ is lowfor Ω. The following characterization of 2-randomness is more general.3.6.19 Proposition. Z is 2-random ⇔ Z is ML-random and low for Ω.Thus, within the ML-random sets, to be 2-random is equivalent to a lownessproperty. The same holds for weak 2-randomness, where the lowness property isforming a minimal pair with ∅ ′ (see page 135).Proof. Since Ω ≡ T ∅ ′ (3.2.30), Z is 2-random ↔ Z is ML-random relative to ∅ ′↔ Z is ML-random relative to Ω. Since Ω is ML-random, by van Lambalgen’sTheorem 3.4.6 the latter is equivalent to: Z is ML-random and Ω is ML-randomrelative to Z.✷By Fact 3.6.18, we have the following.3.6.20 Corollary. (i) The 2-random sets are closed downward under Turingreducibility within the ML-random sets. (ii) Each 2-random set is in GL 1 . ✷In Theorem 8.1.18 states that each set in Low(Ω) is of hyperimmune degree.This yields yet another proof of Corollary 3.6.15.We discuss to which extent the preceding results hold for weak 2-randomness.By Proposition 3.6.4, a weakly 2-random set can be computably dominated,so 3.6.15 fails. Corollary 3.6.20(i) holds for weakly 2-random sets, but (ii) fails:✷

3.6 Notions stronger than ML-randomness 141Theorem 8.1.19 below shows that no weakly 2-random computably dominatedset is in GL 1 . On the other hand, by Exercise 3.6.22 there is a weakly 2-randomset of hyperimmune degree that is not 2-random.Exercises. Show the following.3.6.21. Let A be in ∆ 0 2.IfZ is 2-random then Z△A is 2-random as well.3.6.22. ⋄ There is a weakly 2-random set R of hyperimmune degree such that no set ofthe same Turing degree is 2-random.3.6.23. ⋄ Problem. Is there a characterization of weak 2-randomness via the growthof the initial segment complexity?Demuth randomness ⋆The notion was introduced by Demuth (1988) in the language of analysis. Hiswork was made known to a wider audience by Kučera. Like weak 2-randomness,Demuth randomness lies in between 2-randomness and ML-randomness. We showthat a Demuth random set can be ∆ 0 2, and that all Demuth random sets are inGL 1 . This implies that Demuth randomness and weak 2-randomness are incomparable,even up to Turing degree: a ∆ 0 2 set is not weakly 2-random; on the otherhand, no ML-random set in GL 1 is computably dominated by Theorem 8.1.19,so a weakly 2-random computably dominated set is not Turing equivalent toa Demuth random set. This is an exception to the rule that the randomnessnotions form a linear hierarchy.Demuth tests combine features of ML-tests and Solovay tests. We retain thecondition of ML-tests that S m be c.e. open and λS m ≤ 2 −m , but we relax theuniformity condition: the c.e. index for S m is now given by an ω-c.e. function(Definition 1.4.6), rather than by a computable function. Informally, we canchange the whole open set S m for a computably bounded number of times. Weadopt the failure condition of Solovay tests. In contrast to the previous testnotions, one cannot require for all tests that S m ⊇ S m+1 for each m. This willbecome apparent in the proof of Theorem 3.6.26 below.3.6.24 Definition. A Demuth test is a sequence of c.e. open sets (S m ) m∈N suchthat ∀m λS m ≤ 2 −m , and there is an ω-c.e. function f such that S m =[W f(m) ] ≺ .A set Z passes the test if ∀ ∞ mZ ∉ S m . We say that Z is Demuth random if Zpasses each Demuth test.Each Demuth test is a Solovay test relative to ∅ ′ , so, by Proposition 3.2.19 relativeto ∅ ′ , each 2-random set is Demuth random. On the other hand, each ω-c.e. set Zfails to be Demuth random via the test given by S n =[Z ↾ n ]. In particular, thisapplies to Ω.3.6.25 Theorem. There is a Demuth random ∆ 0 2 set Z.Proof. We write H e for [W e ] ≺ . We use an auxiliary type of tests: a special testis a sequence of c.e. open sets (V m ) m∈N such that λV m ≤ 2 −2m−1 for each mand there is a function g ≤ T ∅ ′ such that V m = H g(m) . Z passes the test if∀ ∞ mZ ∉ V m . Special tests are similar to Demuth tests, but the function g is

142 3 Martin-Löf randomness and its variantsmerely ∆ 0 2, while the bound on the measure of the m-th component is tighterthan 2 −m . It suffices to establish two claims: (1) there is a single special testsuch that each set passing it is Demuth random, and (2) for each special testthere is a ∆ 0 2 set that passes it.Claim 1. There is a special test (V m ) m∈N such that each set Z passing the testis Demuth random.Z is Demuth random iff for each Demuth test (U m ) m∈N , Z passes the Demuthtests (U 2m ) m∈N and (U 2m+1 ) m∈N . Thus it suffices that (V m ) m∈N emulate allDemuth tests (S m ) m∈N such that λS m ≤ 2 −2m for each m. By Fact 1.4.9, thereis a binary function ˜q ≤ T ∅ ′ that emulates all ω-c.e. functions. We can stopthe enumeration of H˜q(e,m) when it attempts to exceed the measure 2 −2m ;thusthere is q ≤ T ∅ ′ such that λH q(e,m) ≤ 2 −2m for each m, and H q(e,m) = H˜q(e,m)if already λH˜q(e,m) ≤ 2 −2m . Now letV m = ⋃ e

3.6 Notions stronger than ML-randomness 143stage s the measure of the clopen set L m,s −C m,s−1 exceeds 2 −m , we put this setinto C m,s and increase g(m) to the stage number, so that it also dominates thevalues Θ Z (m) for these newly added oracles Z. These increases of the currentapproximations to C m , and hence of g(m), can take place at most 2 m times.Thus g is ω-c.e. and C m stabilizes, whence S m = L m −C m determines a Demuthtest as desired.Construction of clopen sets C m = ⋃ s C m,s and an ω-c.e. function g.Let C 0,0 = ∅ and g 0 (0) = 0.Stage s>0. Let C s,s = ∅ and g s (s) = 0. For each m 2 −m let C m,s = L m,s and g s (m) =s;otherwise, let C m,s = C m,s−1 and g s (m) =g s−1 (m).By this construction, if Θ Z (m)↓ for Z ∉ S m then Θ Z (m) ≤ g(m). If Z is Demuthrandom then Z ∉ S m for almost all m, sog dominates Θ Z .✷In Theorem 8.1.19 we will prove that a computably dominated set in GL 1 isnot of d.n.c. degree. Hence no Demuth random set is computably dominated,which strengthens Corollary 3.6.15.Exercises.3.6.27. Show that some low ML-random set is not Demuth random.3.6.28. Suppose Z is Schnorr random relative to ∅ ′ . Show that Z is (i) weakly 2-random and (ii) Demuth random. (iii) Conclude that some weakly 2-random set Y inGL 1 is not 2-random.

4Diagonally noncomputable functionsWe discuss the interactions between computability and randomness. Traditionallythe direction is from computability to randomness. In this direction, twotypes can be distinguished.Interaction 1a: computability theoretic notions are used to obtain mathematicaldefinitions for the intuitive concept of a random set.For instance, in Chapter 3 we introduced ML-randomness and its variants usingtest notions which are based on computable enumerability and other concepts.Interaction 1b: computational complexity is used to analyze randomness propertiesof a set.An example is the result on page 135 that Z is weakly 2-random iff Z is MLrandomand Z, ∅ ′ form a minimal pair.The interaction also goes the opposite way.Interaction 2: concepts related to randomness enrich computability theory.We have already seen examples of this in Chapter 3: the real number Ω and theoperator X ↦→ Ω X . In Section 5.2 we will study K-triviality, a property of setsthat means being far from random. This property turns out to be equivalentto the lowness property of being low for ML-randomness. The class of K-trivialsets has interesting properties from the computability-theoretic point of view. Forinstance, each K-trivial set is superlow. Many equivalent definitions are known,and all touch in some way upon randomness-related concepts.In the spirit of Interaction 2, in this chapter we study diagonally noncomputablefunctions. They arise naturally when one studies ML-random sets. Wehave briefly considered d.n.c. functions in Remark 1.8.30.4.0.1 Definition.(i) A function f : N ↦→ N is diagonally noncomputable, or d.n.c. for short, iff(e) ≠ J(e) for any e such that J(e)↓.(ii) A set D has d.n.c. degree if there is a d.n.c. function f ≤ T D.A d.n.c. function f is incomputable, for otherwise f = Φ e for some e, and hencef(e) =Φ e (e) =J(e). We think of f as far from computable since it effectivelyprovides the value f(e) as a counterexample to the hypothesis that f = Φ e .If Z is ML-random, a finite variant of the function λn.Z ↾ n is diagonally noncomputable,because Z ↾ e = J(e) implies K(Z ↾ e ) ≤ + 2 log e. In Section 4.1 westudy the sets of d.n.c. degree, or, equivalently, the sets that compute a fixedpoint free function g (namely W g(x) ≠ W x for each x).

4.1 D.n.c. functions and sets of d.n.c. degree 145To be of d.n.c. degree is a conull highness property (intuitively speaking, itis weak). Sets of d.n.c. degree are characterized by a property stating that theinitial segment complexity grows somewhat fast. Thus the highness property tobe of d.n.c. degree can also be interpreted as to be “somewhat random”. Thischaracterization is an analog of Schnorr’s Theorem 3.2.9. As a consequence, thesets of d.n.c. degree are closed upwards with respect to the preordering ≤ K whichcompares the degree of randomness of sets (Definition 5.6.1 below.)In Section 4.2 we discuss Kučera’s injury-free solution to Post’s problem, afurther instance of Interaction 2. It is based on a d.n.c. function f< T ∅ ′ , buttakes a particularly simple form when f is a finite variant of the function λn. Z ↾ nfor a ML-random ∆ 0 2 set Z. We use that f is d.n.c. to build a promptly simpleset A ≤ T f. To make f permit changes of A we threaten that f(k) =J(k)for appropriate numbers k. These methods can be extended to an injury-freeconstruction of a pair of Turing incomparable c.e. sets.In Section 4.3 we strengthen the concept of a d.n.c. function in various ways. Weuse the stronger concepts to gain a better understanding of the computationalcomplexity of n-random sets (this is Interaction 1b). Firstly, we show that if aset Z is ML-random and computes a {0, 1}-valued d.n.c. function, then Z alreadycomputes the halting problem. Secondly, we introduce a hierarchy of propertiesof functions strengthening fixed point freeness, and show that an n-random setcomputes a function at level n of that hierarchy.4.1 D.n.c. functions and sets of d.n.c. degreeWe characterize the sets of diagonally noncomputable degree via a growth conditionon the initial segment complexity. Thereafter, we show that the only c.e.sets of diagonally noncomputable degree are the Turing complete ones.Basics on d.n.c. functions and fixed point freeness4.1.1 Proposition. (i) No d.n.c. function f is computable.(ii) ∅ ′ has d.n.c. degree via some function f ≤ tt ∅ ′ .Proof. (i) If f = Φ e is total, then f(e) =J(e), so f is not d.n.c.(ii) A {0, 1}-valued d.n.c. function f ≤ tt ∅ ′ was given in Remark 1.8.30.✷The following provides further examples of d.n.c. functions.4.1.2 Proposition.(i) There is c ∈ N such that∀ ∞ nK(Z ↾ n ) >K(n)+c → Z has d.n.c. degreevia a function f that agrees with λn. Z ↾ n on almost all n (hence f ≤ tt Z).(ii) Each ML-random set Z has d.n.c. degree via a function f ≤ tt Z.Proof. (i) Recall from page 12 that we identify strings in {0, 1} ∗ with numbers.Let c be a constant such that, for each σ, if y = J(U(σ))↓ then K(y) ≤ |σ| + c.

146 4 Diagonally noncomputable functionsIf Z ↾ n = J(n) and σ is a shortest U-description of n, then J(U(σ)) = Z ↾ n andhence K(Z ↾ n ) ≤ |σ| + c = K(n)+c. ThusZ ↾ n = J(n) fails for almost all n.(ii) If Z is ML-random then there is b such that ∀nK(Z ↾ n ) ≥ n − b. SinceK(n) ≤ + 2 log n, this implies K(Z ↾ n ) >K(n)+c for almost all n. ✷Actually, by 8.3.6, every infinite subset of a ML-random set has d.n.c. degree.By the Recursion Theorem 1.1.5, for each computable function g there is x suchthat W x = W g(x) . The functions of the following type are far from computablein the sense that this fixed point property fails.4.1.3 Definition. A function g is fixed point free (f.p.f.) if ∀xW g(x) ≠ W x .The two notions of being far from computable coincide up to Turing degree.4.1.4 Proposition. Let D ⊆ N. Then D has d.n.c. degree ⇔ D computes afixed point free function. The equivalence is uniform.Proof. It suffices to show that each d.n.c. function computes a fixed point freefunction and vice versa.⇒: Suppose that the function f is diagonally noncomputable. We construct afixed point free function g ≤ T f. Letα(x) ≃ the first element enumerated into W x .Let p be a reduction function for α (see Fact 1.2.15). Thus α(x) ≃ J(p(x)) foreach x. Let g ≤ T f be a function such that g(x) is a c.e. index for {f(p(x))}. ThenW x ≠ W g(x) unless #W x =1.If#W x = 1, then W x = {α(x)} = {J(p(x))}.Because f is d.n.c., J(p(x)) ≠ f(p(x)), so W x ≠ W g(x) .⇐: Suppose that the function g is fixed point free. Let h be a computablefunction such that{W J(x) if J(x)↓,W h(x) =∅ otherwise.If J(x)↓ then W g(h(x)) ≠ W J(x) and hence g(h(x)) ≠ J(x). So f = g ◦ h is d.n.c.and f ≤ T g.✷In Definition 4.3.14 we will introduce the hierarchy of n-fixed point free functions(n ≥ 1). The lowest level n = 1 consists of the functions of Definition 4.1.3. Theextendability of this definition to higher levels is one of the reasons why we do notdefine fixed point freeness of a function g by the weaker condition that Φ g(x) ≠ Φ x.By Exercise 8.3.6, A has d.n.c. degree iff A computes an infinite subset of a MLrandomset.Exercises. Show the following.4.1.5. Let f be a d.n.c. function. For each partial computable function ψ, there is afunction ˜f ≤ T f such that ˜f(e) ≠ ψ(e) for any e such that ψ(e)↓.4.1.6. No 1-generic set G (see Definition 1.8.51) is of d.n.c. degree. In particular, no1-generic set computes a ML-random set.

4.1 D.n.c. functions and sets of d.n.c. degree 147By the following, the properties of functions introduced above are independent of theparticular choice of a jump operator, or a universal uniformly c.e. sequence, as far asthe Turing degree of the function is concerned.4.1.7. Suppose that Ĵ is a universal partial computable function (Fact 1.2.15). Theneach d.n.c. function f is Turing equivalent to a d.n.c. function ̂f with respect to Ĵ, andvice versa.4.1.8. Suppose that (Ŵe) e∈N is a universal uniformly c.e. sequence with the paddingproperty, as in Exercise 1.1.12. Then each f.p.f. function g is Turing equivalent to anf.p.f. function ĝ with respect to (Ŵe) e∈N, and vice versa.The initial segment complexity of sets of d.n.c. degreeSchnorr’s Theorem 3.2.9 states that the ML-random sets are the ones with anearly maximal growth of the initial segment complexity in the sense of K.The following result of Kjos-Hanssen, Merkle and Stephan (2006) characterizesthe sets of d.n.c. degree by a growth condition stating that the initial segmentcomplexity grows somewhat quickly. This yields a further proof besides the onein Exercise 4.1.7 that the property to be of d.n.c. degree does not depend of thesomewhat arbitrary definition of the universal partial computable function J.4.1.9 Theorem. The following are equivalent for a set Z.(i) Z has d.n.c. degree.(ii) There is a nondecreasing unbounded function g ≤ T Z such that∀n [g(n) ≤ K(Z ↾ n )].It is useful to think of the function g in (ii) as slowly growing. By 2.4.2 K ∼ C,so we could equivalently take the initial segment complexity of Z via C.Proof. (ii) ⇒ (i): The idea is the same as in the proof of Proposition 4.1.2.Suppose (ii) holds via g ≤ T Z, and let h(r) = min{m : g(m) ≥ r}. (Think of h asa function that grows quickly.) Note that Z computes the function f(r) =Z ↾ h(r) .If Z ↾ h(r) = J(r) and σ is a shortest description of r then J(U(σ)) = Z ↾ h(r) andhence K(Z ↾ h(r) ) ≤ + K(r) =|σ| ≤ + 2 log r. However, by the definition of h wehave K(Z ↾ h(r) ) ≥ r. ThusZ ↾ h(r) = J(r) fails for almost all r.(i) ⇒ (ii): Suppose that (ii) fails. Let Γ = Φ i be a Turing functional such thatΓ Z is total. We show that Γ Z is not a d.n.c. function, namely, Γ Z (e) =J(e) forsome e. For each σ, one can effectively determine an e(σ) such that Φ e(σ) (y) ≃Γ U(σ) (y): the number e(σ) encodes a Turing program implementing a procedurethat on input y first runs U on the input σ; in case of convergence it takes theoutput ρ as an oracle string and attempts to compute Γ ρ (y). Leth(r) = max({r} ∪ {use Γ Z (e(σ)): |σ| ≤ r}).The function g given by g(m) = max{r : h(r) ≤ m} is computable in Z, nondecreasingand unbounded. If (ii) fails, there is an m such that r = g(m) >K(Z ↾ m ). So U(σ) =Z ↾ m for some σ such that |σ| < r. Let e = e(σ). Since

148 4 Diagonally noncomputable functionsh(r) ≤ m we have use Γ Z (e) ≤ m. ThusJ(e) =Φ e (e) =Γ U(σ) (e) =Γ Z (e).✷With a minor modification, the foregoing proof yields a characterization ofthe sets Z such that there is a d.n.c. function f ≤ wtt Z. The characterizingcondition is that K(Z ↾ n ) be bounded from below by an order function, i.e., anondecreasing unbounded computable function.4.1.10 Theorem. The following are equivalent for a set Z.(i) There is a d.n.c. function f ≤ wtt Z.(ii) There is an order function g such that ∀n [g(n) ≤ K(Z ↾ n )].(iii) There is a d.n.c. function f ≤ tt Z.Proof. (ii)⇒(iii) is proved in the same way as the implication (ii)⇒(i) in Theorem4.1.9. Notice that if g is computable then f ≤ tt Z.For (i)⇒(ii), note that in the proof of (i)⇒(ii) above, if there is a computablebound on the use of Γ, we can choose h and hence g computable.✷A completeness criterion for c.e. setsThe only c.e. sets of d.n.c. degree are the Turing complete ones. This result ofArslanov (1981) builds on work of Martin (1966a) and Lachlan (1968).4.1.11 Theorem. (Completeness Criterion) Suppose the set Y is c.e.(i) There is a d.n.c. function f ≤ T Y ⇔ Y is Turing complete.(ii) There is a d.n.c. function f ≤ wtt Y ⇔ Y is wtt-complete.The result can be viewed as a generalization of the Recursion Theorem 1.1.5when stated in terms of fixed point free functions: for every function g< T ∅ ′ ofc.e. degree there is an x such that W g(x) = W x . For instance, if A< T ∅ ′ is c.e.,then the function p Ahas such a fixed point.Proof idea. If Y is a Turing complete set then Y has d.n.c. degree by Proposition4.1.1. If Y is wtt-complete then there is a d.n.c. function f ≤ wtt Y by thesame proposition.Now suppose there is a Turing functional Φ such that f = Φ Y is a d.n.c.function. For an appropriate reduction function p (see 1.2.15), when e enters ∅ ′at a stage s such that Φ Yss (p(e)) ↓, we threaten that this value equal J(p(e)).Then Y s is forced to change below the use of Φ Yss (p(e)). So, once Φ Yss (p(e)) isstable at stage s(e), e cannot enter ∅ ′ any more. Since Y is c.e., s(e) is the firststage where Φ Yss (p(e)) converges and Y s (x) =Y (x) for each x less than the use.So Y can compute such a stage, and ∅ ′ ≤ T Y . If the use of Φ is bounded by acomputable h, then the use of the reduction procedure is bounded by h(p(e)) oninput e, hence ∅ ′ ≤ wtt Y .Proof details. We define an auxiliary partial computable function α. BytheRecursion Theorem we may assume that we are given a computable reductionfunction p such that α(e) ≃ J(p(e)) for each e (see Remark 4.1.12 below).

4.1 D.n.c. functions and sets of d.n.c. degree 149Construction of α.Stage s. Ife ∈∅ ′ s+1 −∅ ′ s and y = Φ Yss (p(e)) ↓, let α(e) =y.We show that ∅ ′ ≤ T Y . Given an input e, using Y as an oracle, we compute thefirst stage s = s(e) such that Φ Yss (p(e)) ↓ and Y is stable below the use of thiscomputation. If e enters ∅ ′ at a stage ≥ s(e), we define α(e) =J(p(e)) = f(p(e)),so f is not a d.n.c. function. Thus e ∈∅ ′ ↔ e ∈∅ ′ s(e) .✷4.1.12 Remark. We justify the seemingly paradoxical argument where we assumethe reduction function p for α is given, even though we are actually constructingα. By Fact 1.2.15, from an index e for a partial computable function Φ eone obtains a reduction function p. Based on p, in our construction we build apartial computable function Φ g(e) . By the Recursion Theorem 1.1.5, there is afixed point i, namely a partial computable α = Φ i such that Φ g(i) = α. Sointheinteresting case that the given e is such a fixed point, p is a reduction functionfor Φ g(e) = Φ e .We provide two applications of the Completeness Criterion.Application 1. By Theorem 3.2.11 the halting probability Ω R is ML-randomfor each optimal prefix-free machine R, and by Proposition 3.2.30, Ω R is wttcomplete(we identify Ω R with its binary representation). We give an alternativeproof of the second fact: Since Ω R is ML-random, a finite variant f of the functionλn. Ω R ↾ n is d.n.c., and f ≤ wtt Ω R . Also, Ω R is left-c.e., namely, the set {q ∈Q 2 :0≤ q

150 4 Diagonally noncomputable functions4.2 Injury-free constructions of c.e. setsOne can solve Post’s problem and even prove the Friedberg–Muchnik Theorem1.6.8 avoiding the priority method with injury. These results of Kučera(1986) are interesting because injury makes sets artificial due to the fact thatone first fulfills tasks and then gets them undone. This does not happen inKučera’s constructions, even if they are a bit more involved.The most direct injury-free solution to Post’s problem relies on ML-randomness.Step 1. Begin with Ω 0 , the set given by the bits of Ω in the even positions, whichis Turing incomplete by Corollary 3.4.8 (and even low by 3.4.11).Step 2. Build a promptly simple set A (Definition 1.7.9) Turing below Ω 0 .The construction of Kučera (1986) in step 2 works for any ∆ 0 2 ML-random setin place of Ω 0 ; see Remark 4.2.4 below.There is no injury because in step 1 there are no requirements, and in step 2 wemerely meet the prompt simplicity requirements, which cannot be injured. Theinjury-free solution to Post’s problem as it is commonly thought of nowadays issomewhat different. Step 1 uses the Low Basis Theorem. Step 2 is a more generalresult of independent interest. Its proof needs the Recursion Theorem.Step 1. Use the Low Basis Theorem 1.8.37 to produce a low set of d.n.c. degree.For instance, take the Π 0 1 class 2 ω −R 1 , which by Theorem 3.2.9 consists entirelyof ML-random, and hence d.n.c. sets. Or take the Π 0 1 class of {0, 1}-valued d.n.c.functions in Fact 1.8.31. The construction in the proof of Theorem 1.8 is relativeto ∅ ′ . One satisfies the requirements one by one in order. Hence they are notinjured.Step 2. Show that Turing below each ∆ 0 2 set Y of d.n.c. degree one can build apromptly simple set.The original proof in Kučera (1986) uses the Low Basis Theorem in Step 1, butavoids the Recursion Theorem in Step 2, as above. It has been criticized that inthe proof of Theorem 1.8.37, injury is merely hidden using ∅ ′ . Indeed, the effectiveversion of the proof in Theorem 1.8.38 has injury to lowness requirements. Thiscriticism does not apply when we avoid the Low Basis Theorem altogether andrather use Ω 0 as the low set of d.n.c. degree.Initially Kučera’s motivation was to disprove the conjecture in Jockusch and Soare(1972a) that each nonempty Π 0 1 class without computable members contains a setY < T ∅ ′ such that each c.e. set A ≤ T Y is computable. Jockusch then pointed out thatKučera’s proof leads to an injury-free solution to Post’s problem.For the injury-free proof of the Friedberg-Muchnik Theorem 1.6.8, Kučera developedhis ideas further. Instead of carrying out two independent steps, he let the constructionrelative to ∅ ′ and the effective construction interact via the Double Recursion Theorem1.2.16. This bears some similarity to the worker’s method of Harrington. A so-calledlevel n argument involves interacting priority constructions relative to ∅, ∅ ′ ,...,∅ (n)(that is, workers at level i for each i ≤ n), and uses an (n + 1)-fold Recursion Theorem.See Calhoun (1993).

4.2 Injury-free constructions of c.e. sets 151Each ∆ 0 2 set of d.n.c. degree bounds a promptly simple setWe construct a promptly simple set Turing below any given ∆ 0 2 set of d.n.c.degree. In the next subsection we describe the orginal argument in Kučera (1986),where the Recursion Theorem is avoided when the given ∆ 0 2-set is in fact MLrandom.However, the more powerful method of the present construction is usedfor instance in the injury-free proof of the Friedberg–Muchnik Theorem. Thehypothesis that Y be ∆ 0 2 is necessary because, say, a weakly 2-random set hasd.n.c. degree but forms a minimal pair with ∅ ′ (page 135).4.2.1 Theorem. (Kučera) Let Y be a ∆ 0 2 set of d.n.c. degree. Then there is apromptly simple set A such that A ≤ T Y .Proof idea. The Completeness Criterion 4.1.11 fails for ∆ 0 2 sets because thereis a low set of d.n.c. degree. The construction of A can be seen as an attemptto salvage a bit of its proof in the case that Y is ∆ 0 2. Where does the proof gowrong? As before, suppose that f = Φ Y is a d.n.c. function. A ∆ 0 2 set Y cannotcompute a stage s(e) such that Φ Yss (p(e)) is stable from s(e) on, only a stagewhere it has the final value for the first time. The problem is that it may changetemporarily after such a stage.On the other hand, we do not have to code the halting problem into Y . Itsuffices to code the promptly simple set A we are building. For each e we meetthe prompt simplicity requirement from the proof of Theorem 1.7.10PS e : #W e = ∞⇒∃s∃x [x ∈ W e,s − W e,s−1 & x ∈ A s ].At stage s we let x enter A for the sake of a requirement PS e only if Φ Y (p(e))has been stable from stage x to s. If we now threaten that J(p(e)) = Φ Y (p(e))then Y has to change to a value not assumed between stage x and s. This allowsus to compute A from Y . It also places a strong restriction on PS e : wheneverΦ Y (p(e)) changes another time at t, then PS e cannot put any numbers less thant into A at a later stage.We work with the effective approximation f s (x) =Φ Y (x)[s]. (Thus, in contrastto 4.1.11, we actually consider changes of the value f s (x) rather than changesof the oracle Y .) For an appropriate computable function p, when we want toput a candidate x ∈ W e into A for the sake of PS e , we threaten that f(p(e))equal J(p(e)). We only put x into A at stage s if f t (p(e)) has been constant sincestage x. (Since f t (p(e)) settles, this holds for large enough x. Thus, PS e is stillable to choose a candidate.) To show A ≤ T f,iff(p(e)) = f t (p(e)), then afterstage t a number x cannot go into A for the sake of PS e . See Fig. 4.1.Proof details. As in the proof of Theorem 4.1.11, we define an auxiliary partialcomputable function α. By the Recursion Theorem we are given a reductionfunction p for α, namely, ∀e α(e) ≃ J(p(e)).

152 4 Diagonally noncomputable functionsxsf(p(e)) is stablex enters W and A eConstruction of A and α. Let A 0 = ∅.Fig. 4.1. Proof of Kučera’s Theorem.Stage s. For each e

4.2 Injury-free constructions of c.e. sets 1534.2.4 Remark. If Y is ∆ 0 2 and ML-random, then a somewhat simpler argumentsuffices to obtain a promptly simple set A ≤ wtt Y (and in fact the use equals theinput in the reduction procedure). We do not need the Recursion Theorem or areduction function. To ensure that A ≤ wtt Y we can directly force Y ↾ e to changeby including the string Y ↾ e in an interval Solovay test G (that is, a certaineffective listing of strings defined in 3.2.22). In other words, if Y is ML-randomwe may let Y ↾ e play the role of f(p(e)) before, so the reduction function p is notneeded any longer. Since the procedure to compute the d.n.c. function λn. Y ↾ nis fixed we obtain A effectively. The reduction procedure to compute A from Yis effectively given by the above, but now it is only correct for almost all inputsbecause σ ⋠ Y merely holds for almost all σ in G.Construction of A and G. Let A 0 = ∅.Stage s>0. For each e

154 4 Diagonally noncomputable functionsProof. Instead of the prompt simplicity requirements PS e we now merely meet therequirements R e : #W e = ∞⇒A∩W e ≠ ∅. To ensure A ≤ wtt C, we ask that C permitthe enumeration of x. Thus, at stage s of the construction, for each e

4.3 Strengthening the notion of a d.n.c. function 155Proof details. Numbers a, b are given (think of them as ∆ 0 2-indices for Y and Z).Let A = W r(a) and B = W r(b) . In a construction relative to ∅ ′ , we build ∆ 0 2 setsY,Z and descending sequences of nonempty Π 0 1 classes (P e ) e∈N and (Q e ) e∈N(they correspond to the Π 0 1 classes in the proof of Theorem 1.8.39 defined ateven stages). We need to be more specific as to how the parallel search is to becarried out. To do so, we divide stages i =0, 1,... into substages t.Construction relative to ∅ ′ of Π 0 1 classes P i ,Q i and strings σ i on P i , τ i on Q i .Let σ 0 = τ 0 = ∅ and P 0 = Q 0 = P .Stage i +1, i =2e. Let Q i+1 = Q i , t = |τ i |, and τ i+1,t = τ i . Go to substage t +1.Substage t +1. Check whether(a) there are σ,k such that |σ|,k

156 4 Diagonally noncomputable functionsSets of PA degreeA highness property of a set D stronger than having d.n.c. degree is obtainedwhen one requires that there be a d.n.c. function f ≤ T D that only takes valuesin {0, 1}. In a typical argument involving a d.n.c. function f (such as the proof ofTheorem 4.1.11) we build a partial computable α and have a reduction function pfor α; when we define α(e) = 0, say, we know that f(p(e)) ≠ J(p(e)) = α(e). If fis {0, 1}-valued, we may in fact conclude that f(p(e)) = 1. Thus we may prescribethe value f(p(e)), while before we could only avoid a value. Since f = Φ D for agiven Turing reduction Φ, we can indirectly restrict D.Up to Turing degree, the {0, 1}-valued d.n.c. functions coincide with the completionsof Peano arithmetic PA; see Exercise 4.3.7. This justifies the followingterminology.4.3.1 Definition. We say that a set D has PA degree if D computes a{0, 1}-valued d.n.c. function.The set ∅ ′ has PA degree by Remark 1.8.30. Exercise 5.1.15 shows that the classof sets of PA degree is null, so this highness property is indeed much strongerthan having d.n.c. degree. Recall from Example 1.8.32 that the {0, 1}-valuedd.n.c. functions form a Π 0 1 class. Thus, by the basis theorems of Section 1.8,there is a set of PA degree that is low, and also a set of PA degree that iscomputably dominated. These examples show that a set can be computationallystrong in one sense, but weak in another.We give two properties that are equivalent to being of PA degree. Both assertthat the set is computationally strong in some sense.4.3.2 Theorem. The following are equivalent for a set D.(i) D has PA degree.(ii) For each partial computable {0, 1}-valued function ψ, there is a total functiong ≤ T D that extends ψ, namely, g(x) =ψ(x) whenever ψ(x)↓. Moreover,one may choose g to be {0, 1}-valued.(iii) For each nonempty Π 0 1 class P , there is a set Z ∈ P such that Z ≤ T D.In other words, the sets Turing below D form a basis for the Π 0 1 classes.Proof. (i)⇒(ii): Suppose f ≤ T D is a {0, 1}-valued d.n.c. function. Let p bea reduction function for ψ, that is, ∀x ψ(x) ≃ J(p(x)). Then ∀x ¬f(p(x)) =J(p(x)). So, if ψ(x) converges, then f(p(x)) = 1 − ψ(x). Let g ≤ T D be the{0, 1}-valued function given by g(x) =1− f(p(x)), then g extends ψ.(ii)⇒(iii): Let P = ⋂ s P s be an approximation of P by a descending effectivesequence of clopen sets; see (1.17) on page 52. For a number x (viewed as abinary string), if s is least such that [xi] ∩ P s = ∅ for a unique i ∈ {0, 1}, thendefine ψ(x) =1− i. (If we see that the extension xi is hopeless, then ψ dictatesto go the other way.) On many strings x, the function ψ makes no decision. Solet g be a total extension as in (ii), and define Z recursively by Z(n) =g(Z ↾ n ).Then Z ≤ T D and Z is in P .

4.3 Strengthening the notion of a d.n.c. function 157(iii)⇒(i): This holds because the {0, 1}-valued d.n.c. functions form a Π 0 1 class.✷Exercises.4.3.3. The equivalence (i)⇔ (ii) above fails for d.n.c. sets without the restriction thatthe function computed by D is {0, 1}-valued: show that the following are equivalentfor a set D. (i) ∅ ′ ≤ T D. (ii) Each partial computable function ψ can be extended to a(total) function g ≤ T D.4.3.4. (Kučera) Show that for each low set Z there is a promptly simple set A suchthat Z ⊕ A is low.4.3.5. ⋄ (Jockusch, 1989) In Definition 4.3.1, the condition that D computes a d.n.c.function with range bounded by a constant would be sufficient: suppose that f is ad.n.c. function with bounded range. Show that there is a {0, 1}-valued d.n.c. functiong ≤ T f.The next two exercises assume familiarity with Peano arithmetic (see Kaye 1991).The sentences in the language of arithmetic L(+, ×, 0, 1) are effectively encoded bynatural numbers, using all the natural numbers. Let α n be the sentence encoded by n.Let ṅ be the effectively given term in the language of arithmetic that describes n.4.3.6. (Scott) Show that a set D has PA degree ⇔ there is a complete extension B ofPeano arithmetic such that B ≤ T D.4.3.7. ⋄ (Scott) Improve the result of previous exercise: D has PA degree ⇔ Peanoarithmetic has a complete extension B ≡ T D.Martin-Löf random sets of PA degreeThe following theorem of Stephan (2006) shows that there are two types of MLrandomsets: the ones that are not of PA degree and the ones that compute thehalting problem. However, such a dichotomy only applies for the particular highnessproperty of having PA degree. A ML-random set can satisfy other highnessproperties, such as being high, without computing ∅ ′ (see 6.3.14).4.3.8 Theorem. If a ML-random set Z has PA degree then ∅ ′ ≤ T Z.Proof. If the proof seems hard to follow, read the easier proof of Theorem 4.3.9first, where a similar technique is used.Let Φ be a Turing functional such that Φ X (n) is undefined or in {0, 1} for eachX, n, and Φ Z is a d.n.c. function. If ∅ ′ ≰ T Z we build a uniformly c.e. sequence(C d ) d∈N of open sets such that λC d ≤ 2 −d and Z ∈ C d for infinitely many d,so Z fails this Solovay test.We define an auxiliary {0, 1}-valued partial computable function α. BytheRecursion Theorem (see Remark 4.1.12) we may assume we are given a reductionfunction p for α, namely, a computable strictly increasing function p such thatα(e) ≃ J(p(e)) for each e. Since Φ Z is {0, 1}-valued, for r ∈ {0, 1}, if we defineα(x) =1− r, we enforce that Φ Z (p(x)) = r.Let (n d ) d>0 be defined recursively by n 1 = 0 and n d+1 = n d + d. The valuesof α on the interval I d =[n d ,n d+1 ) are used to ensure that λC d ≤ 2 −d . When denters ∅ ′ at stage s, consider the set B of strings σ such that Φ σ s ↾ p(nd+1 ) converges.

158 4 Diagonally noncomputable functionsInformally we let C d be the set of oracles Y ∈ [B] ≺ for which Φ Y seems to bea {0, 1}-valued d.n.c. function on p(I d ), namely, for all x ∈ I d , Φ Y (p(x)) ≠J(p(x)) = α(x). Now define α in such a way that λC d is minimal. Then λC d ≤2 −d because there are 2 d ways to define α on I d .If∅ ′ ≰ T Z then for infinitelymany d, Φ Z ↾ p(nd+1 ) converges before d enters ∅ ′ ,soZ ∈ C d .Construction of a uniformly c.e. sequence of open sets (C d ) d>0 and a partialcomputable function α.Stage s. Do nothing unless a number d>0 (unique by convention) enters ∅ ′ at s.In that case let B = {σ : Φ σ s ↾ p(nd+1 )↓}. Let τ d be a string of length d such thatλC d becomes minimal, whereC d =[{σ ∈ B : ∀i

4.3 Strengthening the notion of a d.n.c. function 159Turing degrees of Martin-Löf random sets ⋆We consider ML-randomness in the context of Turing degree structures. Let Ddenote the partial order of all Turing degrees, and let ML ⊆ D denote the degreesof ML-random sets. We know from Theorem 3.3.2 that the degree of A ⊕∅ ′contains a ML-random set for each A. However,ML is not closed upwards in D:4.3.11 Proposition. For each degree c, {x: x ≥ c} ⊆ ML ⇔ c ≥ 0 ′ .Proof. ⇐: This follows from 3.3.2 because each degree d ≥ 0 ′ is in ML.⇒: Suppose that C ≱ T ∅ ′ . Let P be the Π 0 1(C) class of {0, 1}-valued d.n.c.functions f relative to C (i.e., ∀e ¬f(e) =J C (e)). We relativize Theorem 1.8.39to C. Avoiding the cone above ∅ ′ , we obtain a set Y ∈ P such that Y ⊕ C ≱ T ∅ ′ .Since Y ⊕ C has PA degree, it is not in the same Turing degree as a ML-randomset, for otherwise Y ⊕ C ≥ T ∅ ′ by Theorem 4.3.8.✷This yields a natural first-order definition of 0 ′ in the structure consistingof the partial order D with an additional unary predicate for ML. Shore andSlaman (1999) proved that the jump operator is first-order definable in the partialorder D. Their definition uses metamathematical notions and codings of copiesof N with first-order formulas. Even though later on, Shore (2007) found a proofthat does not rely on metamathematics, we still do not know a natural first-orderdefinition of the jump operator (or even of 0 ′ )inD.Next, we study ML within D T (≤ 0 ′ ), the Turing degrees of ∆ 0 2 sets. By Exercise4.2.7, there is an incomputable c.e. set Turing below any pair of ML-random∆ 0 2 sets. So no pair of degrees in ML ∩ D T (≤ 0 ′ ) has infimum 0.4.3.12 Proposition. ML ∩ D T (≤ 0 ′ ) is not closed upwards in D T (≤ 0 ′ ). Also,ML ∩ L is not closed upwards in L, the set of low degrees.Proof. The {0, 1}-valued d.n.c. functions form a Π 0 1 class, which has a low memberD by Theorem 1.8.37. Then the degree of D is not in ML by 4.3.8. On theother hand, by Theorem 4.3.2(iii) there is a ML-random set Z ≤ T D. ✷4.3.13 Remark. Kučera (1988) proved that there is a minimal pair of sets ofPA degree. In fact,0 ′ = inf{a ∨ b: a, b are PA degrees & a ∧ b = 0}.Thus there also is a natural first-order definition of 0 ′ in the structure consistingof the partial order D with an additional unary predicate for being a PA degree.To show that A ⊕ B ≥ T ∅ ′ for each minimal pair A, B of sets of PA degree,one builds a nonempty Π 0 1 class P without computable members such thatX ⊕ Y ≥ T ∅ ′ whenever X, Y ∈ P and X ≠ ∗ Y . Since there are X, Y ∈ P suchthat X ≤ T A and Y ≤ T B, this implies A ⊕ B ≥ T ∅ ′ . To obtain P , take a c.e.set S ≡ T ∅ ′ such that N−S is introreducible. Say, S is the set of deficiency stagesfor a computable enumeration of ∅ ′ (see the comment after Proposition 1.7.6).Let S = U ∪ V be a splitting of S into computably inseparable sets (Soare 1987,

160 4 Diagonally noncomputable functionsThm. X.2.1) and let P = {Z : U ⊆ Z & V ∩ Z = ∅}. IfX, Y are as above, thenX∆Y is an infinite subset of N − S, so that ∅ ′ ≤ T X ⊕ Y .Next, one shows that each Σ 0 2 set C> T ∅ ′ bounds a minimal pair A, B of PAsets by a technique similar to the one in the solution to Exercise 1.8.46. Now letC 0 ,C 1 > T ∅ ′ be a minimal pair of Σ 0 2 sets relative to ∅ ′ , and let A i ,B i ≤ T C i beminimal pairs of sets of PA degree for i ∈ {0, 1}. Then 0 ′ =(a 0 ∨ b 0 ) ∧ (a 1 ∨ b 1 ).Relating n-randomness and higher fixed point freenessDefinition 4.0.1 can be relativized: a function f is d.n.c. relative to C if f(e) ≠J C (e) for any e such that J(e)↓. Forn ≥ 1, we say that f is n-d.n.c. if f is d.n.c.relative to ∅ (n−1) . For example, if n>0 then ∅ (n) computes a {0, 1}-valued n-d.n.c. function, by relativizing Remark 1.8.30 to ∅ (n−1) .By Proposition 4.1.2 relativized to ∅ (n−1) , each n-random set computes an n-d.n.c. function. Thus, for this particular aspect of the computational complexity,a higher degree of randomness implies being more complex. The result is notvery satisfying yet because we would prefer highness properties that are notobtained by mere relativization of a highness property to ∅ (n−1) . For this reasonwe introduce the hierarchy of n-fixed point free functions. It turns out to coincideup to Turing degree with the hierarchy of n-d.n.c. functions. For sets A, B,let A ∼ 1 B if A = B and A ∼ 2 B if A = ∗ B. For n ≥ 3, let A ∼ n B ifA (n−3) ≡ T B (n−3) .4.3.14 Definition. Let n ≥ 1. We say that a function g is n-fixed point free(n-f.p.f. for short) if W g(x) ≁ n W x for each x.For instance, let g ≤ T ∅ ′′ be a function such that W g(x) = ∅ if W x is infiniteand W g(x) = N otherwise, then g is 2-f.p.f. (By Exercise 4.1.7 in relativized formand a slight modification of Exercise 4.1.8, these properties are independent ofthe particular choice of jump operator or universal uniformly c.e. sequence, asfar as the Turing degree of the function is concerned. In Kučera’s notation thehierarchies begin at level 0, but here we prefer notational consistency with thehierarchy of n-randomness for n ≥ 1.)We will need the Jump Theorem of Sacks (1963c). It states that from a Σ 0 2set S one may effectively obtain a Σ 0 1 set A such that A ′ ≡ T S ⊕∅ ′ . (In fact onecan also achieve that C ≰ T A for a given incomputable ∆ 0 2 set C.) For a proofsee Soare (1987, VIII.3.1). In fact we need a version of the theorem for the m-thjump.4.3.15 Theorem. Let m ≥ 0. FromaΣ 0 m+1 set S one may effectively obtain aΣ 0 1 set A such that A (m) ≡ T S ⊕∅ (m) .Proof. The case m = 0 is trivial, and the case m = 1 is the Jump Theoremitself. For the inductive step, suppose that m ≥ 1 and S is Σ 0 m+1. Then Sis a Σ 0 m(∅ ′ ) set. By the result for m relative to ∅ ′ , using ∅ ′ we may obtain aΣ 0 1(∅ ′ ) set B such that (B ⊕∅ ′ ) (m) ≡ T S ⊕∅ (m+1) . Then by the Limit Lemmawe may in fact effectively obtain a Σ 0 2 index for B. By the Jump Theorem, we

4.3 Strengthening the notion of a d.n.c. function 161may effectively obtain a c.e. set A such that A ′ ≡ T B ⊕∅ ′ , and hence A (m+1) ≡ TS ⊕∅ (m+1) .✷The main result of this subsection is due to Kučera.4.3.16 Theorem. Let n ≥ 1. Each n-d.n.c. function computes an n-f.p.f. function,and vice versa.Proof. The case n = 1 is covered by the proof of Proposition 4.1.4, so we mayassume that n ≥ 2. For each set E ⊆ N and each i ∈ N, let(E) i = {n: 〈n, i〉 ∈E}.1. If an n-d.n.c. function f is given, we define an n-f.p.f. function g ≤ T f.Case n =2. Since the index set {e: W e is finite} is c.e. relative to ∅ ′ , there isa Turing functional Φ such that, for each input x, Φ ∅′ (x) is the first i in anenumeration relative to ∅ ′ such that (W x ) i is finite if there is such an i, andΦ ∅′ (x) is undefined otherwise. Let p be a reduction function (Fact 1.2.15) suchthat Φ ∅′ (x) ≃ J ∅′ (p(x)) for each x, and let g ≤ T f be a function such thatW g(x) = {〈y, j〉: j ≠ f(p(x))}. IfΦ ∅′ (x) is undefined then (W x ) i is infinite foreach i while (W g(x) ) f(p(x)) = ∅, soW g(x) ≠ ∗ W x .IfΦ ∅′ (x) =i then (W g(x) ) i = Nsince f(p(x)) ≠ i, so again W g(x) ≠ ∗ W x .Case n =3. We modify the foregoing proof. For each set B and each i ∈ N, let[B]≠i = {〈n, j〉 ∈B : j ≠ i}. By the finite injury methods of Theorems 1.6.4and 1.6.8, there is a low c.e. set B such that (B) i ≰ T [B]≠i for each i. Since Bis low, the relation {〈x, i〉: W x ≤ T [B]≠i } is Σ 0 3 (see Exercise 1.5.7, which isuniform), so there is a Turing functional Φ as follows: on input x, Φ ∅′′ (x) isthe first i in an enumeration relative to ∅ ′′ such that W x ≤ T [B]≠i if there issuch an i, and Φ ∅′′ (x) is undefined otherwise. Let p be a reduction function suchthat Φ ∅′′ (x) ≃ J ∅′′ (p(x)) for each x, and let g ≤ T f be a function such thatW g(x) =[B]≠f(p(x)) .IfΦ ∅′′ (x) is undefined then W x ≰ T [B]≠i for each i andhence W x ≰ T W g(x) .IfΦ ∅′′ (x) =i then W x ≤ T [B]≠i . Since f(p(x)) ≠ i we have(B) i ≤ T W g(x) , and hence W g(x) ≰ T W x because (B) i ≰ T [B]≠i .Case n ≥ 4. Let m = n − 3 and R = ∅ (m) . We run a finite injury constructionof the kind mentioned in the foregoing proof relative to R, and code R intoeach (B) i . In this way we obtain a set B that is c.e. in R such that B ′ ≡ T R ′ ,R ≤ T (B) i and (B) i ≰ T [B]≠i for each i.There is a computable function h such that W x(m) = Wh(x) R , so the relation{〈x, i〉: W x(m) ≤ T [B]≠i } is Σ 0 3(R). Since B ′ ≡ T R ′ , there is a Turing functional Φas follows: on input x, Φ R′′ (x) is the first i in an enumeration relative to R ′′ suchthat W x (m) ≤ T [B]≠i if there is such an i, and Φ R′′ (x) is undefined otherwise.Let p be a reduction function such that Φ R′′ (x) ≃ J R′′ (p(x)) for each x, andlet ˜g ≤ T f be like g before, namely W R˜g(x) =[B] ≠f(p(x)). As before one showsthat WR˜g(x) ≢T W x (m) for each x. Now, by the uniformity of Theorem 4.3.15,

162 4 Diagonally noncomputable functionsthere is a function g ≤ T ˜g such that (W g(x) ) (m) ≡ T WR˜g(x)⊕ R for each x. Since(m)R ≤ T WR˜g(x), this implies that Wg(x) ≢T W x(m) for each x. (This proof actuallyworks for n = 3 as well, in which case we re-obtain the previous one.)2. If an n-f.p.f. function g is given, we define an n-d.n.c. function f ≤ T g.Weusea result of Jockusch, Lerman, Soare and Solovay (1989) which can also be foundin Soare (1987, pg. 273): if ψ is partial computable relative to ∅ (n−1) then thereis a (total) computable function r such that W ψ(x) ∼ n W r(x) whenever ψ(x) isdefined. (Say, if n = 2, fix a Turing functional Φ such that ψ = Φ ∅′ , and letψ s (x) ≃ Φ ∅′ (x)[s]. As long as e = ψ s (x), W r(x) follows the enumeration of W e .)Now let ψ = J ∅(n−1) .Ifψ(x)↓ thenW ψ(x) ∼ n W r(x) ≁ n W g(r(x)) ,soψ(x) ≠ g(r(x)).Hence f = g ◦ r is n-d.n.c. and f ≤ T g.As a consequence we obtain the following result of Kučera (1990).4.3.17 Corollary. If Z is n-random then there is an n-f.p.f. function g ≤ T Z.Proof. By Proposition 4.1.2 relative to ∅ (n−1) , a finite variant f of the functionλn.Z ↾ n is n-d.n.c. There is an n-f.p.f. function g ≤ T f.✷The converse fails because the n-random degrees are not closed upwards. For n =1this follows from Proposition 4.3.11, and for n ≥ 2 it follows because each 2-random setforms a minimal pair with ∅ ′ . Also note that for each n>1 there is an (n − 1)-randomset Z ≤ T ∅ (n−1) . Then Z does not compute an n-f.p.f. function.Our proof of Corollary 4.3.17 is somewhat indirect as it relies on Schnorr’s Theorem3.2.9 relative to ∅ (n−1) . The proofs in Kučera (1990) are more self-contained (butalso longer). The ideas needed in the proof of Theorem 4.3.16 were introduced there.For instance, Kučera’s proof of (1.) in the case n = 2 is as follows. Given x ∈ N, usingthe infinite set Z as an oracle, we may compute n x = µn. #Z ∩ [0,n)=x. Let g ≤ T Zbe a function such that, for each x, W g(x) = {〈y, i〉: i0, once distinct elements a 0,...,a x−1 havebeen enumerated into W ∅′h(x) , let Gx = {Y : ∀i

5Lowness properties and K-trivialityIn this chapter and in Chapter 8 we will study lowness properties of a set A.In particular, we are interested in the question of how they interact with conceptsrelated to randomness. The main new notion is the following: A is K-trivial if∀n K(A↾ n ) ≤ K(n)+bfor some constant b, namely, up to a constant the descriptive complexity of A↾ nis no more than the descriptive complexity of its length. This expresses that Ais far from being ML-random, since ML-random sets have a quickly growing initialsegment complexity by Schnorr’s Theorem 3.2.9. We show that K-trivialitycoincides with the lowness property of being low for ML-randomness.Lowness properties via operators. Recall from Section 1.5 that a lowness propertyof a set A specifies a sense in which A is computationally weak. We always requirethat a lowness property be closed downward under Turing reducibility. Mostly,computational weakness of A means that A is not very useful as an oracle.Lowness properties of a set A are very diverse, as each one represents only aparticular aspect of how information can be extracted from A. They can evenexclude each other for incomputable sets.We have introduced several lowness properties by imposing a restriction onthe functions A computes. For instance, A is computably dominated if eachfunction f ≤ T A is dominated by a computable function (Definition 1.5.9). Inthis chapter we define lowness properties by the condition that the relativizationof a class C to A is the same as C. For example let C = ∆ 0 2. For each A, bytheLimit Lemma 1.4.2 relative to A, wehaveC A = ∆ 0 2(A) ={X : X ≤ T A ′ }.Foran operator C mapping sets to classes, we say that A is low for C if C A = C. Wewrite Low(C) for this class.The condition A ∈ Low(C) means that A is computationally weak in the sensethat its extra power as an oracle does not expand C, contrary to what one wouldusually expect. When C = ∆ 0 2, A is low for C iff A ′ ≡ T ∅ ′ , that is, A is low inthe usual sense of 1.5.2.In some cases, a lowness property implies that the set is in ∆ 0 2. Clearly this isso for the usual lowness. In contrast, being computably dominated is a lownessproperty such that the only ∆ 0 2 sets with that property are the computable ones.One may view the computably dominated sets as a class of the type Low(C) byletting C X be the class of functions dominated by a function f ≤ T X.An operator C is called monotonic if A ≤ T B → C A ⊆ C B . An operator Dis called antimonotonic if A ≤ T B → D A ⊇ D B . The operator X → ∆ 0 2(X)is monotonic, while MLR, the operator given by Martin-Löf randomness relative

164 5 Lowness properties and K-trivialityto an oracle, is antimonotonic. For both types of operators, the correspondinglowness notion is indeed closed downward under Turing reducibility. For instance,A ≤ T B and C B = C implies C ⊆ C A ⊆ C B = C.Let D be a randomness notion. Informally, A is in Low(D) if the computationalpower of A does not help us to find new regularities in a set that is random inthe sense of D. Although D A is ultimately defined in terms of computationswith oracle A, the operator D looks at this information extracted from A in asophisticated way via D-tests relative to A. For this reason, studying the behaviorof D A often yields interesting results on the computational complexity of A. Inthis chapter we focus on lowness for ML-randomness. In Chapter 8 we studylowness for randomness notions weaker than ML-randomness.Existence results and characterization. A computable set satisfies every lownessproperty. Sometimes one is led to define a lowness property and then discoversthat only the computable sets satisfy it. Consider the operator C(X) =Σ 0 1(X):here A ∈ Low(C) implies that A is computable, because both A and N − A arein Σ 0 1(A) =Σ 0 1.It can be difficult to determine whether a lowness property only applies tothe computable sets. If it does so, this is an interesting fact, especially in thecase of a class Low(C); an example is lowness for computable randomness byCorollary 8.3.11. However, it is also the final result.On the other hand, if an incomputable set with the property exists, then oneseeks to understand the lowness property via some characterization. This is especiallyuseful when the property is given in the indirect form Low(C). In thatcase, one seeks to characterize the class by conditions of the following types.(1) The initial segments of A have a slowly growing complexity.(2) The functions computed by A are restricted.A main result of this chapter is a characterization of the first type: A is low forML-randomness iff A is K-trivial. This is surprising, because having a slowlygrowing initial segment complexity expresses that A is far from random, ratherthan computationally weak. This result provides further insight into the classLow(MLR). For instance, we use it to show that Low(MLR) induces an ideal inthe Turing degrees. This fails for most of the other lowness properties we study.Theorem 8.3.9 below is a characterization of the second type: a set is lowfor Schnorr randomness iff it is computably traceable (a strengthening of beingcomputably dominated).Overview of this chapter. In Section 5.1 we introduce several lowness propertiesand show their coincidence with Low(MLR). In Section 5.2 we study K-trivialityfor its own sake, proving for instance that each K-trivial set is in ∆ 0 2. In Section5.3 we introduce the cost function method. It can be used to build K-trivialsets, or sets satisfying certain lowness properties. Different solutions to Post’sproblem can be viewed as applications of this method for different cost functions.There is a criterion on the cost function, being nonadaptive, to tell whether thissolution is injury-free. Section 5.4 contains the proof that each K-trivial set is

5.1 Equivalent lowness properties 165low for ML-randomness, introducing the decanter and golden run methods. Section5.5 applies these methods to derive further properties of the K-trivial sets.In Section 5.6 we introduce the informal concept of weak reducibilities, whichare implied by ≤ T . We study the weak reducibility ≤ LR associated with theclass Low(MLR). We also prove the coincidence of two highness properties: wecharacterize the class {C : ∅ ′ ≤ LR C} by a strong domination property.Some key results of this chapter are non-uniform. For instance, even thoughevery K-trivial set is low, we cannot effectively obtain a lowness index from ac.e. K-trivial set and its constant (Proposition 5.5.5). Corollary 5.1.23 is alsonon-uniform, as discussed in Remark 5.1.25.5.1 Equivalent lowness propertiesWe study three lowness properties that will later turn out to be equivalent: beinglow for K, low for ML-randomness, and a base for ML-randomness.The first two properties indicate computational weakness as an oracle. A is lowfor K if A as an oracle does not help to compress strings any further. A is lowfor ML-randomness if each ML-random set is already ML-random relative to A.The third property, being a base for ML-randomness, is somewhat different: A isconsidered computationally weak because the class of oracles computing A lookslarge to A itself, in the sense that some set Z ≥ T A is ML-random relative to A.The first two implications are easy to verify: A is low for K ⇒ A is low forML-randomness ⇒ A is a base for ML-randomness. The remaining implicationis the main result of this section: A is a base for ML-randomness ⇒ A is lowfor K (Theorem 5.1.22).A fourth equivalent property is lowness for weak 2-randomness. The implicationA is low for weak 2-randomness ⇒ A is low for K is obtained at the end of thissection, the converse implication only in Theorem 5.5.17 of Section 5.5.Being low for K5.1.1 Definition. We say that A is low for K if there is b ∈ N such that∀y [K A (y) ≥ K(y) − b].Let M denote the class of sets that are low for K.Each computable set is low for K. Also, M is closed downward under Turingreducibility, because B ≤ T A implies ∀y [K B (y) ≥ + K A (y)] by Exercise 3.4.4.The notion was introduced by Andrej A. Muchnik in unpublished work datingfrom around 1999. He built an incomputable c.e. set that is low for K. Later inthis section we will reprove his result, but at first in a different, somewhat indirectway: we apply Kučera’s construction in 4.2.1 to obtain a c.e. incomputable set ATuring below a low ML-random set. Then, in Corollary 5.1.23 we prove that eachset A of this kind is low for K. However, in the proof of Theorem 5.3.35 we alsogive a direct construction of a set that is low for K, similar to Muchnik’s.

166 5 Lowness properties and K-triviality5.1.2 Proposition. Each set A ∈ M is generalized low in a uniform way. Moreprecisely, from a constant b one can effectively determine a Turing functional Ψ bsuch that, if A satisfies ∀y [K(y) ≤ K A (y)+b], then A ′ = Ψ b (∅ ′ ⊕ A).Proof idea. Recall that J A (e) denotes Φ A e (e), and Js A (e) denotes Φ A e,s(e). Fora stage s, using A as an oracle we can check whether Js A (e) ↓. So all we need isa bound on the last stage when this can happen: if such a stage s exists thenK A (s) and hence K(s) is at most e + O(1). Thus ∅ ′ can compute such a bound.Proof details. Let M c , c > 1, be an oracle prefix-free machine such thatMc Z (0 e 1) ≃ µs. Js Z (e)[s] ↓ for each Z and each e. Thus Mc Z (0 e 1) convergesiff J Z (e) converges. Since Mc Z (0 e 1) ≃ U Z (0 c−1 10 e 1), we have K Z (Mc Z (0 e 1)) ≤e+c+1 for each Z. Thus, if A is low for K via b then K(Mc A (0 e 1)) ≤ e+c+1+bfor each e. To compute A ′ from ∅ ′ ⊕ A, given input e use ∅ ′ to determinet = max{U(σ) :|σ| ≤ e + c +1+b}. Then J A (e) ↓↔ Jt A (e) ↓, so output 1if Jt A (e) ↓, and 0 otherwise. The reduction procedure was obtained effectivelyfrom b.✷For a c.e. set A, modifying the argument improves the result.5.1.3 Proposition. Each c.e. set A ∈ M is superlow. The reduction procedurefor A ′ ≤ tt ∅ ′ can be obtained in a uniform way.Proof. By Proposition 1.4.4 (which is uniform) it suffices to show A ′ ≤ wtt ∅ ′ .Let M c , c>1, be an oracle prefix-free machine such that for each e,M A c (0 e 1) ≃ µs. J A s (e)[s]↓ & A s ↾ use J A s (e) =A ↾ use J A s (e).As before J A (e) ↓↔ Mc A (0 e 1) ↓, in which case J A (e) has stabilized by stageMc A (0 e 1). Also K Z (Mc Z (0 e 1)) ≤ e + c + 1 for each Z. To compute A ′ from ∅ ′ ,given input e, use ∅ ′ to determine t = max{U(σ) :|σ| ≤ e + c +1+b}; thenJ A (e) ↓↔ J A (e) ↓ [t]. The use of this reduction procedure is bounded by thecomputable function λe. p(2 e+c+1+b ), where p is a reduction function for thepartial computable function U.✷Using a more powerful method, in Corollary 5.5.4 we will remove the restrictionthat A be c.e., by showing that each set A ∈ M is Turing below some c.e. set D ∈ M.This will supersede Proposition 5.1.2 (except for the uniformity statement).We also postpone the result that M is a proper subclass of the superlow sets. By theresults in Section 5.4, M induces a proper ideal in the ∆ 0 2 Turing degrees, while thereare superlow c.e. sets A 0,A 1 such that ∅ ′ ≡ T A 0 ⊕ A 1, Theorem 6.1.4. Alternatively,there is a superlow c.e. set that is not low for K by Proposition 5.1.20 below; the proofrelies on a direct construction.Exercises. The first two are due to Merkle and Stephan (2007).5.1.4. Let A be low for K. Suppose Z ⊆ N and let Y = Z△A be the symmetricdifference. Show that ∀nK(Y ↾ n)= + K(Z ↾ n). In particular, if Z is ML-random then Yis ML-random as well by Schnorr’s Theorem. (Compare this with 3.6.21.)5.1.5. Continuing 5.1.4, suppose that A is incomputable and Z is 2-random.Let Y = Z△A. Show that Y | T Z. (Use that A is ∆ 0 2 by 5.2.4(ii) below.)

5.1 Equivalent lowness properties 1675.1.6. (Kučera) Show that there is a function F ≤ wtt ∅ ′ dominating each functionpartial computable in some set that is low for K.Lowness for ML-randomnessSee the introduction to this chapter for background on lowness for operators.5.1.7 Definition. A is low for ML-randomness if MLR A = MLR.Low(MLR) denotes the class of sets that are low for ML-randomness.Thus, A is low for ML-randomness if MLR A is as large as possible. This propertywas introduced by Zambella (1990). He left the question open whether someincomputable set is low for ML-randomness.The question was answered in the affirmative in work, dating from 1996,of Kučera and Terwijn (1999). They actually built a c.e. incomputable set inLow(MLR). Exercise 5.3.38 below asks for a direct construction of such a set.By the next fact, the existence of such a set also follows from Muchnik’s resultthat some incomputable c.e. set is low for K.Our first proof that a c.e. incomputable set exists in Low(MLR) is actually viaKučera’s Theorem 4.2.1; see page 170.5.1.8 Fact. Each set that is low for K is low for ML-randomness.Proof. Recall that Theorem 3.2.9 can be relativized: Z is ML-random relativeto A ⇔∀nK A (Z ↾ n ) ≥ + n. Thus the class of ML-random sets MLR can becharacterized in terms of K, and MLR A in terms of K A .IfA is low for K thenthe function abs(K − K A ) is bounded by a constant, so MLR = MLR A . ✷In Corollary 5.1.10 we will characterize Low(MLR) using only effective topologyand the uniform measure: A ∈ Low(MLR) ⇔ if G is open, c.e. in A, and λG

168 5 Lowness properties and K-trivialityProof. (iii)⇒(ii): trivial.(ii)⇒(i): If (5.1) holds then non-MLR A ⊆ S. By Fact 1.8.26, there is a computableantichain B such that [B] ≺ = S. By Proposition 3.2.15, non-MLR A isclosed under taking off finite initial segments. Thus non-MLR A ⊆ ⋂ n [Bn ] ≺ .ByExample 3.2.23, ([B n ] ≺ ) n∈N is a Solovay test. Hence non-MLR A ⊆ non-MLR, thatis, A is low for ML-randomness.(i)⇒(iii): Suppose that M is an oracle prefix-free machine for which (5.2) fails.We build a set Z ∈ MLR − MLR A , whence A is not low for ML-randomness. Welet Z = z 0 z 1 z 2 ... for an inductively defined sequence of strings z 0 ,z 1 ... withthe properties (a) and (b) below.(a) We ensure that K M A(z i ) ≤ |z i | − 1. Thus Z ∉ MLR A by Proposition 3.2.17relativized to A.(b) Let H be a c.e. open set such that λH

5.1 Equivalent lowness properties 169is that A is easy to compute in that the class of oracles computing A is large.Given a set A, let S A = {Z : A ≤ T Z}. The set A is considered not complexif S A is large, and complex if S A is small. If smallness merely means being anull class then we can only distinguish between the computable sets A, whereS A =2 N , and the incomputable sets A, where S A is null. In the present formthis result appeared in Sacks (1963a), but it is in fact an easy consequence of aresult in de Leeuw, Moore, Shannon and Shapiro (1956).5.1.12 Theorem. A is incomputable ⇔ the class S A = {Z : A ≤ T Z} is null.Proof. ⇐: If A is computable then S A =2 N ,soλS A =1.⇒: For each Turing functional Φ letS A Φ = {Z : A = ΦZ }.Thus S A = ⋃ Φ SA Φ . It suffices to show that each SA Φ is null. Suppose for acontradiction that λSΦA ≥ 1/r for some r ∈ N. Then A is a path on the c.e.binary treeT = {w : λ[{σ : Φ σ ≽ w}] ≺ ≥ 1/r}.Each antichain on T has at most r elements, for if w 0 ,...,w r is an antichain withr + 1 elements, we have r + 1 disjoint sets [{σ : Φ σ = w i }] ≺ , each of measure atleast 1/r, which is impossible. Then there is n 0 such that for each n ≥ n 0 , A↾ n isthe only string w on T of length n extending A↾ n0 . For otherwise we could pickr + 1 strings on T branching off A at different levels, and these strings wouldform an antichain with r + 1 elements.Now, to compute A(m) for m ≥ n 0 , wait till some w ≽ A ↾ n0 of length m +1is enumerated into T , and output w(m).✷5.1.13 Remark. For n>0 letS A Φ,n =[{σ : A↾ n≼ Φ σ }] ≺ .Then S A Φ,n is Σ0 1(A) uniformly in n. IfA is incomputable then S A Φ = ⋂ n SA Φ,n isnull, so lim n λS A Φ,n = 0. That is, (SA Φ,n ) n∈N is a generalized ML-test relative to A(Definition 3.6.1).If A itself is ML-random, then leaving out the first few components even turns(S A Φ,n ) n∈N into a ML-test relative to A (Miller and Yu, 2008).5.1.14 Proposition. Suppose A is ML-random. Then for each Turing functionalΦ there is a constant c such that (S Φ A,n+c ) n∈N is a ML-test relative to A.The easiest proof is obtained by observing that the c.e. supermartingale (seeDefinition 7.1.5 below) given by L(x) =2 |x| λ[{σ : Φ σ ≽ x}] ≺ is bounded by 2 cfor some c along A. The details are provided on page 266.Exercises.5.1.15. Show that the sets of PA degree (Definition 4.3.1) form a null class.5.1.16. Suppose that A and Y are ML-random and A ≤ T Y . Show that if Y is (i) Demuthrandom, (ii) weakly 2-random, (iii) ML-random relative to a set C, then A hasthe same property.

170 5 Lowness properties and K-trivialityBases for ML-randomnessWe continue to work on the question to what extent high computational complexityof a set A is reflected by smallness of the class S A of sets computing A.We have seen that interpreting smallness by being null is too coarse since thismakes all the incomputable sets complex. Rather, we will consider the case thatS A is null in an effective sense relative to A. From Remark 5.1.13 we obtain thefollowing.5.1.17 Corollary. Let A be incomputable. If Z is weakly 2-random relative to Athen A ≰ T Z.✷Thus S A looks small to an incomputable A even in a somewhat effective sense:S A does not have a member that is weakly 2-random relative to A. On the otherhand, by the Kučera-Gács Theorem 3.3.2 S A always contains a ML-random set.If A is ML-random then by Proposition 5.1.14 S A does not contain a ML-randomset relative to A.Is there any incomputable set A such that S A contains a ML-random set relativeto A? This property was first studied by Kučera (1993).5.1.18 Definition. A is a base for ML-randomness if A ≤ T Z for some set Zthat is ML-random relative to A.Kučera used the term “basis for 1-RRA”. There is no connection to basis theorems.Each set A that is low for ML-randomness is a base for ML-randomness.For, by the Kučera-Gács Theorem 3.3.2 there is a ML-random set Z such thatA ≤ T Z. Then Z is ML-random relative to A.We will prove two theorems. They are due to Kučera (1993) and Hirschfeldt,Nies and Stephan (2007), respectively.Theorem 5.1.19: There is a promptly simple base for ML-randomness.Theorem 5.1.22: Each base for ML-randomness is low for K.This completes the cycle: the classes of sets that are low for K, low for MLrandomness,and bases for ML-randomness are all the same! Moreover, thiscommon class reaches beyond the computable sets:5.1.19 Theorem. There is a promptly simple base for ML-randomness.Proof. Let Z be a low ML-random set, say Z = Ω 0 , the bits of Ω in the evenpositions (see Corollary 3.4.11). By Theorem 4.2.1 there is a promptly simpleset A ≤ T Z. Then, by 3.4.13, Z is in fact ML-random relative to A. ✷For the proof of Theorem 5.1.22 below it might be instructive to begin with a directconstruction of a superlow c.e. set A which is not a base for ML-randomness. (Before5.1.4 we already discussed how to obtain a superlow c.e. set that is not low for K,so this also follows from 5.1.22).5.1.20 Proposition. There is a superlow c.e. set A such that for each set Z, ifA ≤ T Zthen Z is not ML-random relative to A.

5.1 Equivalent lowness properties 171Proof sketch. For each e we build a ML-test (Ce,d) A d∈N relative to A in such a waythat for each j = 〈e, d〉, the following requirement is met:R j : ∀Z [A = Φ Z e⇒ Z ∈ C A e,d].We make A low by meeting the lowness requirements L e from the proof of Theorem1.6.4. When J A (e) newly converges we inititalize the requirements R j for j>e.The strategy for R j is as follows. At each stage let k be the number of times R j hasbeen initialized.1. Choose a large number m j.2. At stage s let S j,s be the clopen set {Z : A↾ mj = Φ Z e ↾ mj [s]}. While λS j,s < 2 −d−k−1 ,put S j,s into C A e,d with use m j on the oracle A. Otherwise put m j − 1intoA, initializethe requirements L e for e ≥ j (we say that R j acts) and goto 1.Thus, the strategy for R j keeps putting S j,s into C A e,d until this makes the contribution(for this value of k) too large; if it becomes too large, the strategy removes the currentcontribution by changing A. Intheconstruction, at stage s let the requirement ofstrongest priority that requires attention carry out one step of its strategy.Verification. Each requirement R j acts only finitely often: λS j,s can reach 2 −d−k−1 atmost 2 d+k+1 times, since the different versions of S j,s are disjoint. Thus A is low andeach m j reaches a limit. Next, for each e, d we have λCe,d A ≤ ∑ k 2−d−k−1 ≤ 2 −d ,so(Ce,d) A d∈N is a ML-test relative to A. IfA = Φ Z e , for each d let j = 〈e, d〉 and consider thefinal value of m j. Since A↾ mj = Φ Z e ↾ mj , Z is in Ce,d. A Thus Z is not ML-random relativeto A. Finally, there is a computable function f such that f(e) bounds the number ofinjuries to L e,soA is superlow.✷5.1.21 Remark. (The accounting method) We describe an important methodto show that a c.e. set L of requests is a bounded request set. We associate arequest 〈r, x〉 with an open set C such that λC ≥ 2 −r . The open sets belongingto different requests are disjoint. Then the total weight ∑ r 2−r [[〈r, y〉 ∈L] is atmost the sum of the measures of those sets, and hence at most 1. Informallyspeaking, we “account” the enumeration of 〈r, x〉 against the measure of theassociated open set. We usually enumerate these open sets actively. When theyreach the required measure we are allowed to put the request into L.We are now ready for the main theorem of this section.5.1.22 Theorem. Each base for ML-randomness is low for K.Proof. Suppose that A ≤ T Z for some set Z that is ML-random relative to A.Given a Turing functional Φ, we define an oracle ML-test (Cd X) d∈N. IfA = Φ Z ,we intend to use this test for X = A. The goal is as follows: if d is a numbersuch that Z ∉ Cd A , then A is low for K via the constant d + O(1). To realizethis goal, we build a uniformly c.e. sequence (L d ) d∈N of bounded request sets.The constructions for different d are independent, but uniform in d, so that foreach X, the sequence of open sets (Cd X) d∈N is uniformly c.e. in X. (This ideawas already used in the proof of Theorem 3.2.29: make an attempt for each d.Let the construction with parameter d succeed when a ML-random set Z is not

172 5 Lowness properties and K-trivialityin the d-th component of a ML-test. Here, the ML-tests are relative to A, and Zis ML-random in A.) We denote by C ⊆ {0, 1} ∗ also the open set [C] ≺ .For each computationU η (σ) =y where η ≼ A(that is, whenever y has a U A -description σ), we want to ensure that there isa prefix-free description of y not relying on an oracle that is only by a constantlonger. Thus we want to put a request 〈|σ| + d +1,y〉 into L d .The problem is that we do not know A, so we do not know which η’s to accept;if we accept too many then L d might fail to be a bounded request set. To avoidthis, the description U η (σ) = y first has to prove itself worthy. Once U η (σ)converges, we enumerate open sets C η d,σ (if Uη (σ) diverges then C η d,σ remainsempty). We let⋃Cd X =C η d,σ . (5.3)η≺X,σ∈{0,1} ∗As long as λC η d,σ < 2−|σ|−d we think of C η d,σas “hungry”, and “feed” it withfresh oracle strings α such that η ≼ Φ α |α| . Since λCη d,σ never exceeds 2−|σ|−d ,wehave λCdX ≤ 2−d Ω X ≤ 2 −d ,so(Cd X) d∈N is an oracle ML test.All the open sets C η d,σ are disjoint. If λCη d,σ exceeds 2−|σ|−d−1 at some stage,then we put the request 〈|σ| + d +1,y〉 into L d . As described in Remark 5.1.21,we may account the weight of those requests against the measure of the sets C η d,σbecause the measure of C η d,σis at least the weight of the request. This shows thateach L d is a bounded request set.Because Z is ML-random relative to A, there is d such that Z ∉ Cd A. Thisimplies that, whenever U η (σ) =y in the relevant case that η ≺ A, then λC η d,σ =2 −|σ|−d , and hence the request 〈|σ| + d +1,y〉 is enumerated into L d . For, ifλC η d,σ < 2−|σ|−d , then once a sufficiently long initial segment α of Z computes η,we would feed α to C η d,σ , which would put Z into CA d. (Intuitively speaking,lots of sets compute A, so we are able to feed all the sets C η d,σfor η ≺ A andσ ∈ dom(U η ).)The open sets Cd A correspond to the open sets Ce,d A in the proof of Proposition 5.1.20.However, in the present proof, a Turing reduction Φ such that A = Φ Z is given inadvance, so one does not need the parameter e. In Proposition 5.1.20, we changed Aactively to keep Ce,d A small, and if A = Φ Z e then Z was “caught” in ⋂ d CA e,d. Here,(Cd A ) d∈N is an ML-test relative to A by definition. The set Z is caught in ⋂ d CA d if Ais not low for K as witnessed by a computation U η (σ) =y for η ≺ A.Construction of c.e. sets C η d,σ ⊆ {0, 1}∗ , σ, η ∈ {0, 1} ∗ , for the parameter d ∈ N.Initially, let C η d,σ = ∅.Stage s. In substages t, 0≤ t

5.1 Equivalent lowness properties 173If α has been declared used for d, as defined below, then go to the next α.Otherwise, see whether there are σ, η such that• U η s(σ) ↓ and η is minimal such, namely, U η′s (σ) ↑ for each η ′ ≺ η,• η ≼ Φ α |α| , and• λC η d,σ [s, t]+2−s ≤ 2 −|σ|−d .Choose σ least and put α into C η d,σ. Declare all the strings ρ ≽ α used for d.Verification. For fixed d, the sets [C η d,σ ]≺ (σ, η ∈ {0, 1} ∗ ) are disjoint, becauseduring each stage s we only enumerate unused strings of length s into these sets.Once λC η d,σ reaches 2−|σ|−d−1 , we enumerate 〈|σ| + d +1,y〉 into L d . Then L d isa bounded request set by the accounting method of Remark 5.1.21.Define CdX by (5.3). Since (CA d ) d∈N is a ML-test relative to A, there is d suchthat Z ∉ Cd A. We verify that L d works. Suppose that U A (σ) =y, and let η ≼ Abe shortest such that U η (σ) =y. We claim that λC η d,σ =2−|σ|−d , so that we areallowed to put the required request 〈|σ| + d +1,y〉 into L d when the measure ofC η d,σ has reached 2−|σ|−d−1 .Assume for a contradiction that λC η d,σ < 2−|σ|−d , and let s be so large thatU η s(σ) =y, η ≼ Φ Z s and λC η d,σ +2−s ≤ 2 −|σ|−d . Then α = Z ↾ s+1 enters C η d,σ atstage s, unless it enters some C η′d,σinstead or is used for d, namely, some β ≺ α′has entered a set C η′′d,σat a previous stage. Because A = Φ Z , in any case, η ≺ A,′or η ′ ≺ A, orη ′′ ≺ A. ThusZ ∈ Cd A contrary to our hypothesis on d. ✷The following shows that a certain class defined in terms of plain ML-randomness,rather than relativized ML-randomness, is contained in the c.e. sets that are lowfor K. In Section 8.5 we will consider subclasses of M in more detail.5.1.23 Corollary. Suppose A is c.e. and there is a ML-random set Z ≥ T Asuch that ∅ ′ ≰ T Z. Then A is low for K.Proof. If A is not low for K, by Theorem 5.1.22, Z is not ML-random relativeto A. Then, by Proposition 3.4.13, ∅ ′ ≤ T A ⊕ Z ≡ T Z.✷We do not know at present whether this containment is strict: the followingquestion is open.5.1.24 Open question. If A is c.e. and low for K, is there a ML-random setZ ≥ T A such that ∅ ′ ≰ T Z?5.1.25 Remark. It is not hard to see that the proof of Theorem 5.1.22 is uniform, inthe sense that if Z is ML-random in A and Φ Z = A, then a constant b such that A islow for K via b can be obtained effectively from an index for Φ and a constant c suchthat Z /∈ R A c . On the other hand, Hirschfeldt, Nies and Stephan (2007) proved thatCorollary 5.1.23 is necessarily nonuniform: One cannot effectively obtain a constant bsuch that A is low for K via b even if one is given c such that Z ∉ R c, Φ, and a lownessindex for Z, that is, an index p such that Z ′ = Φ p(∅ ′ ). (They actually show it for K-triviality instead of being low for K. However, by the proof of Proposition 5.2.3 below,

174 5 Lowness properties and K-trivialitythe implication “low for K ⇒ K-trivial” is uniform in the constants, so the result alsoapplies for the constant via which A is low for K.)5.1.26 Remark. We describe an unexpected application of Theorem 5.1.22. We saythat S ⊆ 2 N is a Scott class (or Scott set) if S is closed downwards under Turingreducibility, closed under joins, and each infinite binary tree T ∈ S has an infinite pathin S. Scott classes occur naturally in various contexts, such as the study of models ofPeano arithmetic and reverse mathematics. The arithmetical sets form a Scott class.On the other hand, Theorem 1.8.37 in relativized form shows that there is a Scott classconsisting only of low sets.Kučera and Slaman (2007) showed that Scott classes S are rich: for each incomputableX ∈ S there is Y ∈ S such that Y | T X (this answered questions of H. Friedman andMcAllister).They choose Y ∈ MLR X . Then Y | T X unless X is a base for ML-randomness, andhence K-trivial. In that case, they build an infinite computable tree T such that a setZ ∈ Paths(T )isnotK-trivial and satisfies Z ≱ T X. For the latter they use the Sackspreservation strategy (see Soare 1987, pg. 122).Exercises. Show the following.5.1.27. If A ∈ Low(Ω) (Definition 3.6.17) and A ∈ ∆ 0 2 then A ∈ Low(MLR).5.1.28. There is an ω-c.e. set A for which Corollary 5.1.23 fails.5.1.29. Each low set A is a “base for being of PA degree” via a low witness: there isa low set D ≥ T A which is of PA degree relative to A, namely, there is {0, 1}-valuedfunction f ≤ T D such that ∀e ¬f(e) =J A (e).5.1.30. If Y and Z are sets such that Y ⊕ Z ∈ MLR, then each set A ≤ T Y,Z is lowfor K. (For instance, let Y be the bits in an even positions and let Z be the bits in anodd position in the binary representation of Ω. Compare this to Exercise 4.2.7.)5.1.31. ⋄ Prove Corollary 5.1.23 directly by combining the proofs of 5.1.22 and 3.4.13.Lowness for weak 2-randomnessA recurrent goal of this book is to characterize the class Low(C) for a randomnessnotion C. Here and in Section 8.3 we will consider, more generally, lowness forpairs of randomness notions such that C ⊆ D (relative to each oracle). RelativizingD to A increases the power of the associated tests, so one would expect thatin general C ⊈ D A . We consider the class of sets A for which, to the contrary,the containment persists when we relativize D.5.1.32 Definition. (Kjos-Hanssen, Nies and Stephan, 2005) A is in Low(C, D)if C ⊆ D A .ThusLow(C) =Low(C, C).We study classes of the form Low(C, D) for various reasons.(a) The proof techniques suggest so.(b) We can deal with more than one randomness notion at the same time.(c) These classes may coincide with interesting computability theoretic classes.For instance, if C is ML-randomness and D is Schnorr randomness thenLow(C, D) coincides with the c.e. traceable sets by Theorem 8.3.3 below.

5.1 Equivalent lowness properties 175We frequently use the fact that, if C ⊆ ˜C ⊆ ˜D ⊆ D are randomness notions,then Low( ˜C, ˜D) ⊆ Low(C, D). That is, the class Low(C, D) is enlarged by eitherdecreasing C or increasing D. In particular, both Low(C) and Low(D) arecontained in Low(C, D).Weak 2-randomness was introduced in Definition 3.6.1, and W2R ⊆ MLR arethe classes of weakly 2-random and ML-random sets, respectively. The followingresult is due to Downey, Nies, Weber and Yu (2006).5.1.33 Theorem. Low(W2R, MLR) =Low(MLR).Thus Low(W2R) ⊆ Low(MLR). Theorem 5.5.17 below shows the converse containment,so that actually Low(W2R) =Low(MLR).Proof. Suppose that A ∉ Low(MLR). By the characterization of Low(MLR)in 5.1.9, there is no c.e. open set R such that λR γ}.Whenever K A (z) ≤ |z| − 1 then z ∈ G. Let S =[G] ≺ . Let (y i ) i

176 5 Lowness properties and K-trivialityλ(G e n e) ≤ 2 −|we|−e−2 .Then λ(G e n e|w e ) ≤ 2 −(e+2) . Since H e+1 = H e ∪ G e n e,λ(H e+1 | w e ) ≤ γ e +2 −(e+2) < γ e+1 .By Claim 5.1.34 for V = H e+1 , w = w e , β = γ e +2 −(e+2) , and γ = γ e+1 > β, wecan choose z = z e such that K A (z) ≤ |z| − 1 and λ(H e+1 | w e z) ≤ γ e+1 .Thus(5.4) holds for e + 1 where w e+1 = w e z.If Z ∈ G e n e, there is m>n e such that [w m ] ⊆ G e n e⊆ H m as G e n eis open.However, since λ(H m |w m ) < 1 by (5.4), we have [w m ] ⊈ H m for each m. ThusZ ∉ G e n e.✷5.2 K-trivial setsWe say that a set A is K-trivial if up to an additive constant the functionλn. K(A↾ n ) grows no faster than the function λn.K(n). Thus, A↾ n has no moreinformation than its length has. It is easily verified that each set that is low for Kis K-trivial, so by Section 5.1, page 170, there is a promptly simple K-trivialset. In Proposition 5.3.11 we will give a direct construction of such a set, afterintroducing the cost function method.We already stated in the introduction to this chapter that the sets that are lowfor ML-randomness (or low for K) actually coincide with the K-trivial sets. Thiswe will prove in Section 5.4. Here we study K-triviality for its own sake, mostlyby combinatorial means. Given the equivalence with the lowness properties ofSection 5.1, this leads to results involving those properties which would be hardto obtain if their definitions were used directly. For instance, it is not too difficultto show that each K-trivial set is ∆ 0 2. A direct proof of this result is possible,but more difficult, for the sets that are low for ML-randomness (Nies, 2005a).A further example of this is the closure of the K-trivial sets under the operation ⊕on sets, where no direct proof is known using the definition of Low(MLR).Some results of this section will be improved in the Sections 5.4 and 5.5 using“dynamic” methods, such as the golden run. For instance, once we know thatbeing K-trivial is the same as being low for K, we may conclude from Proposition5.1.2 that the K-trivial sets are not only ∆ 0 2, but in fact low. Alternatively,in Corollary 5.5.4 we use the golden run method to show directly that each K-trivial set is superlow, and hence ω-c.e.Basics on K-trivial setsEach prefix-free description of a string y also serves as a description for |y|.Thus K(|y|) ≤ + K(y) for each y. Clearly ∀nK(n) = + K(0 n ), so the followingproperty of a set A expresses that the K-complexity of the initial segments of Agrows as slowly as the one of a computable set.5.2.1 Definition. A is K-trivial via b ∈ N if ∀n K(A↾ n ) ≤ K(n)+b.Let K denote the class of K-trivial sets.

5.2 K-trivial sets 177Although we have defined the class of K-trivial sets in terms of the standardprefix-free universal machine, it actually does not depend on this particularchoice: a change of the universal machine would merely lead to a differentconstant b.The intuitive meaning of K-triviality is to be far from ML-random. By Theorem3.2.9 A is ML-random iff ∀n K(A ↾ n ) ≥ n − c for some c. Thus, A isML-random if for each n, the number K(A↾ n ) is within K(n)+c +1≤ + 2 log nof its upper bound n+K(n)+1 given by Theorem 2.2.9. In contrast, A is K-trivialif K(A↾ n ) is within a constant of its lower bound K(n).We say that A is C-trivial if ∀nC(A ↾ n ) ≤ C(n)+b for some b ∈ N (Chaitin,1976). Computable sets are both K-trivial and C-trivial. Chaitin proved thatthere are no further C-trivial sets (Theorem 5.2.20 below). He still managed toshow that all K-trivial sets are in ∆ 0 2. Solovay (1975) constructed an incomputableK-trivial set. These results will be covered in the present and in thenext subsection.Intuitively, an incomputable K-trivial set exists because both sides of the defininginequality ∀nK(A ↾ n ) ≤ K(n)+b are noncomputable, and also because wedo not ask for uniformity in the inequality. In particular, we do not requirethat a short description of A ↾ n can be obtained from a short description of n.Chaitin’s argument gets around this for C-trivial sets because the computableupper bound C(n) ≤ 1+log(n+1) is attained in each interval [2 i −1, 2 i+1 −1). So,roughly speaking, one can replace the right hand side C(n)+b in the definitionof C-triviality by such a computable bound.5.2.2 Fact. Each computable set A is K-trivial.Proof. On input σ, the prefix-free machine M attempts to compute n = U(σ),and outputs A↾ n . Then ∀nK M (A↾ n ) ≤ K(n)+b where b is the coding constantfor M. Alternatively, by Example 2.2.16, W = {〈n +1,A ↾ n 〉: n ∈ N} is abounded request set, so ∀nK(A↾ n ) ≤ + K(n).✷By Theorem 5.1.19, there is a promptly simple base for ML-randomness, andby Theorem 5.1.22 each base for ML-randomness is low for K. Then the followingimplies that there is a promptly simple K-trivial set.5.2.3 Proposition. (Extends 5.2.2) Each set that is low for K is K-trivial.Proof. We actually show that there is a fixed d such that, if A is low for K viaa constant c, then A is K-trivial via the constant c + d. Let M be the oracleprefix-free machine such that M X (σ) ≃ X ↾ U(σ) for each X, σ, and let d be thecoding constant for M. Then K X (X ↾ n ) ≤ K(n)+d for each X and n. HenceK(A↾ n ) ≤ K A (A↾ n )+c ≤ K(n)+c + d for each n.✷K-trivial sets are ∆ 0 2The following theorem of Chaitin (1976) shows that the K-trivial sets are rare:for each constant b there are at most O(2 b ) many. As a consequence, each K-trivial set is ∆ 0 2. In Corollary 5.5.4 we improve this: each K-trivial set is superlow.

178 5 Lowness properties and K-trivialityHowever, the work in this section is not wasted, since the dynamic methods usedthere rely on a computable approximation of the set.5.2.4 Theorem.(i) There is a constant c ∈ N such that for each b, at most 2 c+b sets are K-trivial with constant b.(ii) Each K-trivial set is ∆ 0 2.Proof. (i). The paths of the ∆ 0 2 treeT b = {z : ∀u ≤ |z| [K(z ↾ u ) ≤ K(u)+b]}coincide with the sets that are K-trivial via the constant b. Ifc is the constantof Theorem 2.2.26(ii), then for each n the size of the level {z ∈ T b : |z| = n} isat most 2 c+b .(ii). By (i) each path of T b is isolated, and hence ∆ 0 2 by Fact 1.8.34 relativizedto ∅ ′ .✷We give an affirmative answer to a question of Kučera and Terwijn (1999).After building a c.e. incomputable set in Low(MLR), they asked whether eachset in Low(MLR) isin∆ 0 2. The result was first obtained in a direct manner; seeNies (2005a). Here we use the foregoing result and Theorem 5.1.22.5.2.5 Corollary. Low(MLR) ⊆ ∆ 0 2.Proof. If A ∈ Low(MLR) then A is a base for ML-randomness, and hence lowfor K by Theorem 5.1.22. Each set that is low for K is K-trivial. Then, by theforegoing theorem, A is in ∆ 0 2.✷By Theorem 5.2.4 the class K of K-trivial sets can be represented as an ascendingunion of finite classes (Paths(T b )) b∈N . Note that we do not obtain a uniform listing of∆ 0 2-indices (i.e., of total Turing reductions to ∅ ′ ) for K. An obvious attempt would beto represent a path A of T b by a string σ ∈ T b such that A is the only path extending σ.However, the property of a string σ to be on T b and have a unique path above it is notknown to be ∆ 0 2. Nevertheless, using other methods and the fact that each K-trivial setis ω-c.e., we will see in Theorem 5.3.28 that there is such a listing, which even includesthe constants for K-triviality.As in 1.4.5 let V e be the e-th ω-c.e. set.5.2.6 Fact. {e: V e ∈ K} is Σ 0 3.Proof. V e ∈ K is equivalent to ∃b ∀n ∀s ∃t >s[K t(V e,t ↾ n) ≤ K t(n)+b].As a consequence, the index set {e: W e is K-trivial} is Σ 0 3 as well. Since each finiteset is K-trivial, there is a uniformly c.e. listing of all the c.e. K-trivial sets, usingExercise 1.4.22. Then the index set is Σ 0 3-complete by Exercise 1.4.23. Note that K isalso Σ 0 3 as a class, by a proof similar to the proof of Fact 5.2.6.Exercises.5.2.7. Show that A is low for K ⇔ ∀nK(A↾ n) ≤ + K A (n).5.2.8. Let A be a c.e. set that is wtt-incomplete. Show that ∃ ∞ nK(A ↾ n) ≤ + K(n)and ∃ ∞ nC(A↾ n) ≤ + C(n).✷

5.2 K-trivial sets 1795.2.9. Give a “far-from-random” analog of Proposition 3.2.14. Let R = {r 0 0. On the other hand, theupper bound O(2 b ) is not tight since lim b G(b)/2 b =0.Recall that T b = {x : ∀y ≼ xK(y) ≤ K(|y|)+b}. ThusG(b) =#Paths(T b ).5.2.11 Proposition. Let D : N → N be a nondecreasing function which iscomputably approximable from above and satisfies ∑ b 2−D(b) < ∞. Then thereis ɛ > 0 such that ∀b G(b) ≥⌊ɛ2 b−D(b) ⌋.For instance, if D(b) =2logb, we obtain the lower bound ⌊ɛ2 b /b 2 ⌋ for G(b).Proof. Note that ∀bK(b) ≤ + D(b) by Proposition 2.2.18, and hence ∀xK(x) ≤ +|x| + D(|x|). We may increase D by a constant without changing the validity ofthe conclusion, so let us assume that in fact ∀xK(x) ≤ |x| + D(|x|).There is a constant r ∈ N such that, for each string x and each m ∈ N,K(x0 m ) ≤ K(x)+K(|x|+m)+r, and for each x ′ ≺ x, K(x ′ ) ≤ K(x)+K(|x ′ |)+r,(since x ′ can be computed from x and |x ′ |). Thus, if b ≥ r, then for each x suchthat K(x) ≤ b − r, the set x0 ∞ is K-trivial via b. If|x| ≤ (b − r) − D(b − r)then K(x) ≤ b − r. So the number of such x is at least ⌊2 b−r−D(b−r) ⌋. ThusG(b) ≥⌊2 −r 2 b−D(b) ⌋ for b ≥ r. Since this inequality holds vacuously for b

180 5 Lowness properties and K-trivialityLet S d = {x: K(x) =K(|x|)+d}, and let ˜F (d, n) =#{x: |x| = n & x ∈ S d }.Since {0, 1} ∗ is partitioned into the sets S d , for each n we have∑ ˜F d(d, n)2 −K(n)−d = ∑ |x|=n 2−K(x) .If K(x) ≤ K(|x|)+b then x ∈ S d for a unique d ≤ b. Thus for each n∑F (b, n)2 −K(n)−b = ∑ ∑2 −K(n)−bbb K(x)≤K(n)+b= ∑ ∑ ∑2 −K(n)−bdb≥d|x|=nK(x)=K(n)+d=2 ∑ ˜F (d, n)2 −K(n)−dd=2 ∑2 −K(x) .It now suffices to show that ∑ |x|=n 2−K(x) = O(2 −K(n) ), for then Claim 2 followsafter multiplying by 2 K(n) . Recall from the Coding Theorem 2.2.25 that fora prefix-free machine M we defined P M (y) =λ[{σ : M(σ) =y}] ≺ , and thatP U (y) ∼ 2 −K(y) . Let M be the machine from the proof of Theorem 2.2.26 givenby M(σ) =|U(σ)|. Then P M (n) = ∑ |x|=n P U(x) ∼ ∑ |x|=n 2−K(x) . By 2.2.25,P M (n) =O(2 −K(n) ). Thus ∑ |x|=n 2−K(x) = O(2 −K(n) ) as required.The sequence (G(b)/2 b ) b∈N converges to 0 rather slowly:5.2.13 Proposition. There is no computable function h: N → Q 2 such thatlim b h(b) =0and ∀b [G(b)/2 b ≤ h(b)]. In particular, the function G is not computable.Proof. Assume that such a function h exists. We enumerate a bounded request set L.We assume that an index d for a machine M d is given, thinking of M d as a prefixmachine for L (see Remark 2.2.21).Construction of L. Compute the least b ≥ d such that h(b) < 2 −d . Now enumerate L insuch a way that there are 2 b−d K-trivial sets for the constant b; this is a contradictionsince in that case G(b)/2 b ≥ 2 −d . For each string x of length b − d let A x = x0 ∞ .Whenever s>0 is a stage such that K s(n) 1, is in fact a machine for L, thenK Md (A x ↾ n) ≤ K(n)+b − d for each x, n, whence A x is K-trivial via b.✷Since each A x is finite, the Proposition remains valid when one replaces G by thefunction G fin , where G fin (b) is the number of finite sets that are K-trivial via b.Exercises.5.2.14. There is r 0 ∈ N such that, if A is K-trivial via the constant c, then xA is K-trivial via 2K(x)+c + r 0, for each string x.5.2.15. ⋄ Improve 5.2.13: a function h with the properties there is not even ∆ 0 2.Inparticular G ≰ T ∅ ′ . The same is true for G fin .✷

5.2 K-trivial sets 1815.2.16. ⋄ Problem. It is not hard to verify that G ≤ T ∅ (3) . Determine whetherG ≤ T ∅ (2) . (This may depend on the choice of an optimal machine.)Closure properties of KWe show that K induces an ideal in the ∆ 0 2 weak truth-table degrees. The closureunder ⊕ was proved by Downey, Hirschfeldt, Nies and Stephan (2003).5.2.17 Theorem. If A, B ∈ K then A ⊕ B ∈ K. More specifically, if both Aand B are K-trivial via b, then A ⊕ B is K-trivial via 3b + O(1).Proof idea.By Exercise 5.2.9, it is sufficient to show∀nK(A ⊕ B ↾ 2n ) ≤ K(n)+3b + O(1).The set S n,r = {x: |x| = n & K(x) ≤ r} is c.e. uniformly in n and r. Ifr =K(n)+b, then by Theorem 2.2.26(ii) we have #S n,r ≤ 2 c+b for the constant c.So we may define a prefix-free machine describing a pair of strings in S n,r withK(n)+O(1) bits. It has to describe n only once, using a shortest U-description σ.Thereafter, it specifies via two numbers i, j < 2 c+b in which position the stringsA↾ n and B ↾ n appear in the enumeration of S n,r where r = |σ| + b.Proof details. Let M be the prefix-free machine which works as follows. Oninput 0 b 1ρ search for σ ≼ ρ such that U(σ)↓= n.Ifρ = σαβ where α, β are stringsof length c + b, then let i, j < 2 c+b be the numbers with binary representations1α and 1β. Search for strings x and y, the i-th and the j-th element in thecomputable enumeration of S n,r , respectively. If x and y are found output x ⊕ y.For an appropriate string ρ = σαβ of length K(n)+2(c+b) wehaveM(0 b 1ρ) =A ⊕ B ↾ 2n . Hence K(A ⊕ B ↾ 2n ) ≤ K(n)+3b + O(1).✷Is the class of K-trivial sets closed downward under Turing reducibility? InSection 5.4, with considerable effort, we will answer this question in the affirmativeby showing that K = M. In contrast, an easier fact follows straight from thedefinitions, downward closure under weak truth-table reducibility. This alreadyimplies that K is closed under computable permutations.5.2.18 Proposition.(i) Let A be K-trivial via b. IfB ≤ wtt A then B is K-trivial via a constant ddetermined effectively from b and the wtt reduction.(ii) If A is K-trivial then ∅ ′ ≰ wtt A.Proof. (i) Suppose B = Γ A , where Γ is a wtt reduction procedure with acomputable bound f on the use. Then, for each n,K(B ↾ n ) ≤ + K(A↾ f(n) ) ≤ + K(f(n)) ≤ + K(n).(ii) follows from (i) because Ω is not K-trivial, and Ω ≤ wtt ∅ ′ by 1.4.4.5.2.19 Exercise. If A and B are K-trivial and 0.C =0.A +0.B, then C is K-trivial.✷

182 5 Lowness properties and K-trivialityC-trivial setsIn the next two subsections we aim at understanding K-triviality by varying it.We say that A is C-trivial if ∃b ∀nC(A↾ n ) ≤ C(n)+b (see page 177). A modificationof the proof of Fact 5.2.2 shows that each computable set is C-trivial. Bythe following result of Chaitin (1976) there are no others.5.2.20 Theorem.(i) For each b at most O(b 2 2 b ) sets are C-trivial with constant b.(ii) Each C-trivial set is computable.Proof. We first establish the fact that there are no more than O(b 2 2 b ) M-descriptions of x that are at most C(x)+b long. Thus, surprisingly, the numberof such descriptions depends only on b, not on x.5.2.21 Lemma. Given a machine M, we can effectively find d ∈ N such that,for all b, x,#{σ : M(σ) =x & |σ| ≤ C(x)+b} < 2 b+2 log b+d+5 = O(b 2 2 b ). (5.5)Subproof. The argument is typical for the theory of descriptive string complexity:if there are too many M-descriptions of x, we can find a V-description of xthat is shorter than C(x), contradiction.Recall from page 13 that string(b) is the string identified with b, which haslength log(b + 1). Let ̂b =0 |string(b)| 1string(b) so that |̂b| ≤ 2 log b + 3. We define amachine R. By the Recursion Theorem (with a parameter for M) we may assumethat we are effectively given a coding constant d>0 for R, that is, Φ d = R.b+2 log b+d+5R: For each b, each m ≥ 2 log b + d + 4 and each x, if there are 2strings σ of length at most m + b such that M(σ) =x, let R(̂bρ) =x, forthe leftmost ρ of length m − 2 log b − d − 4 such that ̂bρ is not yet in thedomain of R. Note that |̂bρ|

5.2 K-trivial sets 183which contains at most r strings of each length n. Each set that is C-trivial for bis a path of Tb C , so there are at most r such sets.(ii) Let A be C-trivial via b. The tree TbC is merely ∆ 0 2, so the fact that Ais an isolated path is not sufficient to show that A is computable. Instead, let˜S b ⊇ TbC be the c.e. tree {z : ∀u ≤ |z| [C(z ↾ u ) ≤ 1 + log(u +1)+b]}. For almostevery i, there is a string y of length i such that C(y) =|y| + 1 (Exercise 2.1.19)and therefore (with the usual identifications from page 13) there is a number u,2 i − 1 ≤ u|z| of theform 2 i − 1, A ↾ k is the only string in S b extending z. Since the tree S b is c.e.,we can enumerate it till this string appears. Hence A is computable. ✷Exercises.5.2.22. We say that A is low for C if ∃b∀y [C A (y) ≥ C(y) − b]. Show that each setthat is low for C is computable.5.2.23. Explain why the proof of Theorem 5.2.20(ii) cannot be adapted to show thateach K-trivial set is computable.5.2.24. (Loveland, 1969) Show that the following are equivalent for a set A.(i) A is computable.(ii) There is d ∈ N such that ∀nK(A↾ n| n) ≤ d.(iii) There is b ∈ N such that ∀nC(A↾ n| n) ≤ b.Replacing the constant by a slowly growing function ⋆We replace the right hand side K(n)+O(1) in the definition of K-triviality byK(n)+p(K(n)) + O(1), where p is a function we think of as a slowly growing.For each function p, let K p denote the class of sets A such that∀n K(A↾ n ) ≤ + K(n)+p(K(n)).We prove a result of Stephan which shows that, if p: N → N is unboundedand computably approximable from above (see Definition 2.1.15), the class K pis much larger than K. An example of such a function was given in Proposition2.1.22: p(n) = C(n) = min{C(m): m ≥ n}. The approximability conditionis needed, since Csima and Montalbán (2005) defined a nondecreasingunbounded function f such that A is K-trivial ↔ ∀nK(A↾ n ) ≤ + K(n)+f(n),which implies that A is K-trivial ↔ ∀nK(A↾ n ) ≤ + K(n)+f(K(n)).5.2.25 Theorem. Suppose p is a function such that lim n p(n) =∞ and p iscomputably approximable from above. Then there is a Turing complete c.e. set Esuch that each superset of E is in K p .Proof. For a set X ⊆ N and n ∈ N, we write X for N − X and X ∩ n for X ∩ [0,n).The function g(n) =⌊p(K(n))/2⌋ is computably approximable from above via g s(n) =⌊p s(K s(n))/2⌋. The idea is to make the complement of the c.e. set E very thin, so that

184 5 Lowness properties and K-trivialityfor every A ⊇ E and each n, a description of n and very little extra information yieldsa description of A↾ n. We enumerate E as follows. At stage s,(1) for each rg s(r), put the greatest element ofE s−1 ∩ r into E s.(2) If i ∈∅ ′ s −∅ ′ s−1 then put the i-th element of E s−1 into E.Firstly, we verify that E is Turing complete. By the coding in (2), it is sufficient toshow that E is co-infinite. We prove by induction on q ∈ N that #(E ∩ n) =q forsome n. This is trivial for q =0.Ifq>0, by the inductive hypothesis there is m suchthat #(E ∩ m) =q − 1; let t be the least stage such that E t ∩ m = E ∩ m. Choosek ≥ m, t such that g s(r) >qfor each k ≥ r and each s. Ifq ∈∅ ′ s −∅ ′ s−1 for some s, let vbe the number enumerated in (2) at stage s, else let v = 0. Let n ≥ max{v +1,k+1} beleast such that n − 1 ∉ E k+1 .If#(E ∩ n − 1) >q− 1 we are done. Suppose otherwise,that is, (m, n − 1) ⊆ E. Then n − 1 ∉ E, since n − 1 is not enumerated via (2), nor canany number r ≥ n in (1) demand that #(E ∩ r)

5.3 The cost function method 185In both cases, since the total cost of the enumerations is finite, we can definean auxiliary c.e. object that in some sense has a finite weight. In (I) this objectis a bounded request set showing that A is K-trivial, while in (II) it is a Solovaytest needed to show that A ≤ T Y . Table 5.1 on page 200 gives an overview ofcost functions.Recall that {Y } is a Π 0 2 class for each ∆ 0 2 set Y . A cost function constructionallows us to extend (II): for each Σ 0 3 null class C, there is a promptly simpleset A such that A ≤ T Y for every ML-random set Y ∈ C. This is applied inTheorem 8.5.15 to obtain interesting classes contained in the c.e. K-trivial sets.Most cost functions c(x, s) will be non-increasing in x and nondecreasing in s.That is, at any stage larger numbers are no cheaper, and a number may becomemore expensive at later stages.Note that we have already proved the existence of a promptly simple set thatis low for K and hence K-trivial (see the comment before Proposition 5.2.3).However, the cost function construction in (I) gives a deeper insight into K-triviality. Indeed, we will prove that each K-trivial set can be viewed as beingbuilt via such a construction. For this, it will be necessary to extend the costfunction method to ∆ 0 2 sets: one now considers the sum of the costs c(x, s) ofchanges A s (x) ≠ A s−1 (x). This characterization via a cost function shows thateach K-trivial set A is Turing below a c.e. K-trivial set C, where C is the changeset of A defined in the proof of the Limit Lemma 1.4.2. The only known proofof this result is the one relying on cost functions.To build a promptly simple set A, we meet the prompt simplicity requirementsPS e in the proof of Theorem 1.7.10. Such a requirement acts at most once, andis typically allowed to incur a cost of up to 2 −e . In that case, the sum of thecosts is finite, that is, A obeys the cost function. Instead of 2 −e we could use anyother nonnegative quantity f(e) ∈ Q 2 , as long as the function f is computableand ∑ e f(e) < ∞. We are able to meet each requirement PS e, provided that thecost function satisfies the limit condition, namely, for each e ∈ N, almost all xcost at most 2 −e at all stages s>x.We sketch the construction for (I) above. The standard cost functionc K (x, s) = ∑ x

186 5 Lowness properties and K-trivialityIn the applications (I) and (II) of the method, the cost function is given inadvance. We will also consider the case that c(x, s) depends on A s−1 . Such acost function, called adaptive, is necessary for a direct construction of a promptlysimple set that is low for K (5.3.34). Also adaptive is the cost function used byKučera and Terwijn (1999) to build an incomputable c.e. set that is low for MLrandomness(5.3.38). The latter two cost function constructions merely provethe existence of a promptly simple set in the class K, given that being K-trivialis equivalent to being low for K. However, it is still instructive to study thesedirect constructions because they expose different aspects of K.Adaptive cost functions can be used to hide injury to requirements. For instance,in Remark 5.3.37 we reformulate the construction of a low simple set(which has injury to the lowness requirements) in the language of an adaptivecost function. However, we also argue that a cost function given in advancecannot be used in that way, so the constructions based on non-adaptive costfunctions, such as (I) and (II) above, can be considered injury-free.The basics of cost functions5.3.1 Definition. A cost function is a computable functionc : N × N → {x ∈ Q 2 : x ≥ 0}.We say that c satisfies the limit condition if lim x sup s>x c(x, s) = 0, that is,∀e ∀ ∞ x ∀s >x[c(x, s) ≤ 2 −e ].We say that c is monotonic if c(x +1,s) ≤ c(x, s) ≤ c(x, s + 1) for each x

5.3 The cost function method 187If the computable approximation of a ∆ 0 2 set A is clear from the context, we willalso say that the set A obeys the cost function.We think of a cost function as a description of a class of ∆ 0 2 sets: those setswith an approximation obeying the cost function. For instance, the standard costfunction describes the K-trivial sets. This is somewhat similar to a sentence insome formal language describing a class of structures.We proceed to the general existence theorem. A cost function with the limitcondition has a promptly simple “model”.5.3.5 Theorem. Let c be a cost function with the limit condition. Then thereis a promptly simple set A with a computable enumeration (A s ) s∈N obeying c.Moreover, S ≤ 1/2 in (5.6), and we obtain A uniformly in c.Proof. We meet the prompt simplicity requirements from the proof of 1.7.10PS e : #W e = ∞ ⇒ ∃s ∃x [x ∈ W e,at s & x ∈ A s ](where W e,at s = W e,s − W e,s−1 ). We define a computable enumeration (A s ) s∈Nas follows.Let A 0 = ∅. At stage s>0, for each ex,because c satisfies the limit condition. We enumerate such an x into A at thestage s>xwhere x appears in W e ,ifPS e has not been met yet by stage s.Thus A is promptly simple.If we modify the construction so that each requirement PS e is only allowed tospend 2 −e−2 , we have ensured that S ≤ 1/2. Clearly the construction of A isuniform in an index for the computable function c.✷The c.e. change set C ≥ T A for a computable approximation (A s ) s∈N of a∆ 0 2 set A was introduced in the proof of the Limit Lemma 1.4.2: if s>0 andA s−1 (x) ≠ A s (x) weput〈x, i〉 into C s , where i is least such that 〈x, i〉 ∉ C s−1 .If A is ω-c.e. via this approximation then C ≥ tt A. The following will be used inCorollary 5.5.3, that each K-trivial set is Turing below a c.e. K-trivial set.5.3.6 Proposition. Suppose c is a cost function such that c(x, s) ≥ c(x +1,s)for each x, s. If a computable approximation (A s ) s∈N of a set A obeys c, then thec.e. change set C ≥ T A obeys c as well.Proof. Since x

188 5 Lowness properties and K-trivialityExercises.5.3.7. There is a computable enumeration (A s) s∈N of N in the order 0, 1, 2,... (i.e.,each A s is an initial segment of N) such that (A s) s∈N does not obey c K.5.3.8. Show the converse of Theorem 5.3.5 for a monotonic cost function c :if a computable approximation (A s) s∈N of an incomputable set A obeys c, then csatisfies the limit condition.5.3.9. We view each Φ i as a (possibly partial) computable approximation (A s)byletting A s ≃ D Φi (s). V e is the e-th ω-c.e. set (1.4.5). Let c be a cost function ∀xc(x, s) ≥2(−x). Show that {e: some total computable approximation of V e obeys c} is Σ 0 3.A cost function criterion for K-trivialityWe give a general framework for the cost function construction of a promptlysimple K-trivial set. This construction was already explained on page 185 in theintroduction to this section.5.3.10 Theorem. Suppose a computable approximation (A s ) s∈N of a set A obeysthe standard cost function c K (x, s) = ∑ x0 then the requests in (b)at stage w cause K u (A ↾ w ) ≤ K s (w)+d + 1 for some u>s.IfK s (w) =K(w),we are done. Otherwise, the inequality is caused by a request in (a) at the greateststage t>ssuch that K t (w)

5.3 The cost function method 1895.3.12 Proposition. If a computable approximation (A s) s∈N of a set A obeys the costfunction c K, then for each y, A s(y) for s ≥ h(y) changes at most O(y 2 ) times. Inparticular, A is ω-c.e.Proof. Given y 0. AsS

190 5 Lowness properties and K-trivialityfunctions. A number x can enter A for the sake of PS e only if c(x, s) ≤ 2 −e . Firstly,consider c K(x, s) = ∑ xswe have c(x, t) > 2 −e we may as well assume that the entire interval [x, t) hasbecome unusable for PS e (the numbers with short descriptions at stage t might beclose to t). So PS e will have to look for future candidates x among the numbers ≥ t.For the cost function c Y , whenever Y ↾ e changes at stage t, all the numbers x, listin G all the strings σ of length s such that [σ] ⊆ V x,s . As before, G is an intervalSolovay test by the hypothesis that the approximation of A obeys c.Showing A ≤ T Y is similar to the proof of 5.3.13. Choose s 0 such that σ ⋠ Yfor any σ enumerated into G after stage s 0 . Given an input x ≥ s 0 , using Y asan oracle compute t>xsuch that [Y ↾ t ] ⊆ V x,t . We claim that A(x) =A t (x).Otherwise A s (x) ≠ A s−1 (x) for some s>t, which would cause the strings σ of

5.3 The cost function method 191length s such that [σ] ⊆ V x,s to be listed in G, contrary to Y ∈ V x,t . (In generalthere is no computable bound for the use t on Y ; see Exercise 5.3.19.)Intuitively, we enumerate a Turing functional Γ such that A = Γ Y (see Section6.1). At stage t we define Γ Y (x) =A t (x) for all Y in V x,t . When A s (x) ≠A s−1 (x) for s>twe have to remove all those oracles by declaring them nonrandom.If H is a Σ 0 3 class we slightly extend the argument: by Remark 1.8.58 we haveH = ⋃ ⋂i x V x i for an effective double sequence (Vx) i i,x∈N of Σ 0 1 classes suchthat Vx i ⊇ Vx+1 i for each i, x and Vx i =2 N for i>x. Then the cost functionc(x, s) = ∑ i 2−i λVx,s i is Q 2 -valued and satisfies the limit condition: given k,there is x 0 such that∀i ≤ k +1∀x ≥ x 0 λV i x ≤ 2 −k−1 .Then c(x, s) ≤ 2 −k for all x ≥ x 0 and all s, since the total contribution of terms2 −i λVx,s i for i ≥ k +2toc(x, s) is bounded by 2 −k−1 .Suppose we are given a computable approximation of a set A obeying c. Foreach i, when A s (x) ≠ A s−1 (x) we list the strings σ of length s such that [σ] ⊆ Vx,siin a set G i . Then G i is an interval Solovay test by the definition of c. If Y ∈ Hwe may choose i such that Y ∈ ⋂ x V x.IfY i is also ML-random then we use G ias before to argue that A ≤ T Y .✷As an application we characterize weak 2-randomness within ML-randomness.This was promised on page 135.5.3.16 Theorem. Let Z be ML-random. Then the following are equivalent:(i) Z is weakly 2-random.(ii) Z and ∅ ′ form a minimal pair.(iii) There is no promptly simple set A ≤ T Z.Proof. (i) ⇒ (ii): Suppose the ∆ 0 2 set A is incomputable and A = Φ Z forsome Turing functional Φ. Since {A} is Π 0 2, the class {Y : Φ Y = A} is Π 0 2 byExercise 1.8.65. Also, this class is null by Lemma 5.1.13. Thus Z is not weakly2-random.(ii) ⇒ (iii): Trivial.(iii) ⇒ (i): Suppose Z is not weakly 2-random, then Z is in a null Π 0 2 class H.By Theorem 5.3.15 there is a promptly simple set A ≤ T Z.✷Exercises.5.3.17. Show that Theorem 5.3.15 fails for null Π 0 3 classes.5.3.18. Suppose H = {Y } for a ∆ 0 2 set Y . Recall the representation of the Π 0 2 class Hgiven by Proposition 3.6.2, namely H = ⋂ x Vx for a sequence of uniformly Σ0 1 classes(V x) x∈N where, whenever s>xand r is least such that Y s(r) ≠ Y s−1(r), we put Y s ↾ r+1into V x,s. We may suppose that ∀xY x(x) ≠ Y x+1(x). Let c be the cost function fromthe proof of Theorem 5.3.15. Show that c Y = c.5.3.19. Explain why we merely obtain a Turing reduction in Theorem 5.3.15, and nota weak truth-table reduction as in Remark 4.2.4.

192 5 Lowness properties and K-triviality5.3.20. (i) Show that if Y and Z are sets such that Y ⊕ Z is ML-random and Y isweakly 2-random (for instance, if Y ⊕Z is weakly 2-random), then Y,Z form a minimalpair. (ii) Use this to show there is a minimal pair of computably dominated low 2 sets.K-trivial sets and Σ 1 -induction ⋆In this subsection we assume familiarity with the theory of fragments of Peanoarithmetic; see Kaye (1991). We ask how strong induction axioms are needed toprove that there is a promptly simple K-trivial set. All theories under discussionwill contain a finite set of axioms PA − comprising sufficiently much of arithmeticto formulate number-theoretic concepts, carry out the identifications of binarystrings with numbers on page 13, and formulate and verify some basics on c.e.sets, machines and K. IΣ 1 is the axiom scheme for induction over Σ 1 formulas.Simpson showed that the proof of the Friedberg-Muchnik Theorem 1.6.8 can becarried out in IΣ 1 . See Mytilinaios (1989).The scheme I∆ 0 is induction over ∆ 0 formulas. BΣ 1 is I∆ 0 together withcollection for Σ 1 formulas. BΣ 1 states for instance that f([0,x]) is bounded foreach Σ 1 function f and each x. All formulas may contain parameters. Note thatIΣ 1 ⊢ BΣ 1 but not conversely (see Kaye 1991).Hirschfeldt and Nies proved the following.5.3.21 Theorem. IΣ 1 ⊢ “there is a promptly simple K-trivial set”.Proof sketch. It is not hard, if tedious, to verify that the proofs of Theorems5.3.5 and 5.3.10 can be carried out within IΣ 1 . Thus it suffices to provefrom IΣ 1 that c K satisfies the limit condition in Definition 5.3.1. Let M|= IΣ 1 .Consider the Σ 1 formula ϕ(m, e) given by∃u [ |u| = m +1&∀i (0 ≤ i 2 −e ) ] .Suppose the limit condition fails for c K via e ∈ M. Then, using IΣ 1 , we haveM|= ∀m ϕ(m, e). Now let m be 2 e + 1 (in M), and let u ∈ M be a witnessfor M|= ϕ(m, e). For each i 1,Note that we have actually shown that IΣ 1 suffices for the “benignity” propertyof c K described in Remark 5.3.14.The weaker axiom scheme BΣ 1 is not sufficient to prove that there is a promptlysimple K-trivial set. If M|= I∆ 0 , we say that A ⊆ M is regular if for eachn ∈ M, A ↾ n is a string of M (i.e., A ↾ n corresponds to an element of M viathe usual identifications defined for N on page 12). Each K-trivial set A ⊆ Mis regular because for each n there is in M a prefix-free description of A↾ n . Butthere is a model M|= BΣ 1 in which each regular c.e. set A is computable; seeChong and Yang (2000).Hájek and Kučera (1989) formulated and proved in IΣ 1 a version of the solution toPost’s problem from Kučera (1986) which uses the Low Basis Theorem (see page 150).✷

5.3 The cost function method 193In this context it would be interesting to know to what extent the proof of 5.3.13 canbe carried out in IΣ 1 provided that Y (e) changes sufficiently little, say O(2 e ) times,and whether the alternative solution, avoiding the Low Basis Theorem and using theeven bits of Ω, can be carried out in IΣ 1. It is not known whether IΣ 1 proves thateach K-trivial set A is Turing incomplete (see page 202).Avoiding to be Turing reducible to a given low c.e. setFor many classes of c.e. sets studied in computability theory, given a Turingincomplete c.e. set B, there is a set A in the class such that A ≰ T B. This isthe case, for instance for the class of low c.e. sets (and even of superlow c.e.sets), because there are (super)low c.e. sets A 0 ,A 1 such that A 0 ⊕ A 1 ≡ T ∅ ′ .Forlowness, this follows from the Sacks Splitting Theorem (1.6.10). To extend it tosuperlowness, see Theorem 6.1.4.The class of c.e. K-trivial sets is different: some low 2 c.e. set B is Turingabove all the K-trivial sets by a result of Nies (see Downey and Hirschfeldt20xx). However, one can still build a K-trivial set that is not Turing reducibleto a given low c.e. set B. More generally, this holds for any class of c.e. setsobeying a fixed cost function with the limit condition. To show this, we extendthe construction in the proof of Theorem 5.3.5, by combining it with a methodto certify computations that rely on a given low c.e. set B as an oracle. Thisis known as the guessing method of Robinson (1971). He introduced it to showthat for each pair of c.e. Turing degrees b < a such that b is low, there existincomparable low c.e. Turing degrees a 0 , a 1 such that a 0 ∨a 1 = a and b < a 0 , a 1 .5.3.22 Theorem. Let c be a cost function satisfying the limit condition. Thenfor each low c.e. set B, there is a c.e. set A obeying c such that A ≰ T B.Proof. Recall that N [e] denotes the set of numbers of the form 〈y, e〉. We meetthe requirementsP e : A ≠ Φ B e ,by enumerating a number x ∈ N [e] into A when Φ B e (x) = 0. The problem isthat B may change below the use after we do this, allowing the output of Φ B e (x)to switch to 1. To solve this problem, we use the lowness of B to guess atwhether a computation Φ B e (x)[s] = 0 is correct. Finitely many errors can betolerated. We ask questions about the enumeration of A (involving B), in such away that the answer “yes” is Σ 0 1(B). Since Σ 0 1(B) ⊆ ∆ 0 2, we have a computableapproximation to the answers. Which computable enumeration of A should weuse? We may assume that one is given, by the Recursion Theorem! Formally,we view a computable enumeration (Definition 1.1.15) as an index for a partialcomputable function A defined on an initial segment of N such that, where A(t)isinterpreted as a strong index (Definition 1.1.14) for the part of A enumerated bystage t, wehaveA(s) ⊆ A(s + 1) for each s. Thus we allow partial enumerations.We write A t for A(t). Given any (possibly partial) computable enumeration Ã,we effectively produce an enumeration A, asking Σ 0 1(B)-questions about thegiven enumeration Ã. We must show that A is total in the interesting case

194 5 Lowness properties and K-trivialitythat A = à (by the Recursion Theorem), where these questions are actuallyabout A.The Σ 0 1(B)-question for requirement P e is as follows:Is there a stage s and x ∈ N [e] such that à is defined up to s − 1, and(i) Φ B e (x) = 0[s] &B s ↾ use Φ B e (x)[s] =B ↾ use Φ B e (x)[s] (B is stable up tothe use of the computation), and(ii) c(x, s) ≤ 2 −(e+n) ?Here n =#(N [e] ∩ Ã(s − 1)) is the number of enumerations for the sake of P eprior to s.As B is low, there is a total computable function g(e, s) such that lim g(e, s) =1if the answers is “yes”, and lim g(e, s) = 0 otherwise. (The function g(e, s)actually depends on a further argument which we supress, an index for Ã.)Construction. Let A 0 = ∅. At stage s>0, we attempt to define A s , assumingthat A s−1 has been defined already. Let D = A s−1 . For each e

5.3 The cost function method 195Assume for a contradiction that A = Φ B e . First suppose that lim s g(e, s) =1.Choose witnesses x, s for the affirmative answer to the Σ 0 1(B) question for P eand let u = use Φ B e (x)[s]. Since B ↾ u does not change after s, we search for t tillwe see g(e, t) = 1. Then P e enumerates x at stage s.Now consider the case g(e, s) = 0 for all s ≥ s 0 . Then P e does not enumerateany numbers into A after stage s 0 . Suppose it has put n numbers into A up tostage s 0 . Since A = Φ B e , there is x ∈ N [e] and s ≥ s 0 such that Φ B e (x) =0[s]and c(x, s) ≤ 2 −(e+n) . So the answer to the Σ 0 1(B) question for P e is “yes”,contradiction.✷If B is merely a ∆ 0 2 set, the argument in the proof of Lemma 5.3.25 breaks down inthe case lim s g(e, s) = 1. Otto can now present the correct computation Φ B e (x) =0ata stage s where the g(e, s) has not yet stabilized. To fool us when we try to reconcilethe apparent contradiction, he temporarily changes B below the use at stage t>swhile keeping g(e, t) = 0, and we do not put x into A at s. At a later stage the correctcomputation Φ B e (x) = 0 will return, but now he has increased the cost of x above2 −(e+n) , so we have lost our opportunity.Indeed, Kučera and Slaman (20xx) have shown that some low set B is Turing aboveall the c.e. sets that are low for K (and hence above all the c.e. K-trivial sets bySection 5.4). In fact they prove this for any class inducing a Σ 0 3 ideal in the c.e. degreessuch that there is a function F ≤ T ∅ ′ that dominates each function partial computablein a member of the class. The class of sets that are low for K is of this kind byExercise 5.1.6.5.3.26 Exercise. In addition to the hypotheses of Theorem 5.3.22, let E be a c.e. setsuch that E ≰ T B. Then there is a c.e. set A obeying c such that A ≰ T B and A ≤ T E.Necessity of the cost function method for c.e. K-trivial setsIn the following two subsections we study aspects of the cost function method peculiarto K-triviality. Theorem 5.3.10, the cost function criterion for K-triviality,is actually a characterization: a ∆ 0 2 set A is K-trivial iff some computable approximationof A obeys the standard cost function (as defined in 5.3.4). Thuseach K-trivial set can be thought of as being constructed via the cost functionmethod with c K . However, we cannot expect that every computable approximationof a K-trivial set obeys the standard cost function, for instance because foran appropriate computable enumeration 0, 1, 2,...of N, the total cost of changesis infinite (see Exercise 5.3.7). The total cost is only finite when one views a givenapproximation in “chunks”. One introduces an appropriate computable set E ofstages. In the new approximation changes are only reviewed at stages in E. Inthe calculation of the total cost in (5.6), only the change at the least numbercounts at such a stage.In this subsection we only consider the c.e. K-trivial sets. It is harder to showthe necessity of the cost function method for all the K-trivial sets: this requiresthe golden run method of Section 5.4, and is postponed to Theorem 5.5.2.5.3.27 Theorem. The following are equivalent for a c.e. set A.(i) A is K-trivial.

196 5 Lowness properties and K-triviality(ii) Some computable enumeration (Âi) i∈N of A obeys the standard cost functionc K .Proof. (ii) ⇒ (i): This follows from Theorem 5.3.10.(i) ⇒ (ii): Suppose that the c.e. set A is K-trivial via b. Then there is an increasingcomputable function f such that K(A ↾ n ) ≤ K(n) +b [f(s)] for each sand each n < s, i.e., the inequality holds at stage f(s). The set of stagesE = {s 0

5.3 The cost function method 197in the sequence. Furthermore, there is an effective sequence of the c.e. K-trivialsets with constants.Proof. We effectively transform a pair A, b of an ω-c.e. set and a constant intoan ω-c.e. set à and a constant d such that à is K-trivial via d and A = à incase that A is in fact K-trivial via b. (But even then, d may be larger than b.)To obtain the required listing (B e ,d e ) e∈N , for each e = 〈i, b〉, let A = V i , applythis transformation to the pair A, b, and let B e = à and d e = d.We define a sequence of stages s(i) the same way we did before Theorem 5.3.27,except that now the sequence breaks off if A is not K-trivial via b. The computableapproximation of à follows the approximation of A, but is updated onlyat such stages. In more detail, for each s, letf(s) ≃ µt > s. ∀n

198 5 Lowness properties and K-triviality(b2) Otherwise, i.e., ws[K t(W e,t ↾ n) ≤ K s(n)+b]. Then Theorem 5.3.28 shows:5.3.29 Corollary. The class of c.e. K-trivial sets has a uniformly Σ 0 3 index set. ✷Exercises.5.3.30. Let (B e) e∈N be as in Theorem 5.3.28. Show that there is a computable binaryfunction f such that B f(i,j) = B i ⊕ B j for each i, j ∈ N.5.3.31. Let Q(e, b) be the Π 0 2 relation W e ∪ W b = N & W e ∩ W b = ∅, so that W e iscomputable iff ∃bQ(e, b). Show that Q does not serve as a Π 0 2 relation via which theindex set of the class of computable sets is uniformly Σ 0 3.5.3.32. Anyway, the class of computable sets has a uniformly Σ 0 3 index set.5.3.33. ⋄ Problem. Is every Σ 0 3 index set of a class of c.e. sets uniformly Σ 0 3?Adaptive cost functionsA modification of the proof of Theorem 5.3.11 yields a direct argument thatsome promptly simple set is low for K. Understanding this transition from K-triviality to being low for K may be helpful for the proof in Section 5.4 that thetwo classes are equal (in particular, for Lemma 5.4.10).Given a prefix-free oracle machine M and a ∆ 0 2 set A with a computable approximation(A s ) s∈N , consider the cost functionc M,A (x, s) = ∑ σ2 −|σ| [[M A (σ)[s − 1] ↓ & x

5.3 The cost function method 1995.3.34 Proposition. Suppose that A(x) =lim s A s (x) for a computable approximation(A s ) s∈N that obeys c U,A , namely, there is u ∈ N such thatS = ∑ x,sThen A is low for K.c U,A (x, s)[[s>0&x is least s.t. A s−1 (x) ≠ A s (x)] ≤ 2 u . (5.14)Proof. The proof is somewhat similar to the proof of Theorem 5.3.10. As beforewe enumerate a bounded request set W :at stage s>0, put the request 〈|σ|+u+1,y〉 into W if U A (σ)[s] =ynewly converges, that is, U A (σ)[s] =y but U A (σ)[s − 1]↑.To show that A is low for K, it suffices to verify that W is indeed a boundedrequest set. Suppose a request 〈|σ| + u +1,y〉 is put into W at a stage s via anewly convergent computation U A (σ)[s] =y. Let w = use U A (σ)[s].Stable case. ∀t >sA s ↾ w = A t ↾ w . The contribution to W of such requests isat most Ω A /2 u+1 , since at most one request is enumerated for each descriptionU A (σ) =y once A↾ w is stable.Change case. ∃t >sA s ↾ w ≠ A t ↾ w . Choose t least, and x0, c U,A (x, s) is definedin terms of A s−1 .) The set A is low for K by Proposition 5.3.34. By the sameargument as in the previous proof, we know that A is promptly simple, once wehave shown the following.5.3.36 Claim. c U,A satisfies the limit condition.Given e ∈ N, we will find m such that sup s>m c U,A (m, s) ≤ 2 −e . Note thatif σ ∈ dom U A and t is a stage such that U A (σ)[t] ↓ and each requirementPS j for j ≤ |σ| has ceased to act, then the computation U A (σ)[t] is stable.Let σ 0 ,...,σ k−1 ∈ dom U A be strings such that, where α = ∑ i 2−|σi| ,wehaveΩ A −α ≤ 2 −e−1 . We may choose a stage m ≥ e+1 such that all the computationsU A (σ i )[m] are stable and no PS j , j ≤ e + 1 acts from stage m on. At each stages>mwe have Ω A [s] − Ω A ≤ 2 −e−1 , since the most Ω A [t] can decrease (becausecomputations U A (σ) are destroyed by enumerations into A) is ∑ j>e+1 2−j =2 −e−1 . Thus, Ω A [s] − α ≤ 2 −e for each s>m.Now,fors>m, the sum in (5.13)defining c U,A (m, s) refers to computations other than U A (σ i )[m] as their use isat most m. Soc U,A (m, s) ≤ Ω A [s] − α ≤ 2 −e . This shows the claim. ✷

200 5 Lowness properties and K-trivialityTable 5.1. Overview of cost functions to build a c.e. incomputable set.Cost function Definition Purpose Ref.∑c K (x, s)x

5.4 Each K-trivial set is low for K 201method. Combining this with a further new technique, the golden run method,yields the full result. By Proposition 5.1.2 and Theorem 5.2.4(ii) each set thatis low for K is low, so the full result indeed implies lowness and hence Turingincompleteness for a K-trivial set. In Section 5.5, the core of the combined decanterand golden run methods will be isolated in a powerful (if technical) result,the Main Lemma 5.5.1. It can be used, for instance, to show that Theorem 5.3.10in fact provides a characterization of the K-trivial sets: A is K-trivial iff somecomputable approximation of A obeys the standard cost function c K .Introduction to the proofWe outline the proof of the implication “K-trivial ⇒ low for K”. We also introducesome terminology and auxiliary objects that will be used in the detailedproof. We go through stronger and stronger intermediate results, showing thata K-trivial set is wtt-incomplete, then Turing incomplete, and then low. Eachstep introduces new techniques. Important comments are made in Remarks 5.4.3and 5.4.4.Throughout, we fix a constant b such that the given set A is K-trivial via b, thatis, ∀nK(A ↾ n ) ≤ K(n)+b. By Theorem 5.2.4(ii) we may also fix a computableapproximation (A s ) s∈N of A.1. No K-trivial set A is weak truth-table complete.This was already proved in Proposition 5.2.18(ii). Here we assume ∅ ′ ≤ wtt A andobtain a contradiction. The very basic idea how to use the K-triviality of A is thefollowing: we choose a number n and give it a short description. The opponentOtto claims that A is K-trivial, so he has to respond by giving a short descriptionof A ↾ n . If he later needs to change A ↾ n , he has to provide a description of thenew A ↾ n . Via the short description of n we have made such changes expensivefor him. In this way we limit the changes of A to an extent contradicting itsweak truth-table completeness.As an extreme case, let us assume that for each n, his description of A ↾ n isno longer than our description of n. We choose n large enough so that we canforce Otto to change A ↾ n , using that A is wtt-complete (see below for details).We issue a description of n that has length 0, and wait for the stage where heprovides a description of A ↾ n that also has length 0. He has now wasted allhis capital: if we force him to change A ↾ n , then he has to give up on providingdescriptions.The bounded request set L, and the constant d. Actually, Otto can choose hisdescriptions by a constant longer. To counter this, we force him to change Amore often. His constant is known to us in advance. We issue descriptions ofnumbers n by enumerating requests of the form 〈r, n〉 into a bounded requestset L. By the Recursion Theorem (see Remark 2.2.21), we may assume an index dis given such that M d is a machine for L. To respond to our enumeration of arequest 〈r, n〉 into L, he has to provide a description of A ↾ n that has length atmost r + b + d. For at least 2 b+d times, we want A ↾ n to change after he has

202 5 Lowness properties and K-trivialitygiven such a description. In fact we want A↾ n not to change back, but rather toa configuration not seen before. In that case, if our description of n is of lengthr = 0 (say), his total capital spent is at least (2 b+d +1)2 −(b+d) > 1, contradiction.The hypothesis ∅ ′ ≤ wtt A can be used to force Otto into making sufficientlymany changes. We build a c.e. set B. By the Recursion Theorem, we are given ac.e. index for B, and hence a many-one reduction showing that B ≤ m ∅ ′ (1.2.2).Combining this with the fixed wtt-reduction of ∅ ′ to A, we are given a Turingreduction Γ and a computable function g such that B = Γ A and ∀x use Γ A (x) ≤g(x). (In fact we have used the Double Recursion Theorem 1.2.16, because wealso needed the constant d in advance. This slight technical complication willdisappear when we proceed to showing that all K-trivial sets are low.)Construction of L and the c.e. set B. Let c =2 b+d , and n = g(c). We put thesingle request 〈0,n〉 into L. From now on, at each stage t such that B ↾ c = Γ A ↾ c [t]and K t (A t ↾ n ) ≤ b + d, we force A↾ n to change to a configuration not seen beforeby putting into B the largest number less than c which is not yet in B.In the fixed point case we have B = Γ A , so we can force c such changes. Sinceall the A ↾ n -configurations are different, the measure of their descriptions is atleast (c + 1)2 −(b+d) > 1, which is impossible.2. No K-trivial set A is Turing complete.We assume for a contradiction that A is Turing complete. As before, we buildthe c.e. set B and, by the Recursion Theorem, we are given a Turing functional Γsuch that B = Γ A . Let γ A (m) =use Γ A (m) for each m. We have no computablebound on γ A (m) any longer. If we try to run the previous strategy with n greaterthan the current use γ A (m) for a number m we can put into B, Otto can beatus as follows: first he changes A ↾ n ; when a new computation Γ A (m) appears,then γ A (m) >n. Only then he gives a description of the new version of A ↾ n .Such a premature A-change deprives us of the ability to later cause changes ofA↾ n by putting m into B.To solve the problem we start a new attempt whenever Γ A (m) changes. Butwe have to be careful to avoid spending everything on a failed attempt, so wewill put into L requests of the type 〈r, n〉 for large r and lots of numbers n.Each time we put such a request we wait for the A↾ n description before puttingin the next one. If Γ A (m) changes first then r is increased. Since Γ A is total,Γ A (m) must settle eventually, so r settles as well. On premature A ↾ n -changewe have only wasted 2 −r , but then we increase r. (This is our first example ofa controlled risk strategy. Our risk is only 2 −r for each attempt. We choose thesuccessive values of r so large that their combined risks are tolerable.) On theother hand, if Γ A (m) remains stable for long enough, we can build up in smallportions as much as we would have put in one big chunk. So eventually we canmake an A-change sufficiently expensive for Otto.It is instructive to consider the hypothetical case b = d = 0 once again. Recallfrom Definition 2.2.15 that the weight of E ⊆ N with regard to L is definedby wgt L (E) = ∑ ρ,x 2−(ρ)0 [[ρ ∈ L & x ∈ E &(ρ) 1 = x]. The following procedure

5.4 Each K-trivial set is low for K 203enumerates a set F 2 of weight p =3/4 such that for each n ∈ F 2 , there are U-descriptions of two versions A↾ n . It maintains auxiliary sets F 0 and F 1 .Procedure P 2 (p).(1) Choose a large number m.(2) Wait for Γ A (m) to converge.(3) Let r ∈ N be such that the procedure has gone through (2) for r − 2 times.Pick a large number n. Put 〈r n ,n〉 into L where r n = r, and put n into F 0 .Wait for a stage t such that K(A↾ n )[t] ≤ r+b+d = r, and transfer n fromF 0 to F 1 . (If M d is a machine for L, then t exists.) If wgt L (F 2 ∪ F 1 ) ≤ pgoto (3).If the expression Γ A (m) changes during this loop then all the numberscurrently in F 1 have obtained their A-change, so put F 1 into F 2 and declareF 1 = ∅. (We call this an early promotion of the numbers in F 1 . Thisimportant topic is pursued further in Remark 5.4.3.) Also, the number nin F 0 is garbage as we cannot any longer hope to get descriptions for twoA↾ n configurations, so declare F 0 = ∅. Goto (2).(4) Put m into B. This provides the A-change for the elements now in F 1 ,soput F 1 into F 2 and declare F 1 = ∅. (We call this a regular promotion.)To see that L is a bounded request set, consider what happens to a number nafter a request 〈r n ,n〉 has entered L.• For each r there is at most one n such that r n = r and n is not promotedto F 1 . The total weight of such numbers n is thus at most ∑ r≥3 2−r =1/4.• The weight of the numbers n that get into F 1 is at most p =3/4, sinceat each stage we have wgt L (F 2 ∪ F 1 ) ≤ p, and we empty a set F 1 into F 2before we start a new one (and F 2 never gets emptied).Since Γ A is total, the procedure reaches (4), which causes wgt(F 2 )=3/4. Thisis a contradiction, since for each n ∈ F 2 , two configurations A↾ n have descriptionsof length r n .✸We now remove the hypothesis that b = d = 0. We allow a weight of 1/4of numbers n that are garbage. This extra “space” in L is obtained by forcing2 b+d+1 − 1 many A ↾ n -changes when the request is not garbage (that is, almosttwice as many as in the wtt case). Throughout, we will letk =2 b+d+1 .To force these changes we use k − 1 levels of procedures P i (2 ≤ i ≤ k ). A procedureat level i>2 calls one at level i − 1. The bottom procedure P 2 acts asabove. An argument similar to the one just used shows that the total weight ofnumbers that are not garbage does not exceed 1/2. Before giving more detailsin Fact 5.4.2, we introduce two technical concepts.Stages when A looks K-trivial. The construction is restricted to stages, definedexactly as before Theorem 5.3.27 (except that the given set A was c.e. then).

204 5 Lowness properties and K-trivialityThus, for each s, let f(s) =µt > s ∀n

5.4 Each K-trivial set is low for K 205Case i>2. Call P i−1 (2 −v α, α ′ ), where α ′ = min(α, 2 −i−2 2 −wi−1 ), andw i−1 is the number of P i−1 procedures started so far in the construction.If wgt L (F i ∪ F i−1 )

206 5 Lowness properties and K-trivialityF 0F 1F 2Emptyingdevice F 3Fig. 5.1. Decanter model. Right side: premature A-change. F 2into F 3 , while F 0 and F 1 are spilled on the floor.is emptiedThe use of Γ A (m) acts as the emptying device, which, besides being activatedon purpose, may go off finitely often by itself. This is how we visualize an A-change. It is premature if the device goes off by itself. When F i−1 is emptiedinto F i then F i−2 ,...,F 0 are spilled on the floor.We suppose that a bottle of wine corresponds to a weight of 1. We first pourwine into the highest decanter F 0 . We want to ensure that a weight of at most1/4 of wine we put into F 0 does not reach F k . Recall that the parameter α isthe amount of garbage P i (p, α) allows. If v is the number of times the emptyingdevice has gone off by itself, then for i>2, P i lets P i−1 fill F i−1 in portions of2 −v α. For then, when F i−1 is emptied into F i , a quantity of at most 2 −v α canbe lost because it is in higher decanters F i−2 ,...,F 0 .The procedure P 2 is a special case, but limits its garbage in the same way: itputs requests 〈r n ,n〉 into L where 2 −rn =2 −v α. Once it sees the correspondingdescription of A↾ n , it empties F 0 into F 1 . However, if the hook γ A (m) belongingto P 2 moves before this, F 0 is spilled on the floor while F 1 is emptied into F 2 .5.4.4 Remark. If A is merely ∆ 0 2 then A ↾ n can return to a previous value.However, in the definition of k -sets we need k distinct values. Why can we stillconclude that P k (3/4, 1/4) returns a k -set of weight 3/4?A run of P 2 is the result of recursive calls of runs P k , P k −1 ,...,P 3 ; there arenumbers m k

3: Each K-trivial set is superlow.5.4 Each K-trivial set is low for K 207The construction showing that each K-trivial set A is Turing incomplete is sequentialat each level: at most one procedure runs at any particular level. Toshow that A is superlow, and even low for K, we use a construction where ateach level procedures can run in parallel.Recall that A ′ = dom J A . A procedure P i (p, α) at level i ≥ 2 (with goal p andgarbage quota α, of the form 2 −l , enumerating a set F i ) attempts to providea computable approximation for A ′ , making a guess at each stage. Initially,the guess is “divergent”. When J A (e) newly converges, P i calls a subprocedureQ i−1,e (q, β), where q =2 −e−1 α, in order to test the stability of this computation.When the run of Q i−1,e (q, β) returns an (i − 1)-set G i−1,e of weight q, then P imakes a guess that J A (e) converges.Stable case. If A does not change later below the use of J A (e), the guess is correct.However, G i−1,e is now garbage, since its elements will not reach the i-set F i .This is a one-time event for each e, so the total garbage produced in this way isat most ∑ e 2−e−1 α = α, as required. (Note that this garbage due to the failureof A to change is of a new type. It was not present in the previous constructionswhere we could force A to change.)Change case. If A changes later below the use of J A (e), then all the numbersin G i−1,e are put into F i ,sowgt L (F i ) increases by the fixed quantity 2 −e−1 α.In this case P i changes its guess back to “divergent”. If this happens r times,where r = p2 e+1 /α, then the run of P i reaches its goal p.We consider the hypothetical case that b = d = 0 in some more detail.Procedure P 2 (p, α) (p ∈ Q 2 , α ≤ p of the form 2 −l ).It enumerates a set F 2 , beginning with F 2 = ∅.At stage s, declare e = s available (availability is a local notion for each run ofa procedure). For each e ≤ s, do the following.P1 eP2 eIf e is available and J A (e)↓ [s], call the procedure Q 1,e (2 −e−1 α, β), whereβ = min(2 −e−1 α, 2 −v−1 ) and v is the number of runs of Q 1 type proceduresstarted so far in the construction. Declare e unavailable.If e is unavailable due to a run Q 1,e (q, β) (which may have returned already)and A s ↾ w ≠ A s−1 ↾ w where w is the use of the associated computationJ A (e), declare e available.(a) Put G 1,e into F 2 .(b) If wgt L (F 2 )

208 5 Lowness properties and K-trivialityProcedure Q 1,e (q, β) (β ≤ q, both of the form 2 −l for some l).It enumerates a set G 1,e , beginning with G 1,e = ∅. It behaves similarly to step(3) in the procedure P 2 (p) on page 203.Q1 Pick a number n larger than any number used so far. Put 〈r n ,n〉 into L,where 2 −rn = β. Wait for a stage t>nsuch that K t (n) ≤ r n + d. (If M dis a machine for L then t exists.)Q2 Put n into G 1,e . If wgt L (G 1,e ) 1 behave similar to Q 1,e , but in (Q1)the enumeration of a request 〈r n ,n〉 into L is replaced by a recursive call of aprocedure P j . See Fig. 5.2.Recall that a run of P i enumerates an i-set F i , and a run of Q i−1,e enumeratesan (i − 1)-set G i−1,e . To continue the decanter model, if 2 ≤ i

5.4 Each K-trivial set is low for K 209Q 1,e...Q 1,e +1P 2. . ....Q 1,e +2Q 1,eQ 1,e +1Q 1,e +2Q 1,e............P 2. . ....... ...P k–1Q k–1,e. . . . . .. . . P i. . . P k–1Q k–1,e +1golden run. . . P k–1Q k–1,e +2.........P kQ 1,e +1P 2. . ....Q 1,e +2...C 1D 1C 2. . . D i–1C i. . . D k–1C k –1D k–1C kFig. 5.2. Tree of runs. At the bottom we indicate the promotion of numbers.4. Each K-trivial set is low for K.Now a run P i (p, α) that does not reach its goal tries to build a bounded requestset W showing that A is low for K. It emulates the argument of Proposition 5.3.34that each ∆ 0 2 set A obeying c U,A is low for K. If the run is golden then it succeeds.The hypothesis in 5.3.34 that A obeys c U,A is replaced by the hypothesis thatthe run P i (p, α) does not reach its goal.When a computation U A (σ) =y newly converges, P i (p, α) calls a procedureQ i−1,σ (q, β) with the goal q =2 −|σ| α. As in the foregoing construction, it isnecessary to call in parallel procedures based on different inputs σ. The procedureQ i−1,σ works in the same way as Q i−1,e before: unless it is cancelled, it returnsan (i − 1)-set G i−1,σ of weight q. The numbers in G i−1,σ are no less than theuse of the computation U A (σ). Let u ∈ N be least such that 2 u ≥ p/α. Whenthe run Q i−1,σ (q, β) returns, P i (p, α) hopes that the computation U A (σ) =y isstable, so it puts a request 〈σ + u +1,y〉 into W .Stable case: A does not change later below the use of U A (σ). The enumerationinto W causes K(y) ≤ + |σ|, soifσ is a shortest U A -description of y, we obtainK(y) ≤ + K A (y). The set G i−1,σ is now garbage. The total garbage produced inthis way is at most Ω A α ≤ α, as required. The contribution to W is at most 1/2.Change case: A changes later below the use of U A (σ). Then all numbers in G i−1,σare put into F i ,sowgt L (F i ) increases by q =2 −|σ| α. Since 2 u ≥ p/α, the total

210 5 Lowness properties and K-trivialityweight contributed to W in this way stays below 1/2, or else the run of P i reachesits goal p.If the run P i (p, α) is golden, it succeeds in showing that A is low for K: itisnot cancelled, and does not reach its goal (so W is a bounded request set). Eachrun Q i−1,σ returns unless cancelled, necessarily by an A-change below the useof the computation U A (σ) it is based on. Thus P i (p, α) puts the correspondingrequest into W at a stage when U A (σ) is stable.It is instructive to compare (a) the proof of Theorem 5.1.22 (on bases for MLrandomness)with (b) the proof of Theorem 5.4.1. In both cases, we show that agiven set A is low for K (though in (b) we also need a computable approximationof A). The construction for a fixed parameter d in Theorem 5.1.22 is somewhatsimilar to a golden run P i (p, α). In (a) one feeds hungry sets C η d,σ once Uη (σ) =yis defined. In (b) subprocedures Q i−1,σ are called once U A (σ)[s] =y is defined.The measure 2 −|σ|−d−1 that C η d,σhas to reach so that one can put the request〈|σ|+d+1,y〉 into L d corresponds to the goal q =2 −|σ| α the sub-procedure has toreach, so that it can return and the request 〈|σ|+u+1,y〉 can go into W .However,the verifications that L d and W are bounded request sets are completely different.Exercises.5.4.5. Get hold of a bottle of 1955 Biondi-Santi Brunello wine. Why is a single bottlesufficient for any of the decanter constructions?5.4.6. Sketch a tree of runs as in Figure 5.2 for the case that k = 4. The types ofprocedures are Q 1, P 2, Q 2, P 3, Q 3, and P 4.The formal proof of Theorem 5.4.1We retain the definitions of the previous subsection, in particular of stages s(l)and i-sets (page 203). We modify the computable approximation of A so thatA v (x) =A s(l) (x) for all x, v, l such that s(l) ≤ v1, while it has not reached its goal q, the run Q j,σyw (q, β) keeps calling asingle procedure P j (β, α) (for decreasing garbage quotas α) and waits until this

5.4 Each K-trivial set is low for K 211procedure returns a set F j , at which time it puts F j into G j,σ . Thus the amountof garbage left in C j − D j is produced during a single run of a procedure P jwhich does not reach its goal β, and hence is bounded by β.A run P i (p, α) calls procedures Q j,σyw (2 −|σ| α, β) for appropriate values of β.The weight left in D i−1 − C i by all the returned runs of Q i−1 -procedures whichnever receive an A-change adds up to at most Ω A α since this is a one-time eventfor each σ. The runs of procedures Q i−1 which are cancelled and have so farenumerated a set G do not contribute to the garbage of P i (p, α), since G goesinto F i upon cancellation.To assign the garbage quotas we need some global parameters. At any substageof stage s, letα ∗ i =2 −(2i+5+n P,i) , (5.15)where n P,i is the number of runs of P i -procedures started in the constructionprior to this substage of stage s. Furthermore, letβ ∗ j =2 −(2j+4+n Q,j) , (5.16)where n Q,j is the number of runs of Q j -procedures started so far. Then the sumof all the values of α ∗ i and β ∗ j is at most 1/4. When P i is called at a substageof stage s, its parameter α is at most α ∗ i . Similarly, the parameter β of Q j isat most β ∗ j . This ensures wgt L(C 1 − C k ) ≤ 1/4. We start the construction bycalling P k (3/4, 1/4), so wgt L (C k ) ≤ 3/4. In this way we will show that L is abounded request set.We now give the formal description of the procedures and the construction.Procedure P i (p, α) (1

212 5 Lowness properties and K-triviality(b) If the run Q i−1,σyw has not returned yet, cancel this run and allthe runs of subprocedures it has called.The procedure Q j,σyw (q, β) (0nsuch that n

5.4 Each K-trivial set is low for K 213A↾ w is stable from t to s − 1A s−1 ↾ w ≠ A s ↾ wt〈r, n〉 enters Ls ′G i−1,σ is an (i − 1)-setFig. 5.3. Illustration of Lemma 5.4.7.snot change from t to s − 1, else the run of Q i−1,σyw would have been canceledbefore s. Since A s−1 ↾ w ≠ A s ↾ w , we have a new string z = A s ↾ n as required toshow that C i is an i-set. See Fig. 5.3.✸Next we verify that L is a bounded request set. First we show that no run ofa procedure exceeds its garbage quota. Since the runs are arranged as a tree,each number n enumerated into the right domain of L at a leaf during a stage tcorresponds to a unique run of a procedure at each level at the same stage. Wesay n belongs to this run.5.4.8 Lemma. (a) Let 1 ≤ jt. The only possiblereason that n does not reach C i is that the run of P i did not need n to reach itsgoal in (P2 σ ) (i.e., n ∉ ˜G), in which case the run of P i ends at s.Secondly, assume there is no such s. Then the run of P i , as far as it is concernedwith σ, keeps waiting at (P 2 σ ), and σ does not become available again. Thisproves the claim.As a consequence, for each σ there is at most one run Q i−1,σyw (2 −|σ| α, β)called by P i (p, α) which leaves numbers in D i−1 − C i . If the run of P i returns ata stage s then the sum of the weights of such numbers is bounded by the valueof Ω A α at the last stage before s, otherwise it is bounded by Ω A α. ✸

214 5 Lowness properties and K-trivialityBy the foregoing lemma and the definitions of the values αi ∗, β∗ j at substages,at each stage we havewgt L (C 1 − C k ) ≤ ∑ k −1j=1 wgt L(C j − D j )+ ∑ ki=2 wgt L(D i−1 − C i ) ≤ 1/4.Also wgt L (C k ) ≤ 3/4 since we call P k with a goal of 3/4. Thus wgt L (C 1 ) ≤ 1.Since C 1 is the right domain of L, we may conclude that L is a bounded requestset.From now on we may assume that M d is a machine for L, using the RecursionTheorem as explained in Remark 2.2.21.5.4.9 Lemma. There is a run of a procedure P i , called a golden run, such that(i) the run is not cancelled,(ii) each run of a procedure Q i−1,σyw called by P i returns unless cancelled, and(iii) the run of P i does not return.Subproof. Assume that no such run exists. We claim that each run of a procedurereturns unless cancelled. This yields a contradiction, since we call P k witha goal of 3/4, this run is never cancelled, but if it returns, it has enumerated aweight of 3/4 intoC k , contrary to Fact 5.4.2.To prove the claim, we carry out an induction on levels of procedures of typeQ 1 , P 2 , Q 2 ,...,Q k −1 , P k . Suppose the run of a procedure is not cancelled.Q j,σyw (q, β): If j = 1, the wait at (Q2) always succeeds because M d is a machinefor L. Ifj>1, inductively each run of a procedure P j called by Q j,σyw returns,as it is not cancelled. In either case, each time the run is at (Q2), the weightof D increases by β. Therefore Q j,σyw reaches its goal and returns.P i (p, α): The run satisfies (i) by hypothesis, and (ii) by inductive hypothesis.Thus, (iii) fails, that is, the run returns.✸5.4.10 Lemma. A is low for K.Subproof. Choose a golden run of a procedure P i (p, α) as in Lemma 5.4.9. Weenumerate a bounded request set W showing that A is low for K. Let u ∈ N beleast such that p/α ≤ 2 u .Atstage s, when a run Q i−1,σyw (2 −|σ| α, β) returns,put 〈|σ| + u +1,y〉 into W . We prove that W is a bounded request set, namely,S W = ∑ r 2−r [[〈r, z〉 ∈W ]] ≤ 1. Suppose 〈r, z〉 enters W at stage s due to a runQ i−1,σyw (2 −|σ| α, β) which returns.Stable case. The contribution to S W of those requests 〈r, z〉 where A↾ w is stablefrom s on is bounded by 2 −(u+1) Ω A , since for each σ such that U A (σ) is defined,this can only happen once.Change case. Now suppose that A ↾ w changes after stage s. Then the set Greturned by Q i−1,σyw , with a weight of 2 −|σ| α, went into the set F i of the runP i (p, α). Since this run does not reach its goal p =2 u α,∑s∑2 −|σ| [[Q i−1,σyw returns at s & ∃t >sA t ↾ w ≠ A t−1 ↾ w ]] < 2 u . (5.17)σ,y,w

5.5 Properties of the class of K-trivial sets 215Thus the contribution of the corresponding requests to S W is less than 1/2.Let M e be the machine for W according to the Machine Existence Theorem.We claim that K(y) ≤ K A (y)+u + e +1 for each y. Suppose that s is the leaststage such that a stable computation U A s (σ) =y appears, where σ is a shortestU A -description of y. Let w be the use of this computation. Then σ is availableat s: otherwise some run Q i−1,σy′ w ′ is waiting to be released at (P2 σ). In thatcase, A ↾ w ′ has not changed since that run was started at a stage

216 5 Lowness properties and K-trivialitywhen we verified that A is low for K. In (5.13) on page 198 we introducedthe adaptive cost function c M,A (x, r), the measure of the M A -descriptions atstage r − 1 that are threatened by a potential change A r−1 (x) ≠ A r (x). TheMain Lemma extracts a computable sequence of stages 0 = q(0)

5.5 Properties of the class of K-trivial sets 217Let q(0) = 0. If s = q(r) has been defined, let q(r + 1) be the least stage t>ssuch that the condition of the claim holds (this can be determined effectively).Let u ∈ N be the number such that p/α =2 u . We show that Ŝ

218 5 Lowness properties and K-trivialityc K (x, r) = ∑ x0&x is least s.t. Â r−1 (x) ≠ Âr(x)]≤ ∑ r,x ĉK(x, q(r)) [r >0&x is least s.t. A q(r+1) (x) ≠ A q(r+2) (x)] ≤ Ŝ

5.5 Properties of the class of K-trivial sets 219As a consequence, Theorem 5.4.1 is nonuniform as well.5.5.6 Corollary. One cannot effectively obtain a constant d such that A is lowfor K via d from a pair (A, b), where A is a c.e. set that is K-trivial via b.Proof. If a c.e. set A is low for K, then, by the uniformity of Proposition 5.1.3,a lowness index for A can be computed from a c.e. index for A and a constantvia which A is low for K.✷The nonuniformity in the particular proof of Theorem 5.4.1 given above iseasily detected: the constant via which A is low for K given by that proof isu + e + 1, and both e and u depend on what the golden run is. This actuallyproves that we cannot determine a golden run effectively from the given objectsin the construction.In the following we summarize the properties of the class of K-trivial sets withinthe Turing degrees. Ideals in uppersemilattices were defined in 1.2.27, and theeffective listing (V e ) e∈N of the ω-c.e. sets in 1.4.5.5.5.7 Theorem.(i) The K-trivial sets induce an ideal K in the Turing degrees.(ii) K is the downward closure of its c.e. members.(iii) Each K-trivial set is superlow. K is a Σ 0 3 ideal in the ω-c.e. T-degrees.(iv) K is nonprincipal.Proof. Theorems 5.4.1, 5.2.17 and Corollary 5.5.3 imply (i) and (ii). Fact 5.2.6and Corollary 5.5.4 imply (iii). For (iv), assume that there is a degree b suchthat K =[0, b]. Then b is c.e. by Corollary 5.5.3, and b is low by Theorem 5.5.4.By Theorem 5.3.22 there is a ∈ K such that a ≰ b, contradiction.✷Nies proved that each proper Σ 0 3 ideal in the c.e. Turing degrees has a low 2 upperbound (see Downey and Hirschfeldt 20xx). Then, by Theorem 5.5.3, there is a low 2 c.e.set E such that A ≤ T E for each K-trivial set A.Kučera and Slaman (20xx) built a low set B Turing above all the (c.e.) K-trivial sets.By Theorem 5.3.22, such a set is not computably enumerable. Also see before 5.3.26.5.5.8. ⋄ Problem. Are there c.e. Turing degrees a, b such that K =[0, a] ∩ [0, b]?Such a pair of degrees is called an exact pair for the ideal. Each Σ 0 3-ideal with a boundbelow 0 ′ has an exact pair in the ∆ 0 2 Turing degrees by Shore (1981).5.5.9 Exercise. There is no cost function that is non-increasing in the first argumentand characterizes the superlow sets in the sense of Theorem 5.5.2.5.5.10 Exercise. (i) There is a 1-generic K-trivial set.(ii) No Schnorr random set is K-trivial.The number of changes ⋆Each K-trivial set A is superlow, that is, A ′ is ω-c.e. We ask to what extentthe number of changes in a computable approximation for A can be minimized.Thereafter we do the same for A ′ .

220 5 Lowness properties and K-triviality5.5.11 Proposition. Let A ∈ K. For each order function h there is a computableapproximation (Ãr) r∈N of A such that Ãr(y) changes at most h(y) times.Proof. Proposition 5.3.12 and Theorem 5.5.2 imply this result for some orderfunction h in O(y 2 ). For the full result we modify the proof of (i)⇒(ii) in Theorem5.5.2. Let ˜h(x) =⌊h(x) 1/4 ⌋. Let g be an increasing computable functionsuch that x ≤ g(˜h(x)) for each x, and there is c such that K g(y) (y) ≤ 2 log y + cfor each y. (For instance, if h(x) = log x then g(y) ≥ 2 y4 .)We apply the Main Lemma to the prefix-free oracle machine M such thatM X (σ) attempts to compute y = U(σ); in the case of convergence, it reads g(y)bits of the oracle, thereby making the use g(y), and then halts with output y. Letà r (x) =A q(r+2) (x) ifg(˜h(x)) ≤ q(r), and Ãr(x) = 0 otherwise. Clearly (Ãr) r∈Nis a computable approximation of A. Suppose that r>0 and Ãr−1(x) ≠ Ãr(x).Then A q(r+1) (x) ≠ A q(r+2) (x) and, by the definition of g, x ≤ g(˜h(x)) ≤ q(r)and K q(r) (˜h(x)) ≤ 2 log ˜h(x)+c. ThusM A (σ)[q(r+1)]↓ with use at most q(r) forsome σ such that |σ| ≤ + 2 log h(x). So this change contributes at least ɛ˜h(x) −2to Ŝ for some ɛ > 0 independent of x. Then, since Ŝ

5.5 Properties of the class of K-trivial sets 221for some U-description σ m such that |σ m | ≤ log m + 2 log log m + c. To definethe computable approximation (Z r ) r∈N ,ifr ≥ p(m) andΦ A e (m)[q(r + 1)]↓ with use ≤ q(r)(and hence M A (σ m )[q(r + 1)] ↓ with use ≤ q(r)), then declare Z r (m) = 1,otherwise Z r (m) =0.If Z r (m) =1and A changes below the use of Φ A e (m)[q(r + 1)] between thestages q(r + 1) and q(r + 2), then 2 −|σm| ∼ 1/(m log 2 m) is added to Ŝ. Thuslim r Z r (m) = Z(m) and the number of changes of Z r (m) is in O(m log 2 m).✷For each order function h, the class of c.e. sets such that A ′ can be approximatedwith at most h(x) changes contains a promptly simple set by Theorem 8.4.29 andCorollary 8.4.35. However, if h grows sufficiently slowly, this class is a proper subclassof the c.e. K-trivial sets by Theorems 8.5.1, 8.4.34, and Corollary 8.5.5. Thus, for a c.e.K-trivial set A in general, the number of changes in an approximation to A ′ cannot bedecreased arbitrarily.5.5.13 Exercise. (Barmpalias, Downey and Greenberg) Let h be an order functionsuch that ∑ m1/h(m) < ∞. Extend 5.5.12 to the case of the function h instead ofλm.m log 2 m.Ω A for K-trivial AIn our next application of the Main Lemma, we characterize the K-trivial setsin terms of the operator Ω: if A is ∆ 0 2, then A is K-trivial ⇔ Ω A is left-c.e. (theimplication “⇐” follows from Theorem 5.1.22). The results in this subsectionare due to Downey, Hirschfeldt, Miller and Nies (2005).By Definition 3.4.2, Ω A M = λ[dom M A )] ≺ for a prefix-free oracle machine M.We view U as a universal prefix-free oracle machine (as explained after Definition3.4.2) and write Ω A for Ω A U .5.5.14 Theorem. Let A be a set in ∆ 0 2. Then the following are equivalent.(i) A is K-trivial.(ii) A ≤ T Ω A .(iii) Ω A is left-c.e.(iv) Ω A U is left-c.e. for every optimal prefix-free oracle machine U.(v) Ω A M is difference left-c.e. for every prefix-free oracle machine M.Proof. (i)⇒(v): We will define left-c.e. real numbers α, β such that Ω A M = α−β.Applying the Main Lemma 5.5.1 to the machine M we obtain a computablesequence of stages 0 = q(0)

222 5 Lowness properties and K-triviality{2 −|σ| if M A (σ)↑ [q(r)] & M A (σ)↓ [q(r + 1)]α σ,r =0 otherwise{2 −|σ| if M A (σ)↓ [q(r + 1)] & M A (σ)↑ [q(r + 2)]β σ,r =0 otherwise.Since the sum Ŝ defined in (5.18) is finite, ∀∞ r [α σ,r = β σ,r = 0] for each σ. Letα = ∑ ∑σ r α σ,r and β = ∑ ∑σ r β σ,r,then β ≤ Ŝ

5.5 Properties of the class of K-trivial sets 2235.5.16 Exercise. Let U be a universal prefix-free machine and let A be a set (notnecessarily ∆ 0 2). Then A is K-trivial ⇔ A ≤ T Ω A U .Each K-trivial set is low for weak 2-randomnessIn Theorem 5.1.33 we proved that Low(W2R, MLR) =Low(MLR), which impliesthat Low(W2R) ⊆ Low(MLR). Here we show that actually Low(W2R) =Low(MLR). This is surprising because the two randomness notions are quitedifferent. Two proofs of the containment Low(MLR) ⊆ Low(W2R) were foundindependently. The first, due to Nies, uses the Main Lemma 5.5.1. The second,due to Kjos-Hanssen, Miller and Solomon (20xx), relies on Theorem 5.6.9 belowand will be discussed after having proved that theorem.In Remark 5.6.19 we briefly address lowness for 2-randomness.5.5.17 Theorem. The following are equivalent for a set A:(i) A is low for generalized ML-tests in a uniform way. That is, given anull Π 0 2(A) class U, one can effectively obtain a null Π 0 2 class V ⊇ U.(ii) A is low for weak 2-randomness.(iii) A is K-trivial.Note that the condition (i) appears to be stronger than (ii). It states that eachtest for weak 2-randomness relative to A can be covered by a test for unrelativizedweak 2-randomness. The difference between being lowness for a testnotion and lowness for the corresponding randomness notion will be discussedafter Definition 8.3.1.Proof. (i) ⇒ (ii): Immediate.(ii) ⇒ (iii): This follows from Theorem 5.1.33 and the fact that each set that islow for ML-randomness is K-trivial.(iii) ⇒ (i): In fact, if A is K-trivial, we obtain a conclusion that appears evenstronger than (i) as it applies to Π 0 2(A) classes in general:(i ′ ) Given a Π 0 2(A) class U, one can effectively obtain a Π 0 2 class V ⊇ U suchthat λV = λU.In (i ′ ) it actually suffices to cover Σ 0 1(A) classes.5.5.18 Lemma. (J. Miller) Suppose that for each Σ 0 1(A) class G there is a Π 0 2class S such that S ⊇ G and λS = λG. Then (i ′ ) holds for A.Subproof. Let NeXclasses. Let=[W X e ] ≺ . Then (N X e ) e∈N is an effective listing of the Σ 0 1(X)N X = ⋃ e 0e 1N X e(where for C ⊆ 2 N and x ∈ {0, 1} ∗ we denote by xC the class {xZ : Z ∈ C}). Theclass N X is an effective disjoint union of all the N X e .By the hypothesis for G = N A there is a Π 0 2 class S ⊇ N A of the samemeasure as N A . For each i ∈ N let S i = {Z : 0 i 1Z ∈ S}, then S i is a Π 0 2 classuniformly in i. Given a Π 0 2(A) class U, one can effectively obtain a computable

224 5 Lowness properties and K-trivialityfunction p such that U = ⋂ e N p(e) A by Remark 1.8.58. Further, S p(e) ⊇ Np(e)Aand 2 −p(e)−1 λ(S p(e) − Np(e) A ) ≤ λ(S − N A ) = 0, hence λS p(e) = λNp(e) A . LetV = ⋂ e S p(e), then V ⊇ U and the Π 0 2 class V was obtained effectively from U.Also λV = λU since V − U ⊆ ⋃ e (S p(e) − Np(e) A ).✸To conclude the proof of (iii) ⇒ (i ′ ), we show that for each K-trivial set A thehypothesis of Lemma 5.5.18 holds. By Fact 1.8.26 relative to A there is a c.e.index d such that G =[Wd A]≺ , and WdX is a prefix-free set for each X. Considera prefix-free oracle machine M X such that, for each σ,M X (σ)↓ ↔ σ ∈ W X d (5.20)with the same use on both sides. By Main Lemma 5.5.1 there is a constant u ∈ Nand a computable sequence of stages q(0)

5.6 The weak reducibility associated with Low(MLR) 225Two of the classes introduced in Section 5.1, being low for K and being lowfor ML-randomness, form the least degrees of weak reducibilities denoted ≤ LKand ≤ LR , respectively. For further examples of weak reducibilities, see ≤ cdomin Exercise 5.6.8 (each A-computable function is dominated by a B-computablefunction) and ≤ JT in Definition 8.4.13. An overview of weak reducibilities isgiven in Table 8.3 on page 363. The intuition behind a weak reducibility relationA ≤ W B is that B can only understand a small aspect of the abilities of A.For instance, A ≤ cdom B means that B can emulate the growth of functionscomputed by A, and A ≤ JT B that B can approximate the values of J A .Incontrast, if A ≤ T B then B can emulate everything A does.The K-trivial sets form the least degree of a preordering ≤ K .5.6.1 Definition. For sets A, B, let(i) A ≤ LK B ↔ ∃d ∀y [K B (y) ≤ K A (y)+d](ii) A ≤ LR B ↔ MLR B ⊆ MLR A(iii) A ≤ K B ↔ ∃b ∀nK(A↾ n ) ≤ K(B ↾ n )+b.Informally, A ≤ LK B means that up to a constant, one can compress a stringwith B as an oracle at least as well as with A, and A ≤ LR B means that,whenever A can find “regularities” in a set Z, then so can B. By these definitions,A is low for K iff A ≤ LK ∅, and A is low for ML-randomness iff A ≤ LR ∅.For each X we have X ′ ≰ LR X because Ω X is not ML-random relative to X ′but Ω X is ML-random relative to X. IfA is low for ML-randomness relativeto X, namely, MLR(A ⊕ X) =MLR(X), then A ≤ LR X. The converse fails ingeneral; see Remark 5.6.24(iii) for a counterexample. In particular, ⊕ does notdetermine a join in the LR-degrees.By Exercise 3.4.4, ≤ T implies ≤ LK . The converse fails since some incomputableset is low for K. By Schnorr’s Theorem 3.2.9 in relativized form, ≤ LK implies≤ LR . Note that, while the definition of ≤ LK is in Σ 0 3 form, the definition of≤ LR is not even in arithmetical form as it involves universal quantification oversets. Nonetheless, in Theorem 5.6.5 we will show that ≤ LR is in fact equivalentto ≤ LK . This yields an alternative proof of the result in Section 5.1 that the setsthat are low for K coincide with the sets that are low for ML-randomness.Further results in this section show that ≤ LR is rather different from ≤ T . Forinstance, the set {Z : Z ≤ LR ∅ ′ } is uncountable. (This is possible only becausethere are no reduction procedures.) The equivalence relation ≡ LR is somewhatclose to ≡ T : for example, A ≡ LR B implies that A is K-trivial in B and B is K-trivial in A (5.6.20), and therefore A ′ ≡ tt B ′ . Thus each LR-degree is countable,while the initial interval below the LR-degree of ∅ ′ is uncountable.Each weak reducibility ≤ W determines a lowness property {A: A ≤ W ∅} and,dually, a highness property {C : ∅ ′ ≤ W C}. We will characterize the highnessproperty associated with ≤ LR by a domination property, being uniformly a.e.dominating.5.6.2 Remark. Note that A is K-trivial iff A ≤ K ∅. The preordering ≤ K comparesthe degree of randomness of sets: if A ≤ K B then B is “at least as random”as A. As an alternative, Miller and Yu (2008) compared the degree of randomnessof sets by introducing van Lambalgen reducibility:

226 5 Lowness properties and K-trivialityA ≤ vL B ↔ ∀Y ∈ MLR [A ∈ MLR Y → B ∈ MLR Y ].Intuitively, B is ML-random in at least as many sets Y ∈ MLR as A is (it isnot known whether the condition that Y ∈ MLR actually makes a difference).Interestingly, they showed that ≤ K implies ≤ vL on MLR (see Exercise 8.1.17).The least ≤ vL degree consists of the sets that are not ML-random, because anyML-random set A is ML-random in Ω A by Theorem 3.4.6. If A and B are MLrandomthen A ≤ vL B ↔ A ≥ LR B by 3.4.6., so ≤ vL is actually just anotherway to look at ≥ LR on the ML-random sets.Preorderings coinciding with LR-reducibilityWe characterized the class Low(MLR) byeffective topology in Corollary 5.1.10.This is a special case of characterizing the underlying weak reducibility: A ≤ LR Bmeans that we can cover each Σ 0 1(A) class of (uniform) measure < 1byaΣ 0 1(B)class of measure < 1. We will encounter several variants of such covering procedures.For instance, in Lemma 5.6.4 we cover “small” A-c.e. sets by “small” B-c.e.sets, and in Theorem 5.6.9 we cover each Π 0 2(A) class by a Π 0 2(B) class of thesame measure. Theorem 5.5.17 already involved the special case of this coveringprocedure where B = ∅. A summary of the covering procedures in this chapteris given in Table 5.2 on page 229. Lowness for C-null classes (8.3.1) is anotherexample. See 1.8.59 for the definition of Σ 0 n(C) classes and Π 0 n(C) classes.5.6.3 Lemma. A ≤ LR B ⇔ each Σ 0 1(A) class G such that λG

5.6 The weak reducibility associated with Low(MLR) 227To see this let g(x) =− ln(1 − x) for x ∈ [0, 1) R . Since e y ≥ 1+y for each y ∈ R,we have x ≤ g(x) for each x. On the other hand g ′ (x) =1/(1 − x), so g ′ (0) = 1 =lim x→0(g(x) − g(0))/x. Hence there is ɛ > 0 such that g(x) ≤ 2x for each x ∈ [0, ɛ).If the sequence (a n) n∈N does not converge to 0 then both sides in (5.21) are false.Otherwise, we may as well assume that a n < ɛ for each n. Then∏ ∞n=0 (1 − an) > 0 ↔ ∃c ∈ R ∀k ln ∏ kn=0(1 − an) >c↔ ∃d ∈ R ∀k ∑ kn=0− ln(1 − an) 0. Let R = {n: Q ⊆ E n }. Then Ris B-c.e., I ⊆ R, and R is f-small, by (5.21) and because∏n∈R (1 − 2−f(n) )= ∏ n∈R λ(E n)=λ ⋂ n∈R E n ≥ λQ >0,by the independence of the events E n .5.6.5 Theorem. For each pair of sets A, B we have A ≤ LK B ⇔ A ≤ LR B.Proof. ⇒: This follows from the relativized form of Schnorr’s Theorem 3.2.9:Z ∈ MLR B ↔∀nK B (Z ↾ n ) ≥ + n → ∀nK A (Z ↾ n ) ≥ + n ↔ Z ∈ MLR A .⇐: Let f be the computable function given by f(〈r, y〉) =2 −r . The set I ={〈|σ|,y〉: U A (σ) =y} is a bounded request set relative to A and hence f-small.So by Lemma 5.6.4, I is contained in an f-small B-c.e. set ˜R. Let R ⊆ ˜R bea bounded request set relative to B such that ˜R − R is finite. Then, applyingto R the Machine Existence Theorem 2.2.17 relative to B, we may conclude that∀yK B (y) ≤ + K A (y).✷Exercises.5.6.6. Show that A ≤ LK B ⇔ ∀nK B (A↾ n) ≤ + K A (n).5.6.7. Prove the converse of Lemma 5.6.4.5.6.8. Consider the weak reducibility defined by A ≤ cdom B if each A-computablefunction is dominated by a B-computable function. Show that ∅ ′ ≤ cdom C ⇔∅ ′ ≤ T C.✷

228 5 Lowness properties and K-trivialityA stronger result under the extra hypothesis that A ≤ T B ′If one can approximate A computably in B, then A ≤ LR B implies that eachΣ 0 1(A) class G can be approximated from above by a sequence of uniformly Σ 0 1(B)classes (U n ) n∈N , in the sense that G ⊆ U n and λ(U n − G) tends to 0. In otherwords, G ⊆ S for a Π 0 2(B) class S of the same measure as G. This result ofKjos-Hanssen, Miller and Solomon (20xx) will be applied in Theorem 5.6.30 toprove the coincidence of two highness properties.Note that A ≤ LR B does not in general imply A ≤ T B ′ . (For instance, let B =∅ ′ , and note that #{A: A ≤ LR ∅ ′ } =2 ℵ0 by Theorem 5.6.13 proved shortly.)However, A ≤ LR B does imply A ≤ T B ′ if B is low for Ω (Corollary 8.1.10), andthus for almost all sets B in the sense of the uniform measure.5.6.9 Theorem. The following are equivalent for sets A and B.(i) A ≤ LR B and A ≤ T B ′ .(ii) For each Σ 0 1(A) class G there is a Π 0 2(B) class S ⊇ G such that λS = λG.(iii) Given a Π 0 2(A) class U one can effectively obtain a Π 0 2(B) class V ⊇ Usuch that λV = λU.By Exercise 5.6.10 A ≤ LR B and A ≤ T B ′ implies A ′ ≤ T B ′ . Thus the theoremcharacterizes the intersection of the weak reducibility relations A ≤ LR B andA ′ ≤ T B ′ for sets A and B.The implication (iii)⇒(i) in Theorem 5.5.17 states that each K-trivial set islow for generalized ML-tests. Since each set in Low(MLR) is∆ 0 2, the implication(i)⇒(iii) in the foregoing theorem for B = ∅ shows that each set in Low(MLR) islow for generalized ML-tests. Thus we have obtained an alternative proof of thisimplication (iii)⇒(i) in Theorem 5.5.17: instead of using the Main Lemma 5.5.1it relies on Lemma 5.6.4.Proof. (ii)⇒(iii): this is similar to the proof of Lemma 5.5.18, replacing Π 0 2 classesby Π 0 2(B) classes.(iii)⇒(i): A ≤ LR B by Lemma 5.6.3. To show A ≤ T B ′ , recall from page 12that ≤ L is the lexicographical order on sets, and note that U = {Z : Z ≤ L A}is a Π 0 1(A) class such that λU =0.A. Choose a Π 0 2(B) class V ⊇ U such thatλV = λU. By Exercise 1.9.22 λV is left-Π 0 2(B), and hence 0.A = λV is right-c.e.relative to B ′ . Applying the same argument to N − A instead of A shows that1 − 0.A is right-c.e. relative to B ′ . Therefore A ≤ T B ′ .(i)⇒(ii): Although the two proofs were found independently, the present prooffollows the same outline as the proof of (iii)⇒(i) in Theorem 5.5.17. (In fact, thatproof shows (ii) of the present theorem under the hypothesis that A is K-trivialin B, which is stronger than (i) of the present theorem by 6.3.16.)Recall that WeX = dom Φ X e . Let d be a c.e. index such that G =[Wd A]≺ andis an antichain for each oracle X. The plan is as follows:W X d(a) translate the setting of Σ 0 1(A) classes into the setting of A-c.e. sets,(b) apply Lemma 5.6.4 to obtain a B-c.e. set,

5.6 The weak reducibility associated with Low(MLR) 229(c) translate the setting of B-c.e. sets back into the setting of Σ 0 1(B) classes todefine a sequence (U n ) n∈N of uniformly Σ 0 1(B) sets such that for the Π 0 2(B)class S = ⋂ n U n we have S ⊇ G and λS = λG.We carry out the steps in detail.(a) The set I = {〈A↾ u ,x〉: Φ A d(x)↓ with use u} is A-c.e.(b) Note that ∑ σ,x 2−|x| [[〈σ,x〉∈I]] = λG ≤ 1, so by Lemma 5.6.4 for thefunction f(〈σ,x〉) =2 −|x| , there is a B-c.e. set R ⊇ I such that∑σ,x 2−|x| [[〈σ,x〉∈R]] < ∞.Since A ≤ T B ′ there is a B-computable approximation (A r ) r∈N of A. For each nletF n = {〈σ,x〉∈R: ∃r ≥ n [σ ≺ A r & Φ σ d,r(x)↓ with use |σ|]}.Then the sequence (F n ) n∈N is uniformly c.e. in B and I ⊆ F n ⊆ R for each n.To see that I = ⋂ n F n, given 〈σ,x〉, let n be a stage such that A r ↾ |σ| = A n ↾ |σ|for each r ≥ n. If〈σ,x〉∈F n then σ ≺ A and hence 〈σ,x〉∈I.(c) Let U n =[{x: ∃σ 〈σ,x〉∈F n }] ≺ be the open set generated by the projectionof F n on the second component (these open sets correspond to the sets U n inthe proof of (iii)⇒(i) of Theorem 5.5.17). Since I ⊆ F n for each n, wehaveG ⊆ S := ⋂ n U n. Note that S is Π 0 2(B). To show λS = λG, fixɛ > 0. Let k beso large that∑σ,x 2−|x| [[〈σ,x〉∈R & 〈σ,x〉≥k]] < ɛ,and let n be so large that, for all 〈σ,x〉

230 5 Lowness properties and K-trivialityThe size of lower and upper cones for ≤ LR ⋆If “size” is interpreted by “measure” then ≤ LR behaves like ≤ T in that eachlower cone and each nontrivial upper cone is null (the second result is an analogof Theorem 5.1.12). In contrast, if size means cardinality, some ≤ LR lower conehas the maximum possible size 2 ℵ0 , while of course each Turing lower cone iscountable.5.6.11 Fact. The class {X : X ≤ LR A} is null for each A.Proof. If X ∈ MLR A then X ≰ LR A since X is not ML-random relative toitself. This suffices since MLR A is conull.✷5.6.12 Fact. If A ∉ Low(MLR) then {Z : A ≤ LR Z} is null.Proof. (Stephan) Let X ∈ MLR − MLR A .IfZ ∈ MLR X then X ∈ MLR Z byTheorem 3.4.6, so A ≰ LR Z. Again this suffices since MLR X is conull. ✷Note that Z ≤ LR ∅ ′ iff each 2-random set is in MLR Z . This class is much largerthan the class of ∆ 0 2 sets by a result of Barmpalias, Lewis and Soskova (2008).5.6.13 Theorem. #{Z : Z ≤ LR ∅ ′ } =2 ℵ0 .Proof. We prove the result for the equivalent reducibility ≤ LK . We may assumethat Ω X ≤ 3/4 for each X by arranging that the oracle prefix-free machine M 2never halts (see Theorem 2.2.9). For the duration of this proof, contrary to thedefinition on page 14, we will write U α (σ) = y if U α |α|(σ) = y, namely thecomputation halts within |α| steps. Let Ω α = ∑ σ 2−|σ| [[U α (σ)↓ & |σ| ≤ |α|]].5.6.14 Fact. Let k ∈ N and γ ∈ {0, 1} ∗ . Then ∃α ≽ γ ∀β ≽ α [Ω β − Ω α ≤ 2 −k ].Subproof. If the fact fails for k, γ, we can build a sequence γ ≼ α 0 ≺ α 1 ≺ ...such that Ω αi+1 − Ω αi > 2 −k for each k, which contradicts Ω ⋃ i αi ≤ 1. ✸We define a ∆ 0 2 tree B ⊆ 2 N such that #Paths(B) = 2 ℵ0 and Z ≤ LK ∅ ′for each Z ∈ Paths(B). We represent B by a monotonic one-one ∆ 0 2 functionF : {0, 1} ∗ → {0, 1} ∗ in the sense that B = {α: ∃η α≼ F (η)}. Thus, the classPaths(B) is perfect, and the range of F is the set of branching nodes of B.We build a single bounded request set L relative to ∅ ′ that can simulate theU Z -descriptions for any path Z of B. For this we have to choose the branchingnodes sufficiently far apart. Define F inductively:Let F (∅) be the least string α such that ∀β ≽ α [Ω β − Ω α < 2 −4 ]. If n>0 andF (η) has been defined for all strings η of length n − 1, then, for a ∈ {0, 1}, letF (ηa) be the least string α ≽ F (η)a such that ∀β ≽ α [Ω β − Ω α < 2 −2n−4 ].Note that F ≤ T ∅ ′ since the function α → Ω α is computable. Now let L = ⋃ n L n,where L n is the set of pairs 〈|σ|,y〉 such that U F (η) (σ) = y, for some η oflength n such that U F (η′) (σ)↑ for each η ′ ≺ η. Clearly L is c.e. in ∅ ′ . For any setS ⊆ N × {0, 1} ∗ , let µ(S) = ∑ r 2−r [[〈r, y〉 ∈S ]. We have to show that µ(L) ≤ 1.Now µ(L 0 ) ≤ 3/4, and for n>0, µ(L n ) ≤ 2 n 2 −2n−2 =2 −n−2 , since for each

5.6 The weak reducibility associated with Low(MLR) 231string η of length n − 1, if γ = F (η), then for each extension δ a = F (ηa),a ∈ {0, 1}, wehaveΩ δa − Ω γ ≤ 2 −2n−2 .Thusµ(L) ≤ 3/4+ ∑ n>0 2−n−2 ≤ 1.If Z ∈ Paths(B) and U Z (σ) =y then 〈|σ|,y〉∈L.Thus∀y [K Z (y) ≤ K ∅′ (y)+d]where d is the coding constant for L, and hence Z ≤ LK ∅ ′ .✷Stronger results have been obtained: #{Z : Z ≤ LR B} =2 ℵ 0whenever B ∉ GL 2(Barmpalias, Lewis and Soskova, 2008) and whenever B ∈ ∆ 0 2 − Low(MLR) (Barmpalias,20xx). On the other hand, if B ∈ Low(Ω) then Z ≤ LR B implies Z ≤ T B ′ byCorollary 8.1.10, so #{Z : Z ≤ LR B} ≤ℵ 0 .The following lowness property may be interesting: A is an LR-base for ML-randomnessif A ≤ LR Z for some Z ∈ MLR A .IfA is ML-random then A is not such a basebecause Z ∈ MLR A implies A ∈ MLR Z ,soA ≰ LR Z. Barmpalias, Lewis and Stephan(2008) have built a perfect Π 0 1 class P such that Y ≤ LR ∅ ′ for each Y ∈ P , and alsoP ∩ Low(MLR) =∅. Barmpalias has pointed out that P contains a set A that is lowfor Ω by 8.1.3, so A is an LR-base for ML-randomness while A ∉ Low(MLR). Thiscontrasts with Theorem 5.1.22.5.6.15. ⋄ Exercise. (Barmpalias, Lewis and Stephan 2008, Thm. 16) Show that foreach incomputable W there is A ≡ LR W such that A | T W .5.6.16. ⋄ Exercise. Show that some weakly 2-random set Z satisfies Z ≤ LR ∅ ′ .Operators obtained by relativizing classesWe have introduced preorderings associated with the classes Low(MLR),M (being low for K) and K (the K-trivial sets). Now we will view these classesrelative to an oracle X, thereby turning them into operators C : P(N) → P(P(N)).The classes M, Low(MLR), and K, respectively, yield the operatorsM(X) = {A: ∃d ∀yK X (y) ≤ K A⊕X (y)+d} = {A: A ⊕ X ≡ LK X},Low(MLR X )={A: MLR A⊕X = MLR X } = {A: A ⊕ X ≡ LR X}, andK(X) = {A: ∃b ∀n [K X (A↾ n ) ≤ K X (n)+b]}.(Note that K X (A ⊕ X ↾ 2n )= + K X (A ↾ n ) and K X (2n) = + K X (n), so in thedefinition of K(X) we could as well replace A by A ⊕ X for a proper relativization.)The classes M(X) and Low(MLR X ) are closed downward under ≤ T foreach X. All the three operators C are degree invariant, namely, X ≡ T Y impliesC(X) =C(Y ). This is clear for M and Low(MLR), since ≤ T implies ≤ LR .In fact, the three operators coincide, because the results that the correspondingclasses coincide can be relativized. This yields further information on ≤ LR .Forinstance, each LR-degree is countable. The only known proof of this result isvia relativized K-triviality. Note that X ≡ LR Y implies that K(X) =K(Y ).Therefore C(X) only depends on the LR-degree of X.We summarize some of the previous results in their relativized forms.5.6.17 Theorem.(i) K(X) is closed under ⊕ for each X.(ii) There is a c.e. index e such that, for each X, We X ∈ K(X) and We X ≰ T X,and therefore X< T X ⊕ WeX ∈ K(X).

232 5 Lowness properties and K-triviality(iii) We have M(X) =Low(MLR X )=K(X); in particular, K(X) is closeddownward under ≤ T .Proof. It is sufficient to note that the proofs of 5.2.17, 5.3.11, 5.1.22 and 5.4.1can be carried out relative to the oracle X.✷In contrast, the following is more than a relativization of the statements of 5.5.3and 5.5.4. Such a relativization would merely yield relativized truth-table reductions,where the oracle X is used first to compute the truth table and then toanswer queries in the truth-table. To prove versions of these corollaries for K(X)with plain truth-table reductions, we have to look at their proofs.5.6.18 Theorem.(i) A ∈ K(X) ⇒ A ≤ tt C for some C ∈ K(X) which is c.e. in X.(ii) A ∈ K(X) ⇒ A ′ ≤ tt X ′ .Proof. (i) By Proposition 5.3.12 relative to X, there is an X-computable approximation(A s ) s∈N such that A s (y) changes at most O(y 2 ) times. Then thechange set C in the proof of Corollary 5.5.3 is in K(X), and C ≥ tt A.(ii) The proof of Corollary 5.5.4 relative to X shows that the change set C forA satisfies (C ⊕ X) ′ ≤ tt X ′ , because in the proof of Proposition 5.1.3 relativeto X, the use of the weak truth-table reduction for A ′ ≤ wtt ∅ ′ is bounded by acomputable function independent of X. Since A ≤ tt C we have A ′ ≤ m C ′ ≤ tt X ′ .Alternatively, one can view the proof of Proposition 5.5.12 relative to X. ✷5.6.19 Remark. Note that Z is 2-random relative to A iff Z is ML-random in A ′(see 3.4.9). Thus, A is low for 2-randomness ⇔ A ′ ≡ LR ∅ ′ . By Theorem 5.6.17 thisclass also coincides with {A: A ′ ∈ K(∅ ′ )}. Each low set is low for 2-randomness.See Exercise 5.6.23 for more.Studying ≤ LR by applying the operator KLet S LR be the operator given byS LR (X) ={A: A ≤ LR X}.If A ⊕ X ≤ LR X then A ≤ LR X,soK(X) ⊆ S LR (X) for each X becauseK(X) =Low(MLR X ). Note that K(X) can be proper subclass of S LR (X): forinstance S LR (∅ ′ ) is uncountable by Theorem 5.6.13 while K(X) is countable foreach X (also see 5.6.24(iii) below). In contrast, we have:5.6.20 Lemma. A ≡ LR B ⇔ A ∈ K(B) &B ∈ K(A).Proof. ⇐: This is immediate since K(X) ⊆ S LR (X) for each X.⇒: Note that K A (A↾ n ) ≤ + K A (n) for each n. By Theorem 5.6.5 A ≡ LK B,sowe may replace the oracle A by B in this inequality, which shows that A ∈ K(B).Similarly B ∈ K(A).✷Together with Theorem 5.6.18(ii) we obtain:

5.6 The weak reducibility associated with Low(MLR) 2335.6.21 Corollary. For each pair of sets A, B, ifA ≡ LR B then A ′ ≡ tt B ′ .Inparticular, each LR-degree is countable.✷This contrasts with Theorem 5.6.13 that the LR-lower cone below ∅ ′ is of cardinality2 ℵ0 . Since the union of fewer than 2 ℵ0 countable sets has cardinality lessthan 2 ℵ0 , the cone below the degree of ∅ ′ in the LR-degrees is of cardinality 2 ℵ0 .In the Turing degrees each lower cone {b: b ≤ a} is countable (Fact 1.2.26).Exercises. Show the following5.6.22. (Barmpalias and Nies) The LR-degrees of the ML-random and the c.e. setshave a singleton as an intersection, namely, the LR-degree of Ω: if Z is ML-random, Ais c.e., and Z ≡ LR A, then ∅ ′ ∈ K(Z). In particular, Z ≡ LR ∅ ′ .5.6.23. (i) Some nonlow c.e. set A is low for 2-randomness (See 5.6.19.).(ii) If A is low for 2-randomness then A ′′ ≡ tt ∅ ′′ (A is superlow 2)Comparing the operators S LR and K ⋆Recall from the chapter introduction that an operator S : P(N) → P(P(N)) ismonotonic if X ≤ T Y → S(X) ⊆ S(Y ) for each X, Y . This property is strongerthan degree invariance. We say that an operator S is closed under ⊕ if S(X) isclosed under ⊕ for each X, and S is Σ 0 n if the relation “Z ∈ S(X)” is Σ 0 n.5.6.24 Remark. (i) The operator K is Σ 0 3 and closed under ⊕. Shore providedan example of non-monotonicity for K. By Theorem 5.3.11, let A be a promptlysimple set in K(∅) =K. Then A is low cuppable, i.e. there is a low c.e. set G suchthat ∅ ′ ≡ T A⊕G, by Theorem 6.2.2 below. Hence A ∈ K(∅)−K(G), for otherwiseA ⊕G ∈ K(G) and therefore ∅ ′′ ≡ T (A⊕G) ′ ≤ T G ′ ≡ T ∅ ′ by Theorem 5.6.18(ii),which is a contradiction.(ii) By Theorem 5.6.5, the lower cone operator S LR is Σ 0 3 as well, but unlike K,S LR is monotone. The same example can be used to show that S LR is not closedunder ⊕: since A ∈ K =Low(MLR) wehaveA ≤ LR G. Trivially G ≤ LR G.A⊕G ≤ LR G would imply ∅ ′ ≡ LR G (since A⊕G ≡ T ∅ ′ ≥ T G), which contradictsCorollary 5.6.21 because G is low. Alternatively, pick B ∈ S LR (∅ ′ ) − K(∅ ′ ), thenB ⊕∅ ′ ≰ LR ∅ ′ . By these examples, ≤ LR behaves differently from ≤ T in a furtheraspect: the operation ⊕ does not determine a supremum in the ≤ LR -degrees.(iii) We have also obtained a low c.e. set G such K(G) is a proper subclass ofS LR (G), because A ∈ S LR (G) − K(G).Slaman (2005) studied monotonic operators S that are Borel (in the sense that therelation “Z ∈ S(X)” is Borel) and for each X, S(X) ≠ P(N) contains X, is closeddownward under ≤ T , and closed under ⊕. He proved that every such operator is givenby (possibly transfinite) iterates of the jump on an upper cone in the Turing degrees.Some possibilities for such an operator S(X) are {Y : Y ≤ T X}, {Y : Y ≤ T X ′ },or{Y : ∃n ∈ N Y ≤ T X (n) }. Slaman’s result can also be used to derive the properties inRemark 5.6.24. The operator K is Σ 0 3 and hence Borel. By Theorem 5.6.18, K is notgiven by iterates of the jump, so K fails to be monotonic. S LR is not given by iteratesof the jump because X ′ ∉ S LR(X) for each X; see the beginning of this section. Since

234 5 Lowness properties and K-trivialityS LR(X) is downward closed under ≤ T for each X, this yields a further, indirect, proofthat S LR(X) is not for all X closed under ⊕.5.6.25 Exercise. Show that if A ∈ K and X is ML-random, then A ∈ K(X).Uniformly almost everywhere dominating setsA highness property of a set C expresses that C is close to being Turing above ∅ ′ .Our first example of such a property was that C is high 1 , namely, ∅ ′′ ≤ T C ′ ;equivalently, some function f ≤ T C dominates all the computable functions(Theorem 1.5.19). Dobrinen and Simpson (2004) introduced a stronger highnessproperty when they investigated the strength of axiom systems in the languageof second-order arithmetic. In the following, we say that a property Q of a setholds for almost every Z if the class {Z : Q(Z)} is conull.5.6.26 Definition. We say that a set C is uniformly almost everywhere dominating,oru.a.e.d. for short, if there is a function f ≤ T C such that for all e,for almost every Z [ Φ Z e total → Φ Z e is dominated by f ] .Clearly each u.a.e.d. set is high. The word “uniform” reminds us of the fact thatthe same f works for almost all Z (and all e). However, in Proposition 5.6.31 wewill show that the seemingly weaker notion of being almost everywhere dominating,where f can depend on Z and e, is in fact equivalent.It is not necessary to require domination of the total Φ Z e for all Turing functionalsΦ e as a particular one is sufficient: for each Z, e, n, letΘ 0e 1Z (n) ≃ Φ Z e (n).5.6.27 Lemma. Suppose a function f ≤ T C dominates Θ Y for almost every Y .Then C is uniformly almost everywhere dominating.Proof. Fix e. Then λ{Z : Φ Z e total & Φ Z e not dominated by f}≤ 2 e+1 λ{Y : Θ Y total & Θ Y not dominated by f} =0.We will keep Θ fixed for the rest of this section.Our main goal is to show that C ≥ LR ∅ ′ if and only if C is uniformly almosteverywhere dominating. This is an analog for highness properties of the coincidencesof lowness properties obtained in Section 5.1: C ≥ LR ∅ ′ is the dual ofC ≤ LR ∅, namely, C ∈ Low(MLR). Unlike the case of Low(MLR), the propertycharacterizing C ≥ LR ∅ ′ is not directly related to randomness. Rather, it is adomination property defined in terms of the uniform measure. It is instructiveto begin with the special case C = ∅ ′ , due to Kurtz (1981), which will be usedin the next subsection to prove the full result.5.6.28 Proposition. ∅ ′ is uniformly almost everywhere dominating.Proof. The class {Y : Θ Y ↾ n+1 ↓} is Σ 0 1 uniformly in n. Hence ∅ ′ can determinethe greatest q n of the form i2 −n ,0≤ i

5.6 The weak reducibility associated with Low(MLR) 235Define a function f ≤ T ∅ ′ as follows. On input n, using ∅ ′ , determine t n andoutput the maximum of values Θ σ t n(n) over all such strings σ of length t n .For each n let C n = {Y : Θ Y ↾ n+1 ↓ & Θ Y (n) >f(n)}. Note that λC n < 2 −n bythe definition of q n . Hence 2 −k ≥ λ ⋃ n>k C n for each k. The intersection of theclasses ⋃ n>k C n over all k is {Y : Θ Y is total and not dominated by f}, whichis a null class.✷Dobrinen and Simpson (2004) pointed out that ∅ ′ ≤ T C ⇒ C is u.a.e.d. ⇒C is high 1 , and asked whether the class of u.a.e.d. sets coincides with one of thesetwo extremes. The answer is negative. The first examples of u.a.e.d. sets C ≱ T ∅ ′appeared in Cholak, Greenberg and Miller (2006). One of their constructions usesa method inspired by the forcing method from set theory, and builds a u.a.e.d.set C such that C ≱ T E for a given incomputable set E. The other constructionyields a c.e. set C. Here we first prove that C is u.a.e.d. iff C ≥ LR ∅ ′ . Then, inSection 6.3, we give examples of sets C ≥ LR ∅ ′ such that C< T ∅ ′ . Such a set Ccan be chosen to be either c.e., or ML-random.Binns, Kjos-Hanssen, Lerman and Solomon (2006) showed that some high setis not almost everywhere dominating. In Theorem 8.4.15 we will prove a strongerresult: ∅ ′ ≤ LR C implies that C is superhigh, namely, ∅ ′′ ≤ tt C ′ . This propertyonly depends on the Turing degree of C. The superhigh degrees form a propersubclass of the high degrees.∅ ′ ≤ LR C if and only if C is uniformly a.e. dominatingThe first step towards proving this equivalence was made by Dobrinen and Simpson(2004). They expressed the property of being u.a.e.d. in terms of a coveringprocedure (see before Lemma 5.6.3, or Table 5.2 on page 229): each Σ 0 2 class,and in fact, each Π 0 3 class (by Exercise 5.6.32) can be covered by a Π 0 2(C) class ofthe same measure. This states indeed that C is computationally strong since theΠ 0 2(C) class emulates the given Π 0 3 class. In comparison, (ii) of Theorem 5.5.17says that A is weak, because each Π 0 2(A) class can be emulated by a Π 0 2 class.Since the domain of a Turing functional is a Π 0 2 class, the notation becomessimpler when one rephrases the covering procedure using complements.5.6.29 Lemma. C is uniformly a.e. dominating ⇔for each Π 0 2 class H there is a Σ 0 2(C) class L ⊆ H such that λL = λH.Proof. ⇒: Suppose C is u.a.e.d. via the strictly increasing function f ≤ T C.Let H = ⋂ n S n where the S n are Σ 0 1 classes uniformly in n. Let Ψ be the Turingfunctional such that Ψ Z (n) ≃ µt. Z ∈ S n,t for each Z, n. The required Σ 0 2(C)class isL = { Z : ∃m [ ∀n

236 5 Lowness properties and K-triviality⇐: Let Θ be the Turing functional of Lemma 5.6.27 and let H be the Π 0 2class {Z : Θ Z total}. Choose a Σ 0 2(C) class L ⊆ H such that λL = λH. ThusL = ⋃ n P n where P n is Π 0 1(C) uniformly in n and P n ⊆ P n+1 for each n.Uniformly in n we have a C-computable sequence (P n,t ) t∈N of clopen sets suchthat P n = ⋂ t P n,t, by (1.17) on page 55.We define a function f ≤ T C that dominates the total function Θ Z for almostevery Z. On input n, using C as an oracle find t ∈ N such that∀σ [ (|σ| = t &[σ] ⊆ P n,t ) → Θ σ t (n)↓ ] .Note that t exists since P n ⊆ H. Let f(n) be the maximum of all the valuesΘ σ t (n) where |σ| = t and [σ] ⊆ P n,t . Almost every set Z ∈ H is in P k for some k.In that case Θ Z (n) ≤ f(n) for all n ≥ k.✷The coincidence result is due to Kjos-Hanssen, Miller and Solomon (20xx).5.6.30 Theorem. ∅ ′ ≤ LR C ⇔ C is uniformly a.e. dominating.Proof. ⇒: Firstly, ∅ ′ is u.a.e.d. by 5.6.28, so by Lemma 5.6.29 every Π 0 2 classcontains a Σ 0 2(∅ ′ ) class of the same measure (also see Exercise 5.6.33).Secondly, we apply the implication (i)⇒(iii) in Theorem 5.6.9 for A = ∅ ′ andB = C. Note that (i) holds by our hypothesis ∅ ′ ≤ LR C, so taking complementsin (iii) we obtain that each Σ 0 2(∅ ′ ) class contains a Σ 0 2(C) class of the samemeasure.We may conclude that each Π 0 2 class contains a Σ 0 2(C) class of the samemeasure. Hence C is uniformly a.e. dominating by a further application ofLemma 5.6.29.⇐: Each Π 0 1(∅ ′ ) class is a Π 0 2 class by Exercise 1.8.67, so by Lemma 5.6.29 eachΠ 0 1(∅ ′ ) class contains a Σ 0 2(C) class of the same measure. The implication (ii)⇒(i)of Theorem 5.6.9 now shows that ∅ ′ ≤ LR C.✷In the following we consider a further property of a set C introduced by Dobrinenand Simpson (2004) that appears to be weaker than being uniformly a.e.dominating. We say that C is almost everywhere dominating if for almost everyset Z, every function g ≤ T Z is dominated by a function f ≤ T C. The differenceis that now the dominating function f ≤ T C can depend on the oracle Z andon g. However, by the following result of Kjos-Hanssen (2007), together with theforegoing Theorem, this property is equivalent to being u.a.e.d.5.6.31 Proposition. ∅ ′ ≤ LR C ⇔ C is almost everywhere dominating.Proof. ⇒: Immediate by Theorem 5.6.30.⇐: By Lemma 5.6.3 it suffices to show that each Π 0 1(∅ ′ ) class H of positivemeasure contains a Σ 0 2(C) class L of positive measure (and hence a Π 0 1(C) class ofpositive measure). We proceed similar to the implication “⇒” of Lemma 5.6.29.Let H = ⋂ n S n, where S n is uniformly Σ 0 1, and let the functional Ψ be defined asbefore. Since C is a.e. dominating, there is e such that 0 < λ{Z : Φ C e dominatesΨ Z }. Define a Σ 0 2(C) class L as before, but based on f = Φ C e . Then L ⊆ H andλL >0.✷

5.6 The weak reducibility associated with Low(MLR) 237Exercises.5.6.32. Show that if C is u.a.e.d., then in fact each Π 0 3 class G can be covered by aΠ 0 2(C) class of the same measure.5.6.33. Give a direct proof (not via Proposition 5.6.28) for the fact used in the proofof Theorem 5.6.30 that each Π 0 2 class H contains a Σ 0 2(∅ ′ ) class L of the same measure.Thus Z is weakly 3-random iff Z is weakly 2-random relative to ∅ ′ .

6Some advanced computability theoryWe outline the plan for the next three chapters. In the present chapter we mainlydevelop computability theory, having its interaction with randomness in mind.Chapter 7 is mostly about randomness. We discuss betting strategies, and introducethe powerful notion of martingales as their mathematical counterpart.Like Chapter 5, Chapter 8 is on the interactions of computability and randomness.It combines the technical tools of the foregoing two chapters in order tostudy lowness properties and highness properties of sets. For instance, in Theorem8.3.9, lowness for Schnorr randomness is characterized by a computabilitytheoretic property called computable traceability. The proof relies on computablemartingales.A main technical advance of this chapter is a method for building Turingfunctionals: they can be viewed as ternary c.e. relations of a particular kind.An appropriate language is introduced. As a first application of this method, webuild superlow c.e. sets A 0 and A 1 such that A 0 ⊕ A 1 is Turing complete.Prompt simplicity is a highness property within the c.e. sets we introducedin 1.7.9. (However, this property is compatible with being low for ML-randomnessby Theorem 5.1.19.) In Theorem 6.2.2 we use the language for building Turingfunctionals to characterize the degrees of promptly simple sets as the low cuppabledegrees. The promptly simple degrees also coincide with the non-cappabledegrees, as mentioned at the beginning of Section 1.7. Historically, these resultsof Ambos-Spies, Jockusch, Shore and Soare (1984) constituted the first exampleof a coincidence result for degree classes that had been studied separately before.In Chapter 5 we obtained such coincidence results for the K-trivial degrees andfor the degrees of uniformly a.e. dominating sets.C.e. operators are functions of the form Y → W Y e , first mentioned in (1.4) onpage 10. Here we use the new language in order to build such operators. Wealso present inversion results for c.e. operators V that are increasing in the senseof ≤ T . For instance, there is a c.e. set C, as well as a ML-random ∆ 0 2 set C,such that V C ≡ T ∅ ′ .IfV is the c.e. operator corresponding to the constructionof a c.e. K-trivial set, then C ≡ LR ∅ ′ , whence C is uniformly a.e. dominating byTheorem 5.6.30. Inversion results can be used to separate highness properties.For instance, we will build a superhigh set that is not uniformly a.e. dominating.The language for building Turing functionals introduced here will also be usedin the proof of Theorem 8.1.19 that a computably dominated set in GL 1 doesnot have d.n.c. degree, and when we show near the end of Section 8.4 that thestrongly jump traceable c.e. sets form a proper subclass of the c.e. K-trivial sets.

6.1 Enumerating Turing functionals 2396.1 Enumerating Turing functionalsBy Remark 1.6.9, constructions using the priority method can be viewed as agame between us and an evil opponent called Otto. It is important that Otto andwe have the same types of objects. For instance, he enumerates the c.e. sets W e ,and we also build c.e. sets. How about the Turing functionals Φ e ? In the proofs ofresults such as the Friedberg–Muchnik Theorem 1.6.8, the functionals belongedto him while we had to diagonalize against them. In subsequent constructionswe often build a set Turing below another, so we have to provide a Turing functional.Usually we give an informal procedure that relies on an oracle Y . Then,by the oracle version of the Church–Turing thesis a Turing program P e formalizingthe procedure exists; this determines the required Turing functional Φ e .Wehave already introduced functionals in such a way, for instance in the proof ofthe implication from right to left of Proposition 1.2.21, and in the proof of Theorem1.7.2. In more complex constructions it is convenient to view a functionalbelonging to us as a particular kind of a c.e. relation, since its enumeration canbe made part of the construction.Basics and a first exampleA computation Φ Y (x) =w relies on the finite initial segment η of the oraclesuch that |η| = use Φ Y (x) (recall from 1.2.18 that the use is 1 + q where q isthe maximum oracle question asked during the computation). We will view theconvergence of such a computation as the enumeration of a triple 〈η, x, w〉 intoa c.e. set, where η is the initial segment of the oracle, x is the input and w theoutput. We choose η shortest possible, namely, no substring of η yields a haltingcomputation on input x.6.1.1 Fact. The Turing functionals Φ e correspond in a canonical way to the c.e.sets Γ ⊆ {0, 1} ∗ × N × N such that(F1) 〈η, x, v〉, 〈η, x, w〉 ∈Γ → v = w (output uniqueness), and(F2) 〈η, x, v〉, 〈ρ, x, w〉 ∈Γ → (η = ρ ∨ η | ρ) (oracle incomparability).Proof. Given Φ e , enumerate Γ as follows. Run the computation of the Turingprogram P e with η on the oracle tape on input x. If an oracle question q isasked such that q ≥ |η| then stop. Otherwise, if the computation halts with themaximum oracle question having been |η| − 1, or no questions are asked andη = ∅, then put 〈η, x, w〉 into Γ. Clearly Γ has the properties (F1) and (F2).Conversely, given Γ we use the oracle version of the Church–Turing thesis toobtain a Turing program P e and hence Φ e . In the informal procedure, supposethe oracle is Y , and the input is x. Whenever 〈η, x, w〉 is enumerated into Γ, testwhether η ≺ Y . If so, output w and stop.✷Terminology used previously only for the functionals Φ e can now be appliedto any Turing functional Γ as in Fact 6.1.1. For instance, Γ Y (x) =w meansthat Φ Y e (x) =w for the corresponding Φ e , which is equivalent to 〈η, x, w〉 ∈Γfor some η ≺ Y . In this case, we let use Γ Y (x) ≃ use Φ Y e (x) =|η|. We write γ Y (x)

240 6 Some advanced computability theory= use Γ Y (x). More generally, if an upper case Greek letter denotes a functional,the corresponding lower case Greek letter denotes its use function.If a ∆ 0 2 set A is given by a computable approximation (A s ) s∈N , we let Γ A (x)[s] =w if 〈η, x, w〉 ∈Γ s for some η ≺ A s .Theorem 1.7.2 states that for each c.e. incomputable set C, there is a simpleset A ≤ wtt C. In the proof we implicitly built a Turing functional Γ to show thatA ≤ wtt C. Let us reformulate the construction in this new language to define Γexplicitly.Construction of A and Γ. Let A 0 = ∅ and Γ 0 = ∅.Stage s>0.1. For each i γ s−1 (x) → C s ↾ γs−1(x)≠ C s−1 ↾ γs−1(x), (6.1)because then η = C s ↾ γs(x) satisfies η | ρ for any ρ such that 〈ρ, x, w〉 ∈Γ s−1(a c.e. set never turns back). To make Γ C total, we also require that∀x lim s γ s (x) < ∞, (6.2)so that for each x there is v such that 〈η, x, v〉 ∈Γ, where γ(x) = lim s γ s (x) andη = C ↾ γ(x) .ThusΓ C (x)↓.

6.1 Enumerating Turing functionals 241It would be safest to keep γ s (x) from changing all-together, as in the rephrasedproof of Theorem 1.7.2 above. We usually “correct” the output of Γ C (x) afterputting γ(x) − 1intoC. In more complex constructions, this conflicts with constraintson C, sowehavetochooseγ(x) larger than these constraints. Certainstrategies declare γ(x) to be undefined. It is redefined at some later stage to bea large number y, that is, y − 1 is larger than any number mentioned so far. (Inparticular, y is not yet in C.) Informally, this process is called lifting the use. Toachieve (6.2) we must ensure that γ(x) is lifted only finitely often.6.1.3 Remark. In the next two applications of the method we build reductionsto a set C of the form E 0 ⊕ E 1 . A proper formal treatment would require to viewfunctionals as c.e. sets of quadruples 〈η 0 , η 1 ,x,v〉, where η i denotes an oraclestring that is an initial segment of E i . Similarly, in the treatment via markerswe would have to keep the left use γ 0 (x) and the right use γ 1 (x) separate. Wecan usually avoid this by requiring that the use be the same on both sides (anexception is the proof of Theorem 6.3.1 below). This common use will also bedenoted by γ(x). A change of either of the two sets below γ(x) allows us toredefine the computation Γ E0⊕E1 (x).The Sacks Splitting Theorem 1.6.10 implies that there are low c.e. sets A 0and A 1 such that ∅ ′ ≤ T A 0 ⊕ A 1 . Bickford and Mills (1982) strengthened this:6.1.4 Theorem. There are superlow c.e. sets A 0 ,A 1 such that ∅ ′ ≤ T A 0 ⊕ A 1 .Proof. We enumerate the sets A 0 and A 1 , and a Turing functional Γ such that∅ ′ = Γ(A 0 ⊕ A 1 ). The use of Γ(A 0 ⊕ A 1 ; p) is denoted by γ(A 0 ⊕ A 1 ; p) andpictured as a movable marker. For the duration of this proof, k and l denotenumbers in {0, 1}, p and q denote numbers in N, and [p, k] stands for 2p + k.Also, j(X, n) stands for use J X (n). To avoid that J A k(p) change too often, weensure that at each stage s, for each p and k such that [p, k] ≤ s, wehaveJ A k(p)[s]↓ → γ(A 0 ⊕ A 1 )([p, k]) >j(A k ,p)[s] (6.3)To do so, when J A k(p)[s] newly converges for the least [p, k], we change A 1−k .(This idea to change the other side stems from the proof of the Sacks SplittingTheorem 1.6.10.)Construction of A 0 ,A 1 and Γ. Let A 0 = A 1 = ∅. Define Γ(A 0 ⊕ A 1 ; 0) = 0 withuse 2.Stage s>0. Define Γ(A 0 ⊕ A 1 ; s) = 0 with large use. Do the following.(a) If there is [p, k] such that J A k(p)[s − 1]↑ and J A k(p)↓ at the beginning ofstage s, then choose [p, k] least. Put γ(A 0 ⊕ A 1 ;[p, k]) − 1intoA 1−k andredefine Γ(A 0 ⊕ A 1 ; q), s ≥ q ≥ [p, k], with value ∅ ′ s(q) and large use.(b) If n ∈∅ ′ s −∅ ′ s−1 then put γ(A 0 ⊕ A 1 ; n) − 1intoA 0 .A typical set-up is shown in Figure 6.1, where J A1 (p) converged after J A0 (p).Claim 1. The condition (6.3) holds for each s.

242 6 Some advanced computability theoryγ(2p)use J A0 (p)use J A1 (p)γ(2p +1).Fig. 6.1. Typical set-up in the proof of 6.1.4.We use induction on s. The condition holds for s =0.Ifs>0, we may supposethere is a least [p, k] such that J A k(p)[s − 1] ↑ and J A k(p)[s] ↓ (else there isnothing to prove), in which case we put y = γ(A 0 ⊕ A 1 ;[p, k]) − 1intoA 1−k .• If [q, l] < [p, k] then y ≥ γ(A 0 ⊕ A 1 ;[q, l]) >j(A l ,q)[s] by the inductivehypothesis, so (6.3) remains true for J A l(q)[s].• Since we enumerate y into A 1−k , the computation J A k(p)[s] remains convergent,so we ensure that γ(A 0 ⊕ A 1 ;[p, k]) >j(A k ,p)[s].• For [q, l] > [p, k], the use of computations J A l(q)[s] is below the new valueof γ([q, l]).✸Claim 2. A 0 and A 1 are superlow.Similar to the construction of a superlow simple set in 1.6.5, if J A k(p)[s] ↓ letf k (p, s) = 1, and otherwise let f k (e, s) = 0. We define a computable function hsuch that h([p, k]) bounds the number of times J A k(p) becomes defined. Thenb k (p) =2h([p, k]) + 1 bounds the number of times f k (p, s) changes.By (6.3), J A0 (p) becomes undefined at most 2p times due to a change of ∅ ′ ↾ 2p .Otherwise, J A k(p) becomes undefined only when some computation J A 1−k(q)becomes defined where [q, 1 − k] < [p, k]. Thus the function h given by h(0) = 1and h([p, k]) = 1 + 2p + ∑ qh([q, 1 − k]) [[q, 1 − k] < [p, k]] is as desired. ✸By the proof of Claim 2 each marker γ(A 0 ⊕ A 1 ; m) reaches a limit. Therefore∅ ′ = Γ(A 0 ⊕ A 1 ).✷6.1.5 Remark. In contrast to the case of the Sacks Splitting theorem, we cannotachieve that the use function of Γ is computably bounded for the oracle A 0 ⊕A 1 :Bickford and Mills (1982) showed that each superlow c.e. set A is non-cuppablein the c.e. wtt-degrees, namely, if ∅ ′ ≤ wtt A ⊕ W for a c.e. set W then already∅ ′ ≤ wtt W . Thus some low c.e. set is not superlow. One can also obtain such aset by a direct construction (see Exercise 1.6.7).6.2 Promptly simple degrees and low cuppabilityWe say that a c.e. set A is cuppable if ∅ ′ ≤ T A ⊕ Z for some c.e. set Z ≱ T ∅ ′ .Informally speaking, A is helpful for Z to compute ∅ ′ . This class contains incomputablesets, and even high sets, by a result in Miller (1981) with Harrington.Variants of this concept are obtained by asking that the set Z ≱ T ∅ ′ be in someother class instead of the c.e. sets. For instance, A is low cuppable if ∅ ′ ≤ T A ⊕ Z

6.2 Promptly simple degrees and low cuppability 243for some low c.e. set Z. Being non-cuppable in a particular sense is a lownessproperty within the c.e. sets. In Remark 8.5.16 we show that any c.e. set that isnot cuppable by a ML-random set Z ≱ T ∅ ′ is K-trivial.The main result of this section, due to Ambos-Spies et al. (1984), is that A haspromptly simple Turing degree iff A is low cuppable. Thus the correspondinglowness property, to be not low cuppable, is equivalent to being cappable (seepage 37). On the other hand, it is known that the non-cuppable c.e. sets form aproper subclass of the cappable sets.C.e. sets of promptly simple degreeBy Definition 1.7.9, a co-infinite c.e. set E is promptly simple if it has a computableenumeration (E s ) s∈N such that for each e the requirementPS e : #W e = ∞⇒∃s>0 ∃x [x ∈ W e,at s ∩ E s ]is met. We say that A has promptly simple degree if A ≡ T E for some promptlysimple set E, and A has promptly simple wtt-degree if A ≡ wtt E for some promptlysimple set E. We prove an auxiliary result of Ambos-Spies et al. (1984) characterizingthe c.e. sets of promptly simple degree in terms of so-called delayfunctions.6.2.1 Theorem. Let A be c.e. Then the following are equivalent.(i) A has promptly simple degree.(ii) A has promptly simple wtt-degree.(iii) There is a computable delay function d with ∀sd(s) ≥ s such that foreach ePSD e :#W e = ∞ ⇒ ∃s∃x [ x ∈ W e,at s & A d(s) ↾ x ≠ A s ↾ x]. (6.4)If A d(s) ↾ x ≠ A s ↾ x we say that A promptly permits below x.Proof. (i)⇒(iii): we prove (iii) under the weaker hypothesis that there is apromptly simple set E ≤ T A. Suppose E is promptly simple via a computableenumeration (E s ) s∈N , and E = Φ A for some Turing functional Φ. As in theproof of Theorem 1.7.14, we enumerate auxiliary sets G e uniformly in e. BytheRecursion Theorem we are given a computable function g such that G e = W g(e)for each e (this is similar to the proof of 1.7.14). We show that PSD e is met foreach e, where the delay function isd(s) =µt > s. ∀e

244 6 Some advanced computability theoryVerification. Consider PSD e .IfW e is infinite then G e = W g(e) is infinite as well.Since E is promptly simple via the given enumeration, there is a stage s wherewe attempt to meet PSD e via x, y, and y ∈ W g(e),t for some t ≥ s such thaty ∈ E t . Since E s (y) =0=Φ A (y)[s] and use Φ A (y)[s] ≤ x, by the definition ofthe function d this means that A d(s) ↾ x ≠ A s ↾ x .(iii)⇒(ii): We use the methods of Theorem 1.7.3 to build a promptly simple setE ≡ wtt A. We meet the requirements PS e above.Construction of E. Let E 0 = ∅.Stage s>0.• For each e < s, if PS e is not yet met and there is x ≥ 3e such thatx ∈ W e,at s , then check whether A d(s) ↾ x ≠ A s ↾ x . If so, put x into E s .• If y ∈ A at s , put p Es−1(3y) intoE.Clearly E is promptly simple and E ≤ wtt A. Note that #E ∩ [0, 3i) ≤ 2i foreach i, since at most i elements less than 3i enter due to the requirements PS eand at most i for the coding of A into E. Hence p E(y) ≤ 3y for each y. ThenE s ↾ 3y+1 = E ↾ 3y+1 implies A s (y) =A(y), so that A ≤ wtt E.✷A c.e. degree is promptly simple iff it is low cuppableWe now prove the main result of this section. It was applied already in Remark5.6.24.6.2.2 Theorem. Let A be c.e. Then the following are equivalent.(i)(ii)A has promptly simple degree.A is low cuppable, namely, there is a low c.e. set Z such that ∅ ′ ≤ T A⊕Z.Proof. (i) ⇒ (ii):Idea. We enumerate Z and build a Turing functional Γ such that ∅ ′ = Γ A⊕Z .To ensure Z is low, we meet the usual requirements from the proof of 1.6.4L e : ∃ ∞ sJ Z (e)[s]↓ ⇒ J Z (e)↓.To aid in meeting L e we enumerate a c.e. set G e . As in Theorem 1.6.4, whenJ Z (e) converges we would like to preserve Z up to the use of this computation.However, now we also have to maintain the correctness of Γ. When a numberenters ∅ ′ this may force us to change Z. We would like to achieve the situationwhere γ A⊕Z (e) > use J Z (e) after ∅ ′ ↾ e has settled, for in that case we do nothave to change Z below the use of J Z (e). For this we use the hypothesis that Ahas promptly simple degree via a delay function d as in Theorem 6.2.1. Whenwe see a new computation J Z (e) at stage s, it will generally be the case thatx = γ(e) ≤ use J Z (e)[s]. In this case we put x into G e hoping that A willpromptly permit below x.If A does so, we may compute the stage r by when it has permitted, and weonly resume the construction then. The A-change allows us to lift the use γ r (e).Nothing happened from stage r to s−1. In particular, Z was not enumerated, so

6.2 Promptly simple degrees and low cuppability 245J Z (e)[s − 1] remains unchanged at stage r. Any computation Γ A⊕Z (e)[t] definedat a stage t ≥ r has use γ t (e) >r≥ use J Z (e)[r]. We say that we have clearedthe computation J Z (e) of the use γ(e).If A does not permit promptly then instead we put x into Z at stage s, and liftγ(e) via this Z-change. Then the next computation Γ A⊕Z (e) will have a largeruse. However, this case only applies finitely often: otherwise, G e is infinite, soeventually we would get the prompt permission, contradiction. (Putting x into Zis often called the “capricious destruction of the computation J Z (e)[s]”. Theactual purpose is that we threaten to make G e infinite each time we give up onthis computation.)Details. As in the proof of Theorem 1.7.14, we are given a computable function gand think of W g(e) as G e by the Recursion Theorem.Construction of Z, Γ, and an auxiliary u.c.e. sequence (G e ) e∈N .Let Z 0 = ∅ and G e,0 = ∅ for each e. Let Γ = ∅.Stage s>0.(1) Coding ∅ ′ . If v

246 6 Some advanced computability theoryFor (b), choose s 1 ≥ s 0 such that either J Z (e) is stable from s 1 on, or J Z (e)[s]↑for all s ≥ s 1 , and also ∅ ′ (e) is stable at s 1 . Then γ(e) reaches its limit by stage s 1 .✸By the remarks above, (b) implies that Γ A⊕Z is total. Also ∅ ′ = Γ A⊕Z sincewe keep each Γ A⊕Z (i) correct at each stage s>i.(ii) ⇒ (i):Idea. We use the lowness of Z via the Robinson guessing method already appliedin the proof of Theorem 5.3.22. Fix a Turing reduction for ∅ ′ ≤ T A ⊕ Z. Weenumerate a c.e. set C and a Turing functional ∆. By the Double RecursionTheorem 1.2.16, we may assume that we are given in advance a Turing functionalΓ such that C = Γ(A ⊕ Z), and a reduction function p for ∆ in the sense of Fact1.2.15. Keep in mind that Γ now belongs to Otto, while ∆ is ours.In more detail, given indices for a c.e. set ˜C and a Turing functional ˜∆, weeffectivelyobtain a many-one reduction showing ˜C ≤ m ∅ ′ and a reduction function p for ˜∆ becauseProposition 1.2.2 and Fact 1.2.15 are uniform. The reduction for ∅ ′ ≤ T A⊕Z is fixed inadvance, so combining it with the many-one reduction we obtain a Turing functional Γsuch that ˜C = Γ(A ⊕ Z). Based on Γ and p we run the construction to build C and ∆.By the Double Recursion Theorem we may assume that C = ˜C and ∆ = ˜∆. ThenC = Γ(A ⊕ Z) and p is a reduction function for ∆.The so-called length-of-agreement function associated with Γ isl Γ (t) = max{y : C ↾ y = Γ A⊕Z ↾ y [t]}.We enumerate a number w ∈ N [e] into C to enforce the prompt change of Aneeded in order to meet the requirement PSD e . Suppose w is not in C yet andΓ A⊕Z (w) = 0. If we put w into C, then one of A or Z must change belowthe use γ A⊕Z (w) in order to correct the Γ computation at input w. When x ≥γ A⊕Z (w)[s] enters W e at stage s, a reasonable strategy would be to enumerate winto C at the same stage, hoping that the correction of Γ A⊕Z (w) will be via A.In this case we would have a prompt permitting by A via the delay functiond(s) = min{t >s: l Γ (t) ≥ s}.Namely, d(s) is the first stage greater than s where the length of agreement isat least s again.If the correction is always via a change of Z, we waste all the numbers win N [e] without ever getting the desired prompt permission. To avoid this, beforeenumerating w we ask whether Z is stable up to the use u = γ A⊕Z (w)[s]. This ispossible since Z is low. We define a new value of the auxiliary functional ∆ Z (e)with the same use u, which allows us to control the jump J Z at the input p(e).Only if the answer is “yes” do we put w into C at s. IfZ ↾ u changes after allthen the ∆ Z (e) computation goes away. We have to try again when the next(sufficiently large) number enters W e , using the next w ∈ N [e] . We can affordfinitely many incorrect answers. Eventually the answer will be correct, so we getthe desired prompt permission by A.

6.3 C.e. operators and highness properties 247Details. Since Z is low, by the Limit Lemma 1.4.2 there is a computable binaryfunction g such that Z ′ (m) = lim s g(m, s) for each m.Construction of C and ∆, given Γ and the reduction function p. Let C 0 = ∅.Stage s>0. For each e, if the requirement PSD e has not yet been met, letw = min(N [e] − C s ). If l Γ (s) >wand ∆ Z (e)[s] is undefined, then see if there isan x ∈ W e,s − W e,s−1 such that x ≥ u = γ A⊕Z (w)[s]. If so, define ∆ Z (e) =0with use u. Search for the least t ≥ s such thatZ t ↾ u ≠ Z s ↾ u or g(p(e),t)=1.(In the fixed point case, one of the two alternatives applies, since p is a reductionfunction for ∆.)If g(p(e),t) = 1 do the following: enumerate w into C (at this stage s). IfA d(s) ↾ u ≠ A s ↾ u , then we got the prompt permission, so PSD e is met. OtherwiseZ d(s) ↾ u ≠ Z s ↾ u , so we may declare ∆ Z (e)[d(s)] undefined. (Later we may tryagain with the next w in N [e] and a new x.)Verification. Assume for a contradiction that W e is infinite but PSD e is not met.Case 1: ∆ Z (e) is defined permanently from some stage s on. This is so becauseat s we put the number w into C when Z s ↾ u was stable already, where u =γ A⊕Z (w)[s]. Then A d(s) ↾ x ≠ A s ↾ x ,soPSD e is met, contradiction.Case 2: ∆ Z (e) is undefined at infinitely many stages. Since ∆ Z (e) ≃ J Z (p(e)),this implies p(e) ∉ Z ′ , so lim s g(p(e),s)=0.ThusN [e] ∩ C is finite; let s 0 = d(t)where t is the greatest stage such that t = 0 or an element of N [e] enters C. Letw = min(N [e] − C) and suppose that Γ A⊕Z (w) has settled by stage s 1 ≥ s 0 . Lets ≥ s 1 be least such that some x ≥ u = γ A⊕Z (w)[s 1 ] enters W e at stage s. Then∆ Z (e)[s] is undefined (else PSD e would have been met by stage d(t)). So wedefine ∆ Z (e)[s] = 0 with use u. Since Z ↾ u is stable from s 1 on, this contradictsthe case hypothesis.✷In the proof of (i)⇒(ii) the number of injuries to L e depends on the element of G ethat is promptly permitted by A. Thus the set Z we build is not superlow for anyobvious reason. We say that A is superlow cuppable if there is a superlow c.e. set Zsuch that ∅ ′ ≤ T A ⊕ Z. Indeed, Diamondstone (20xx) has proved that some promptlysimple set fails to be superlow cuppable. Also see Exercise 8.5.24.6.3 C.e. operators and highness propertiesFor each c.e. operator W we will build a c.e. set C, and also a ML-random set C,such that W C ⊕ C ≡ T ∅ ′ .IfW is the c.e. operator given by the construction ofa low simple set, then C< T ∅ ′ and C is high. For a stronger result, we take theoperator given by the cost function construction of a c.e. K-trivial set, (5.3.11)and obtain a set C< T ∅ ′ such that C ≡ LR ∅ ′ .The basics of c.e. operatorsDefinitions of c.e. sets can usually be viewed relative to an oracle C. For instance,we have done this for the halting problem ∅ ′ : in Definition 1.2.9 we defined

248 6 Some advanced computability theoryC ′ = dom(J C )={e: e ∈ We C }. Given a particular definition of a c.e. set A andan oracle C, we will write A C for the set defined in the same way relative to C.In this way, the definition of A yields a c.e. operator. For a further example,let A = {q ∈ Q 2 : q

6.3 C.e. operators and highness properties 249The intuition is as follows: a number i enters W C at stage s with a certain use.From then on it stays in W C as long as the approximation to C does not changebelow that use. The set W C is Σ 0 2 via the approximation (W C [s]) s∈N in the senseof Proposition 1.4.17, since W C (i) = lim inf s W C (i)[s].Pseudojump inversionThe jump operator is increasing in the sense that C ′ ≥ T C for each C. This failsfor a c.e. operator W in general. For instance, let W C = {q ∈ Q 2 : q T C and (V C ) ′ ≡ T C ′ for each C.Recall that the jump operator is degree invariant and even monotonic (see 1.2.14 andthe discussion on page 34). Degree invariance fails for a pseudojump operator in general.For a trivial counterexample, we view the construction in the proof of Theorem 1.6.8relative to an oracle. We now build c.e. operators A and B such that A C ⊕C | T B C ⊕Cfor each C. Let{W C A C−{0} if 0 ∈ C,=B C−{0} else.The operator given by V C = W C ⊕ C is not degree invariant since V {0} | T V {1} .By Downey, Hirschfeldt, Miller and Nies (2005) the operator C → Ω C is not degreeinvariant.The pseudojump inversion theorem, due to Jockusch and Shore (1984), statesthat no matter what W is, there exists a c.e. set C such that W C ⊕ C ≡ T ∅ ′ .This result can be used, for instance, to build a high c.e. set C< T ∅ ′ . For, a c.e.set C is high iff ∅ ′ is low relative to C. Let W be the c.e. operator given by theconstruction of a low simple set, then C< T ∅ ′ and ∅ ′ is low relative to C.6.3.1 Theorem. For each c.e. operator W there is a c.e. set C such thatW C ⊕ C ≡ T ∅ ′ .Proof idea. We enumerate a Turing functional Γ such that ∅ ′ = Γ(W C ⊕ C).We also ensure that W C is ∆ 0 2 (and not merely Σ 0 2, which is automatic), so thatW C ⊕ C ≤ T ∅ ′ by the Limit Lemma 1.4.2. The requirements for W C ∈ ∆ 0 2 areQ e : ∃ ∞ s>0 [ e ∈ W C (e)[s − 1] ] ⇒ e ∈ W C .If we meet all the Q e then W C is in ∆ 0 2 via the computable approximationλe, s. W C (e)[s]. Note that the Q e are similar to the lowness requirements L efrom the proof of Theorem 1.6.4, but with the c.e. operator W instead of thejump operator.The construction parallels the one in the proof of (i)⇒(ii) of Theorem 6.2.2that each c.e. set A of promptly simple degree can be cupped to ∅ ′ by a low c.e.set Z. In both cases we build a reduction Γ computing ∅ ′ with the appropriate

250 6 Some advanced computability theoryoracle. Here we want to achieve W C ⊕ C ≡ T ∅ ′ , there we ensured A ⊕ Z ≡ T ∅ ′ .The set C here corresponds to Z there. Both C and Z are built by us. The setW C plays the role of the given set A there.Recall the discussion in Remark 6.1.3. Here we let γ s (e) >edenote the use onthe right side and view it as a movable marker. It matters that the actual use onthe left side is e + 1. However, to simplify notation we artificially increase thisuse to γ(e).To maintain ∅ ′ = Γ(W C ⊕ C), when i enters ∅ ′ we put γ(i) − 1intoC. Thestrategy for Q e is not to restrain W C (which is impossible), but rather to stabilizeW C (e), by ensuring for all stages s and for each e use (e ∈ W C [s]). (6.5)For then, if i ≥ e enters ∅ ′ , the Γ-correction via a C-change does not remove efrom W C .Thuse can leave W C at most e times, till ∅ ′ ↾ e is stable. Hence Q eis met. The potential problem is that oracle incomparability (F2) in Fact 6.1.1might fail for Γ because W C (e) can change back and forth. But e can onlyleave W C at a stage s when C ↾ r changes, where r = use (e ∈ W C [s]). Soby (6.5), without violating (F2) we can put a new computation into Γ when ereappears in W C .It may help our understanding to introduce further requirementsT i : ∅ ′ (i) =Γ(W C ⊕ C; i).When i enters ∅ ′ then T i puts γ(i) − 1intoC, but otherwise it wants to keep γ(i)from changing. Q e is allowed to move γ(e), which injures T i for i ≥ e. IfT i enumeratesinto C and this removes e from W C , then Q e is injured for e>i. This is reflected bythe priority ordering Q 0

6.3 C.e. operators and highness properties 251(a) If e enters W C at a stage s 1 ≥ s 0 then γ s (e) > use (e ∈ W C [s]) for s ≥ s 1 ,so a later enumeration into C will not remove e from W C .(b) By (a) let t ≥ s 0 be a stage by which W C (e) has reached its limit. Thenγ s (e) =γ t (e) for all t ≥ s.✸Next we show that Γ is a Turing functional. The output uniqueness (F1) inFact 6.1.1 follows from the coding in (1): suppose 〈η, i, v〉 ∈Γ t , so that |η| =2γ t (i). We only change the output when i enters ∅ ′ at stage s. In that caseγ s−1 (i) − 1 entered C, so we do not put a further triple 〈η, i, w〉 into Γ.Fix e. For each stage u, let η u = W C ⊕ C ↾ 2γ(e) [u]. In order to establish oracleincomparability (F2) it suffices to show the following.6.3.3 Claim. Let siat each stage. ✸At each stage u, we ensure that 〈η u ,e,∅ ′ u(e)〉 ∈Γ where η u = W C ⊕C ↾ 2γ(e) [u].By Claim 6.3.2, γ(e) and W C ↾ γ(e) settle. Thus ∅ ′ = Γ(W C ⊕ C).✷Applications of pseudojump inversionA set C is called high if ∅ ′′ ≤ T C ′ (1.5.18). For instance, Ω ∅′ is high by 3.4.17.So far we have not given an example of a Turing incomplete high ∆ 0 2 set.6.3.4 Corollary. There is a high c.e. set C< T ∅ ′ .Proof. We obtain a c.e. set C by applying Theorem 6.3.1 to the c.e. operator Wgiven by the construction of a low simple set in the proof of Theorem 1.6.4.Then ∅ ′ is low relative to C, that is, (∅ ′ ⊕ C) ′ ≡ T C ′ . Since C is c.e., this implies∅ ′′ ≡ T C ′ . Also, W C is simple relative to C. Hence N − W C is not c.e. relativeto C, and therefore W C ≰ T C by Proposition 1.2.8. Since W C ∈ ∆ 0 2, this showsthat C< T ∅ ′ .✷A direct proof of 6.3.4 uses the priority method with infinite injury (see Soare 1987).A requirement for ∅ ′ ≰ T C may be injured infinitely often by a stronger priorityrequirement for making C high. While the construction becomes more complex, this

252 6 Some advanced computability theorymethod is also more flexible than the one in the foregoing proof: for instance we canachieve C ≱ T B for a given incomputable c.e. set B. A particular implementation ofthe method is by using trees of strategies. This even yields a minimal pair of high c.e.sets (see Exercise 7.5.12).By Definitions 1.5.2 and 1.5.18, a set Z is called low n if Z (n) ≡ T ∅ (n) , and Z ishigh n if Z (n) ≥ T ∅ (n+1) . The downward closed classes low n and non-high n forma hierarchy displayed in (1.8) on page 28. We will see that Theorem 6.3.1 can beused to separate all the classes of c.e. degrees given by this hierarchy: for eachn ≥ 0 there is a c.e. set in low n+1 − low n , and a c.e. set in high n+1 − high n .For each e we let Ve X = We X ⊕X. Notice that Theorem 6.3.1 is uniform, namely,there is a computable function ˜f such that, given a c.e. operator W = W e , thec.e. set C = W ˜f(e)satisfies W C ⊕ C ≡ T ∅ ′ . The construction can be viewedrelative to an oracle Z, so we may regard C as a c.e. operator. Here we adaptthe notation slightly in the statement and proof by joining the oracle Z to theset C Z we construct. In this way we obtain a computable function f such that,for each oracle Z,V e (V f(e) (Z)) ≡ T Z ′ . (6.6)The classes low n and high n can be relativized to an oracle Z:low Z n = {A: (A ⊕ Z) (n) ≡ T Z (n) }, andhigh Z n = {A: (A ⊕ Z) (n) ≥ T Z (n+1) }.The proof of Corollary 6.3.4 shows that, if V Ze ∈ low Z 1 − low Z 0 for each Z, thenthe set C obtained via Theorem 6.3.1 is high but not Turing complete. This is(i) of the following Lemma for n =0.6.3.5 Lemma. Let n ≥ 0.(i) ∀X [V e (X) ∈ low X n+1 − low X n ] ⇒ ∀Y [V f(e) (Y ) ∈ high Y n+1 − high Y n ].(ii) ∀Y [V e (Y ) ∈ high Y n+1 − high Y n ] ⇒ ∀X [V f(e) (X) ∈ low X n+2 − low X n+1].Proof. (i) Given Y , let X = V f(e) (Y ). By hypothesis V e (X) (n+1) ≡ Twhile V e (X) (n) > T X (n) . Also V e (X) =V e (V f(e) (Y )) ≡ T Y ′ by the definitionof f. ThusY (n+2) ≡ T V f(e) (Y ) (n+1) while Y (n+1) > T V f(e) (Y ) (n) , as required.(ii) is similar: given X, let Y = V f(e) (X). By hypothesis V e (Y ) (n+1) ≡ T Y (n+2)while V e (Y ) (n) < T Y (n+1) . Also V e (Y ) ≡ T X ′ . Thus X (n+2) ≡ T V e (Y ) (n+1) ≡ TV f(e) (X) (n+2) while X (n+1) ≡ T V e (Y ) (n) < T V f(e) (X) (n+1) , as required. ✷6.3.6 Theorem. All the classes of c.e. sets in the hierarchy (1.8)X (n+1)computable ⊆ low 1 ⊆ low 2 ⊆ ...⊆ non-high 2 ⊆ non-high 1 ⊆ {Z : Z ≱ T ∅ ′ }are distinct. In fact, there is a uniformly c.e. sequence (A k ) k∈N + such that foreach i ∈ N we have A 2i+1 ∈ low i+1 − low i and A 2i+2 ∈ high i+1 − high i .Proof. Let e be such that W e is the c.e. operator given by the construction ofa low simple set (see page 248). For k ≥ 1 let A k = V f (k−1) (e)(∅). (For instance,

6.3 C.e. operators and highness properties 253A 1 = V f (0) (e)(∅) =W e ⊕∅is low but not computable, and A 2 = V f(e) (∅) is highbut not Turing complete.) By Lemma 6.3.5 the sequence (A k ) k∈N + is as required.✷The operators W of interest in Theorem 6.3.1 satisfy W X < T X ′ for each X, soCis automatically incomputable. However, one can also meet additional requirementsin the construction to ensure this actively. One can even build C of promptly simpledegree. This extension will be needed for the next theorem.6.3.7 Exercise. Show that one can choose the set C in Theorem 6.3.1 incomputable,and even of promptly simple degree.6.3.8 Theorem. There is a c.e. set that is not low n or high n for any n.Proof. Let f be as in (6.6) for the c.e. operator W e given by the extendedconstruction of Exercise 6.3.7. By the Recursion Theorem 1.1.5 relative to anoracle, there is e such that V e (X) =V f(e) (X) for each X. Let V = V e . ThenV (V (X)) ≡ T X ′for each X. Moreover X < T V (X) < T V (V (X)) since we used the extendedconstruction. (Such a c.e. operator V is called a half-jump.)If n ≥ 1 then ∅ (n) ≡ T V (2n) (∅) < T V (2n+1) (∅) < T V (2n+2) (∅) ≡ T ∅ (n+1) . SinceV (2n+1) (∅) ≡ T (V (∅)) (n) , this shows that ∅ (n) < T (V (∅)) (n) < T ∅ (n+1) ,soV (∅)is a c.e. set as desired.✷No half-jump V is degree invariant (see page 34). The foregoing proof shows in factthat X (n) < T (V (X)) (n) < T X (n+1) for each X. On the other hand, Downey and Shore(1997) have proved that if an increasing c.e. operator V is degree-invariant then V isrelatively low 2 or relatively high 2 on an upper cone. That is, there is a set D such thateither V (X) ′′ ≡ T X ′′ for all X ≥ T D, orV (X) ′′ ≡ T X ′′′ for all X ≥ T D.Inversion of a c.e. operator via a ML-random setThe following result is an analog of Theorem 6.3.1.6.3.9 Theorem. For each c.e. operator W and each Π 0 1 class P of positivemeasure there is a set C ∈ P such that W C ⊕ C ≡ wtt ∅ ′ .6.3.10 Corollary. For each c.e. operator W there is a ML-random set C suchthat W C ⊕ C ≡ wtt ∅ ′ .Proof. We apply the theorem to the Π 0 1 class P = {Z : ∀nK(Z ↾ n ) ≥ n}. Notethat λP ≥ 1/2 by 3.2.7, and each member of P is ML-random by 3.2.9. ✷This inversion result has similar applications as Theorem 6.3.1: there is a high MLrandomset C< T ∅ ′ as in Corolllary 6.3.4, and in fact the hierarchy in Theorem 6.3.6can be separated by ML-random ∆ 0 2 sets. This was shown by Kučera (1989) in a directway. Downey and Miller (2006) proved a stronger result already announced by Kučera:for each Σ 0 2 set S ≥∅ ′ there is a ML-random ∆ 0 2 set C such that C ′ ≡ T S. This isanalogous to the Jump Theorem of Sacks (1963c) discussed on page 160.

254 6 Some advanced computability theoryProof of Theorem 6.3.9. By Theorem 1.9.4 we may suppose that λP ≥ 1/2.We build a set C ∈ P in such a way that W C ⊕C ≡ wtt ∅ ′ .ForW C ≤ wtt ∅ ′ , we willprovide an effective approximation to W C . In the construction we have a strategyQ e (without an explicit requirement) making a guess at whether e ∈ W C .Itdefines a Π 0 1 class P e such that either e ∈ W Y for no Y ∈ P e , or for all. Thisresembles the strategy to guess the value of J Y (e) in the proof of the Low BasisTheorem 1.8.37. As in that proof, Q e may change from the guess that e ∉ W Cto the guess that e ∈ W C . Thereafter it only returns to the guess that e ∉ W Cwhen it is initialized, which may happen finitely often. At stage s, Q e determinesa clopen set P e,s . The actual Π 0 1 class on which Q e succeeds is P e = ⋂ s≥s 1P e,s ,where s 1 is the first stage from which on Q e is no more initialized and does notchange its guess. At each stage s, τ e,s is an approximation of W C ↾ e+1 given bythe guesses of the strategies Q i for i ≤ e.As in the proof of Theorem 6.3.1, we build a functional Γ, and a strategy T eis responsible for ∅ ′ (e) =Γ(W C ⊕ C; e). At stage s this strategy determines anapproximation σ = σ e,s to C ↾ e+1 , and defines Γ(τ e,s ⊕ σ e,s ; e) =∅ ′ s(e). In theend we verify that σ e = lim s σ e,s exists and show that C = ⋃ e σ e is as required.The framework for the construction is as follows: for each stage s, strategiesQ e ,T e for e ≤ s act at substages e =0,...,s. The strategy Q e defines P e,s andτ e,s . The strategy T e defines σ e,s . Both τ e,s and σ e,s have length e +1.The strategies choose their objects in such a way that P 0,s ⊇ ... ⊇ P s,s andσ 0,s ≺ ...≺ σ s,s . More precisely, Q e chooses P e,s inside P e−1,s ∩ [σ e−1,s ], and T edefines σ e,s as the leftmost string σ of length e + 1 such that [σ] ∩ P e,s ≠ ∅.The priority ordering is Q 0

6.3 C.e. operators and highness properties 255Stage s ≥ 0. At substage −1, let P −1,s = P s − U s−1 and let σ −1,s = τ −1,s = ∅.Carry out substage e for e =0, 1,...,s.Substage e.Q e : If G ≠ ∅, whereG = P e−1,s ∩ [σ e−1,s ] ∩ [{ρ: |ρ| = s & e ∉ W ρ s }] ≺ ,then let P e,s = G and τ e,s = τ e−1,s 0. Otherwise, let P e,s = P e−1,s ∩[σ e−1,s ]and τ e,s = τ e−1,s 1.T e : If e ≥ 2 and e ∈∅ ′ s −∅ ′ s−1 then put [σ e,s−1 ]intoU.Let σ e,s be the leftmost string σ ∈ {σ e−1,s 0, σ e−1,s 1} such thatP e,s ∩ [σ] ≠ ∅. Enumerate 〈τ e,s ⊕ σ e,s ,e,∅ ′ s(e)〉 into Γ.Verification. We have λU ≤ 1/4 as explained above, and thus ∀sP s − U s−1 ≠ ∅.6.3.11 Claim. Let e ≥−1. (i) τ e = lim s τ e,s exists. (ii) σ e = lim s σ e,s exists.The claim holds trivially for e = −1. Now suppose that e ≥ 0. Inductively, let s 0be the stage by which τ e−1,s and σ e−1,s have reached their limits. Let P e−1 bethe Π 0 1 class ⋂ s≥s 0P e−1,s . Then P e−1 ∩ [σ e−1 ] ≠ ∅ by the definition of σ e−1 atstage t.(i) If P e−1 ∩ [σ e−1 ] ∩ {Y : e ∉ W Y } ≠ ∅, then from s 1 := s 0 on we defineτ e,s = τ e−1 0. Otherwise, from some s 1 ≥ s 0 on we define τ e,s = τ e−1 1.(ii) Let P e = ⋂ s≥s 1P e,s where s 1 is as in (i) above. Then P e ∩ [σ e−1 ] ≠ ∅. Thusthere is a least a ∈ {0, 1} such that P e ∩ [σ e−1 a] ≠ ∅, and we eventually defineσ e,s to be σ e−1 a.✸Let C = ⋃ e σ e, then C ∈ P e for each e. ThusW C = ⋃ e τ e.6.3.12 Claim. Γ is a Turing functional such that ∅ ′ − {0, 1} = Γ(W C ⊕ C).We verify the conditions (F1) and (F2) in Fact 6.1.1.We only enumerate triples〈η, e, v〉 into Γ such that |η| =2e+2. Thus oracle incomparability (F2) is triviallysatisfied. For output uniqueness (F1), we give a formal argument for the automaticΓ-recovery mentioned above. Assume for a contradiction that at stagess

256 6 Some advanced computability theorypossibilities contradict τ e,s = τ e,t = and σ e,s = σ e,t . So Γ is a Turing functional.Now ∅ ′ − {0, 1} = Γ(W C ⊕ C) is immediate by the enumeration into Γ of thestrategies T e .✸The use of Γ(W C ⊕ C; e) is bounded by 2e +2. Thus ∅ ′ ≤ wtt W C ⊕ C. Theproof of Claim 6.3.11 allows us to compute recursively a bound on the numberof times τ e,s and σ e,s can change. Thus W C ⊕ C ≤ wtt ∅ ′ .✷An alternative proof of Theorem 6.3.9 due to Kučera is sketched in Simpson (2007).It uses ideas from the proof of Theorem 3.3.2 in a construction relative to ∅ ′ . Theconstruction in the present proof is not relative to ∅ ′ , and thereby provides directly acomputable approximation to W C ⊕ C. Further, it yields weak truth-table equivalenceW C ⊕ C ≡ wtt ∅ ′ . We do not know whether the conclusion in Theorem 6.3.1 can beimproved to weak truth-table equivalence. The proof of 6.3.1 suggests a negative answer.Separation of highness propertiesThe inversion theorems can be used to separate highness properties impliedby high 1 , both via c.e. sets and via ML-random ∆ 0 2 sets. We will consider twoproperties between high 1 and {C : ∅ ′ ≤ T C}. The property of being uniformlya.e. dominating was already introduced in Definition 5.6.26. In Theorem 5.6.30we showed that C is uniformly a.e. dominating iff C ≥ LR ∅ ′ . Mohrherr (1986)introduced a further property and separated it from highness within the c.e. sets.6.3.13 Definition. We say that C is superhigh if ∅ ′′ ≤ tt C ′ .It is equivalent to require ∅ ′′ ≤ wtt C ′ . For by Exercise 1.4.8 due to Mohrherr(1984), if E is a set such that ∅ ′ ≤ tt E, then X ≤ wtt E implies X ≤ tt E.A further equivalent formulation is ∀x [∅ ′′ (x) = lim s f(x, s)] for some functionf ≤ T C such that the number of changes of f(x, s) is bounded by a computablefunction g(x).In the framework of weak reducibilities, superhighness is the highness propertydual to superlowness (see Table 8.3 on page 363). The basic idea is to applypseudojump inversion to the c.e. operator W given by the construction of a c.e.incomputable set with a lowness property to obtain a set C< T ∅ ′ that satisfiesthe dual highness property. In Theorem 6.3.14 we extend this method in orderto separate highness properties.The implications between these highness properties are the following: for eachset C,∅ ′ ≤ T C ⇒ ∅ ′ ≤ LR C ⇒ C is superhigh ⇒ C is high. (6.7)Only the second implication is nontrivial; it will be proved in Theorem 8.4.17.In Proposition 8.5.11 below we show that high 1 is a null class.6.3.14 Theorem. The converse implications in (6.7) fail. Moreover, a counterexampleC can be chosen to be either c.e., or ML-random and ∆ 0 2.Proof. We apply the inversion results Theorem 6.3.1 and Corollary 6.3.10.Throughout, C denotes either a c.e. set, or a ML-random ∆ 0 2 set.

6.3 C.e. operators and highness properties 2571. To obtain a set C ≥ LR ∅ ′ such that C< T ∅ ′ , let W be the c.e. operator givenby the construction of an incomputable K-trivial set in Proposition 5.3.11. Bythe inversion results, there is a set C such that W C ⊕ C ≡ T ∅ ′ and W C ≰ T C.Hence C< T ∅ ′ . Moreover, ∅ ′ ∈ K(C), whence ∅ ′ ≤ LR C.2. To build a superhigh set C such that ∅ ′ ≰ LR C, we apply one of the inversionresults to the c.e. operator given by the construction in Proposition 5.1.20 of ac.e. superlow set that is not a base for ML-randomness. (The solution to Exercise5.2.10 could be used as well.) If we carry out this construction relative to Cwe obtain a binary function q ≤ T C such that e ∈ (W C ⊕ C) ′ ↔ lim s q(e, s) =1for each e.By the remark at the end of the proof of 5.1.20, the number of changes inthis approximation is bounded by a computable function (and not merely by afunction computable in C). Therefore (W C ⊕ C) ′ ≤ tt R ≤ m C ′ where R is thechange set for q as in the proof of Proposition 1.4.4. If C is obtained via one ofthe inversion results then ∅ ′ ≡ T W C ⊕ C, so that ∅ ′′ ≤ tt C ′ .The set ∅ ′ is not low for ML-randomness relative to C (see page 231). Since Cis ∆ 0 2 this implies ∅ ′ ≰ LR C.3. To obtain a set C that is high but not superhigh, let W be the c.e. operatorgiven by the construction of a low but not superlow set in the solution on page 385to Exercise 1.6.7. If C is obtained via one of the inversion results, then on theone hand ∅ ′ is low relative to C, namely (∅ ′ ⊕ C) ′ ≡ T C ′ . On the other hand, ∅ ′is not superlow relative to C, namely, (∅ ′ ⊕ C) ′ ≰ tt(C) C ′ ; here ≤ tt(C) denotesthe reducibility where C can be used as an oracle to compute the truth table.Since C is ∆ 0 2 we have ∅ ′′ ≡ m (∅ ′ ⊕ C) ′ , so this implies ∅ ′′ ≰ tt C ′ .✷In Remark 5.6.24 we compared the operators K(X) and S LR (X) ={A: A ≤ LRX}. Let us now compare the classes {C : ∅ ′ ∈ K(C)} and {C : ∅ ′ ≤ LR C}. Theformer is not closed upward under ≤ T . The two classes coincide on the ∆ 0 2 setsbut can be separated by a Σ 0 2 set. From (iii) below we obtain a further proofthat the operator K is not monotonic.6.3.15 Proposition.(i) If ∅ ′ ∈ K(C) then ∅ ′ ≤ LR C.(ii) If C is ∆ 0 2, then, conversely, ∅ ′ ≤ LR C implies ∅ ′ ∈ K(C).(iii) There is a c.e. set D and a Σ 0 2 set S, S ′ ≡ T ∅ ′′ , D ≤ T S, such that∅ ′ ∈ K(D) but ∅ ′ ∉ K(S).(iv) If S is as in (iii) then ∅ ′ ≤ LR S while ∅ ′ ∉ K(S).Proof. (i) ∅ ′ ∈ K(C) ↔∅ ′ ⊕ C ∈ K(C) ↔∅ ′ ⊕ C ≡ LR C → ∅ ′ ≤ LR C.(ii) If C is ∆ 0 2 then ∅ ′ ≤ LR C implies that ∅ ′ ⊕ C ≡ LR C.(iii) We combine the argument in Remark 5.6.24 with pseudojump inversion.Let W be the c.e. operator given by the construction of a promptly simple K-trivial set in Proposition 5.3.11. Let D be a c.e. set such that W D ⊕ D ≡ T ∅ ′ ,then ∅ ′ ∈ K(D). Because ∅ ′ is of promptly simple degree relative to D, by

258 6 Some advanced computability theoryTheorem 6.2.2 relativized to D there is a set S ≥ T D such that ∅ ′ ⊕ S ≡ TD ′ ≡ T S ′ . Since D ≥ LR ∅ ′ we have S ≥ LR ∅ ′ . But also ∅ ′ ∉ K(S), for otherwise∅ ′ ⊕ S ∈ K(S) and hence (∅ ′ ⊕ S) ′ ≤ T S ′ , contrary to the fact that ∅ ′ ⊕ S ≡ T S ′ .Since S is c.e. in D, we have that S is Σ 0 2 and S ′ ≡ T ∅ ′′ .(iv) See the proof of (iii).6.3.16 Exercise. Find sets A, B such that A ≤ LR B, A ≤ T B ′ , and A ∉ K(B).6.3.17. ⋄ Problem. Decide whether ∅ ′ ≤ LR C ⇔∃B ≤ T C [∅ ′ ∈ K(B)].Minimal pairs and highness properties ⋆Recall minimal pairs for Turing reducibility from Definition 1.7.13. Given a highnessproperty H, it is interesting to determine whether there is a minimal pair ofsets A, B satisfying H: as we remarked already at the beginning of Section 1.5,this would show that H is not too close to being Turing above ∅ ′ . In fact, for allthe highness properties we study there is such a minimal pair.Next, one can ask whether there is in H a minimal pair of sets of a particularkind, for instance of c.e. sets, or of ML-random sets. Lachlan (1966) proved thatthere is a minimal pair of c.e. high sets (see the comment after 6.3.4). Shore(unpublished) and Ng (2008b) independently improved this by showing thatthere is a minimal pair of c.e. superhigh sets.6.3.18. ⋄ Problem. Is there a minimal pair of c.e. sets that are u.a.e.d.?A minimal pair of high ML-random sets A, B can be obtained as follows: the2-random set A = Ω Ω is high by Proposition 3.4.17. Let B = Ω, then A ⊕ B isML-random by Theorem 3.4.6, and A, B form a minimal pair by Exercise 5.3.20.On the other hand, the superhigh sets are contained in a null Σ 0 3 class by 8.5.12below. So by Theorem 5.3.15 there is a promptly simple set Turing below all theML-random superhigh sets (see page 357 for details).By Remark 4.3.13 there is a minimal pair of sets of PA degree. However, wecannot find such a pair among the PA sets of c.e. or of ML-random degree,because they compute ∅ ′ by Theorems 4.1.11 and 4.3.8 respectively.A condition on a highness property H stronger than containing a minimalpair is that for each set F there is B ∈ H such that F and B form a minimalpair. Simpson observed that the class of u.a.e.d. sets satisfies this condition. ForCholak, Greenberg and Miller (2006) proved that for each incomputable set E,there is a u.a.e.d. set B ≱ T E. Their forcing construction can be extended slightlyso that B ≱ T E i for each i, where (E i ) i∈N is any sequence of incomputable sets.We may assume that F is incomputable. Let (E i ) i∈N be a list of the incomputablesets computed by F . Then B and F form a minimal pair.✷

7Randomness and betting strategiesSchnorr (1971) criticized that Martin-Löf randomness is too strong a notionto be considered algorithmic. He suggested a weaker notion, nowadays calledSchnorr randomness (Section 3.5). In this chapter we study randomness notionsin between ML-randomness and Schnorr randomness.The main notion is computable randomness. Recall that we identify subsets Zof N with infinite sequences of bits. For n ≥ 0, we refer to Z(n) as the bit of Z inposition n. Our test concept is (the mathematical counterpart of) a computablebetting strategy B that tries to gain capital along Z by predicting Z(n) afterhaving seen Z(0),...,Z(n − 1). The set Z is computably random if each suchbetting strategy B fails in the sense that the capital B(Z ↾ n ) is bounded fromabove. Our test concept can be viewed as a special case of the martingale notionfrom probability theory (see Shiryayev 1984).If it is allowed to leave B(x) undefined for strings x that are not prefixes of Z,we obtain a notion called partial computable randomness.If B may also bet on the bits in an order it chooses, we obtain the even strongernotion of Kolmogorov–Loveland (KL) randomness. Each ML-random set is KLrandom.It is unknown at present whether the two notions coincide. In this case,ML-randomness could in fact be characterized by a computable test concept.(Most researchers think at present that KL-randomness is weaker than MLrandomness.)The approach to randomness via unpredictability is useful because we have anintuitive understanding of betting strategies. On the other hand, this approachis still within the framework given in the introduction to Chapter 3. A bettingstrategy is a particular type of test, and the null class it describes is the class ofsets on which it succeeds.Schnorr did not even view computable randomness as the right answer to hiscritic, because a computable betting strategy B may succeed on a set very slowly,so slowly that B(Z ↾ n ) cannot be bounded from below by an order functionfor infinitely many n. If our tests are computable betting strategies togetherwith such an order function and we require this lower bound on the capital forinfinitely many n, we re-obtain Schnorr randomness defined in 3.5.8, as we willsee in Section 7.3. This gives further evidence that Schnorr randomness is theright answer.A questions we ask frequently in this book is how the strength of a randomnessnotion is reflected by the growth rate of the initial segment complexity. Forinstance, for a computably random set it can be as low as O(log n), but not for

260 7 Randomness and betting strategiesa partial computably random set. KL-random sets behave more like ML-randomsets. Their growth rate is closer to the growth rate for ML-random sets, wherea lower bound of λn.n − b for some b is given by Schnorr’s Theorem.We also show that the computational complexity is more varied for weakerrandomness notions. For instance, there is a computably random set in eachhigh degree.We have seen in the previous chapters that Martin-Löf-randomness interactswell with computability. Not only is Martin-Löf randomness defined as an algorithmicnotion, it also leads to objects such as Ω, classes such as Low(MLR), andmethods such as the elegant solution to Post’s problem outlined in Remark 4.2.4.There are fewer interactions of this kind for Schnorr randomness and the otherrandomness notions studied in this chapter. However, in Section 8.3 we will characterizethe corresponding lowness notions and see that they determine classesof considerable interest in computability theory. For instance, a set is low forSchnorr randomness if and only if it is computably traceable, a property strongerthan being computably dominated.7.1 MartingalesWe introduce martingales as a mathematical counterpart of the intuitive conceptof a betting strategy. At first we do not make any assumptions on the effectivity.It takes some effort to develop the basics, but we will be rewarded withsimple proofs of facts that have been used already, such as Proposition 5.1.14.The advantage of martingales over Martin-Löf tests and their variants is thatmartingales have useful built-in algebraic properties. For instance, we can definea new martingale as the sum of given ones.Formalizing the concept of a betting strategyImagine a gambler in a casino is presented with bits of a set Z in ascendingorder. So far she has seen x ≺ Z, and her current capital is B(x) ≥ 0. She betsan amount α, 0≤ α ≤ B(x), on her prediction that the next bit will be 0, say.Then the bit is revealed. If she was right, she wins α, else she loses α. Thus,B(x0) = B(x)+α and B(x1) = B(x) − α,and hence B(x0) + B(x1) = B(x)+α + B(x) − α =2B(x). The same considerationsapply if she bets that the next value will be 1: the betting is fair in thatthe expected capital after the next bet is equal to the current capital.7.1.1 Definition. A martingale is a function B : {0, 1} ∗ → R + ∪ {0} that satisfiesfor every x ∈ {0, 1} ∗ the equalityFor a martingale B and a set Z, letB(x0) + B(x1) = 2B(x). (7.1)B(Z) = sup n B(Z ↾ n ).We say that the martingale B succeeds on Z if the capital it reaches along Z isunbounded, that is, B(Z) =∞. Let Succ(B) ={Z : B succeeds on Z}.

7.1 Martingales 261Martingales are equivalent to measures on 2 N via the measure representationsof Definition 1.9.2. We prefer martingales here because they formalize bettingstrategies, for which we have an intuitive understanding.7.1.2 Fact. Let B : {0, 1} ∗ → R + ∪ {0} and let r = λx.2 −|x| B(x). ThenB is a martingale ⇔ r is a measure representation.Proof. Given x ∈ {0, 1} ∗ , let n = |x|. Then(7.1) ⇔ 2 −(n+1) B(x0) + 2 −(n+1) B(x1) = 2 −n B(x)⇔ r(x0) + r(x1) = r(x).✷7.1.3 Example. Recall that r(x) =2 −|x| is the representation of the uniformmeasure λ on 2 N . It corresponds to the martingale B with constant value 1. Notethat B formalizes the strategy where the player bets the amount α = 0 on anyprediction.More generally, for a measurable class C ⊆ 2 N , one defines the measure λ C byλ C (A) =λ(C ∩ A) for measurable A. Its representation is r(x) =λ(C ∩[x]). ThenB C (x) =2 |x| r(x) =λ(C |x) (7.2)is a martingale by Fact 7.1.2. B C (x) is a conditional probability, namely thechance to get into C when starting from x. In particular, its initial capital is λC.If [x] ⊆ C then B C (x) =1.IfC is closed the converse implication holds as well,because the only conull closed class is 2 N .7.1.4 Example. Given a string x, the elementary martingale E x starts withcapital 1. It bets its whole capital on the next bit of x till x is exhausted. Thus,E x is the martingale given by⎧⎪⎨ 2 |y| if y ≼ xE x (y) = 2⎪⎩|x| if x ≺ y0 if x | y.E x coincides with 2 |x| B C for C =[x].Note that B C is bounded by 1 and hence does not succeed on any set. It is usedto build up more complex martingales via infinite sums, which may then succeed onsome set Z. IfλC > 0 then B C(x) gets arbitrarily close to 1 for appropriate x byTheorem 1.9.4.SupermartingalesWe often work with a notion somewhat broader than martingales.7.1.5 Definition. A supermartingale is a function S : {0, 1} ∗ → R + ∪ {0} thatsatisfies for every x ∈ {0, 1} ∗ the inequalityS(x0) + S(x1) ≤ 2S(x). (7.3)

262 7 Randomness and betting strategiesThus, the average of S(x0) and S(x1) is at most S(x). This implies thatS(x0) ≤ S(x) orS(x1) ≤ S(x). (7.4)Extending Definition 7.1.1, for a set Z we let S(Z) = sup n S(Z ↾ n ). We say Ssucceeds on Z if S(Z) =∞, and Succ(S) denotes the class of sets Z on which Ssucceeds.A supermartingale can still be viewed as the mathematical counterpart of abetting strategy. If the supermartingale inequality (7.3) is proper, i.e.,d(x) =S(x) − 1/2(S(x0) + S(x1)) > 0, (7.5)the gambler first donates the amount d(x) to charity. Thereafter she bets withthe rest S(x)−d(x). Of course, she could as well keep d(x) and never bet with it.Thus, each supermartingale is below a martingale with the same start capital:7.1.6 Proposition. For each supermartingale S there is a martingale B suchthat B(∅) =S(∅) and B(x) ≥ S(x) for each x.Proof. Let B(x) =S(x)+ ∑ z≺xd(z), where the function d is defined by (7.5).Clearly ∀x B(x) ≥ S(x) and B(∅) =S(∅). We verify the martingale equalityfor B by induction on |x|:1/2(B(x0) + B(x1)) = 1/2(S(x0) + S(x1)) + ∑ z≼xd(z)=1/2(S(x0) + S(x1)) + d(x)+ ∑ z≺xd(z)= S(x)+ ∑ z≺xd(z)= B(x).Some basics on supermartingalesIn the following fact, parts (i) and (ii) are used to assemble complex (super)martingalesfrom simple ones, and (iii) is a localization principle which allows us totransfer results on supermartingales we have proved in {0, 1} ∗ to a set of stringsof the form {x: x ≽ v}.7.1.7 Fact.(i) If α ∈ R + and B and C are (super)martingales then so are αB and B +C.(ii) If N i is a (super)martingale for each i ∈ N and ∑ i N i(∅) < ∞ thenN = ∑ i N i is a (super)martingale.(iii) If B is a (super)martingale and v ∈ {0, 1} ∗ then λx.B(vx) is a (super)martingale.Proof. (i) and (iii) are easily verified. For (ii), it suffices to show that N(x) < ∞for each x. We use induction on |x|.Forx = ∅ this is a hypothesis. Now N i (xa) ≤2N i (x) for each i and each a ∈ {0, 1}. SoN(x) < ∞ implies N(xa) < ∞. ✷

7.1 Martingales 263Measure representations can be used via Fact 7.1.2 to generalize the supermartingaleinequality (7.3).7.1.8 Lemma. (i) Let S be a supermartingale and let x 0 ,x 1 ... be a finite orinfinite antichain of strings. Then ∑ i 2−|xi| S(x i ) ≤ S(∅).(ii) If S is a martingale and ⋃ i [x i]=2 N then equality holds.Proof. (i) By 7.1.6 we may assume that S is a martingale. Let r be the correspondingmeasure representation: r(x) =2 −|x| S(x). Then ∑ i r(x i) ≤ r(∅) sincethe clopen sets [x i ] are pairwise disjoint. This implies the required inequality.(ii) is clear by the definition of measure representations.✷The lemma can be generalized using the localization principle 7.1.7(iii): for eachstring v, ifx 0 ,x 1 ... is a finite or infinite antichain of strings extending v then∑i 2−|xi|−|v| S(x i ) ≤ S(v).A frequently used fact is that, if b>S(∅), then S(∅)/b bounds the measureof the class of sets along which S reaches at least b. For instance, if S(∅) =1then S reaches the capital 4 along at most 1/4 of the sets.7.1.9 Proposition. Let S be a supermartingale, b ∈ R, and S(∅)

264 7 Randomness and betting strategiesRecall that Q 2 is the set of rationals of the form z2 −n where z ∈ Z and n ∈ N.The undergraph of a supermartingale S is the set of pairs{〈x, q〉 : q ∈ Q 2 & q 2 i ]}. Further, G i is open and λG i ≤ 2 −ifor each i.Proof. (i) Note that ∑ i B G i(∅) ≤ 2, so B is a martingale by Fact 7.1.7. We haveB Gi (σ) = 1 whenever [σ] ⊆ G i .ThusifZ ∈ ⋂ i G i then B(Z ↾ n ) is unbounded.(ii) Clearly G i is open; λG i ≤ 2 −i follows from Proposition 7.1.9.✷7.2 C.e. supermartingales and ML-randomnessIn the following we impose effectivity conditions on supermartingales. In thissection we consider computably enumerable supermartingales. They yield a furthercharacterization of ML-randomness besides Theorem 3.2.9, which helps usto understand the growth rate of the function λn.K(Z ↾ n ) − n for a ML-randomset Z. In the next section we will proceed to computable supermartingales.

7.2 C.e. supermartingales and ML-randomness 265Computably enumerable supermartingales7.2.1 Definition. A supermartingale L is called computably enumerable if L(x)is a left-c.e. real number uniformly in x. Equivalently, the undergraph U ={〈x, q〉: q ∈ Q 2 & q < L(x)} is computably enumerable. A c.e. index for Userves as an index for L. We write L s (x) >qif 〈x, q〉 ∈U s .We provide a version of Fact 7.1.7 for c.e. supermartingales.7.2.2 Fact. (i) If α > 0 is left-c.e. and B and C are c.e. (super)martingales,then so are αB and B + C.(ii) If (N i ) i∈N is a uniformly c.e. sequence of (super)martingales and∑i N i(∅) < ∞, then N = ∑ i N i is a c.e. (super)martingale.Proof. (i) is easily verified. For (ii), N is a (super)martingale by 7.1.7. To seethat N is c.e., note that for each q ∈ Q 2 we have N(x) >q ↔∃n∃s ∃q 0 ...q n−1 ∈ Q 2[ ∑iq& ∀i q i]. ✷7.2.3 Definition. A supermartingale approximation is a uniformly computablesequence (L s ) s∈N of Q 2 -valued supermartingales such that L s+1 (x) ≥ L s (x) foreach x, s. We say that (L s ) s∈N is an approximation of L if L(x) = sup s L s (x) foreach x. In this case, L is a c.e. supermartingale.For instance, if R is c.e. open and R = ⋃ s [R s] ≺ as in (1.16) then the martingaleL = B R from Example 7.1.3 has a supermartingale approximation given byL s = B ≺.ThusB [Rs] R is computably enumerable.7.2.4 Fact. Each c.e. supermartingale L has a supermartingale approximation(L s ) s∈N . Moreover, it can be obtained uniformly.Proof. Let U ⊆ {0, 1} ∗ ×Q 2 be the undergraph of L. At stage s let the variablesq, q ′ range over numbers of the form i2 −s . Let ˜L s (y) be the largest q such that〈y, q〉 ∈U s . Define L s (x) by induction on |x|. IfL s (x) has been defined for all xsuch that |x|

266 7 Randomness and betting strategiesNote that (B Gi ) i∈N is a uniformly c.e. sequence of martingales. For each i wehave B Gi (∅) =λG i ≤ 2 −i . Hence B is c.e. by Fact 7.2.2. If Z ∈ ⋂ i G i then Bsucceeds on Z.(i) ⇒ (iii): Given is a c.e. supermartingale S, we may assume that S(∅) ≤ 1.Then (G i ) i∈N is a ML-test where G i = {Z : ∃x ≺ Z [S(x) > 2 i ]} as in 7.1.15(ii).If S succeeds on Z then Z fails this test.✷We are now in the position to prove Proposition 5.1.14 that if A is ML-random,then for each Turing functional Φ there is c ∈ N such that (S Φ A,n+c ) n∈N is aMartin-Löf test relative to A. LetL(x) =2 |x| λ[{σ : Φ σ ≽ x}] ≺ .Then L(x0) + L(x1) ≤ 2L(x) for each x. Further, {〈q, x〉: q ∈ Q 2 & q

7.2 C.e. supermartingales and ML-randomness 267value r(Z) ∈ [0, ∞] to a set Z in such a way that r(Z) indicates the degree ofnonrandomness of Z. In each case Z ML-random if and only if r(Z) < ∞, andthe larger r(Z), the less random is Z. Any universal c.e. supermartingale is afunction of that kind. Actually, we have studied such functions earlier. Schnorr’sTheorem 3.2.9 states that Z is ML-random iff Z ∉ R b for some b, and ourintuition is that the larger b must be chosen, the less random is Z. To obtain afunction r as above, let CPS(Z) = sup n 2 n−K(Z↾n) . ThenCPS(Z) < 2 b ↔ ∀nK(Z ↾ n ) >n− b ↔ Z ∉ R b .In particular, Z is ML-random ↔ CPS(Z) < ∞.The function CPS is somewhat crude for gauging nonrandomness. The followingfunction due to Miller and Yu (2008) is finer. One takes the sum of all the2 n−K(Z↾n) rather than their supremum. Thus, letJM(Z) = ∑ n 2n−K(Z↾n) .Let us compare JM with the universal c.e. martingale FS defined in (7.6). Fix Zand let n ∈ N. If y = Z ↾ n then 2 n−K(Z↾n) =2 −K(y) E y (y). Since E y (z) =2 |y|for any z ≽ y, it follows that for each k,∑n≤k 2n−K(Z↾n) = ∑ y≼Z↾ k2 −K(y) E y (Z ↾ k ) ≤ FS(Z ↾ k ).Hence JM(Z) ≤ FS(Z) for each Z.We summarize the preceding discussion. The implication (iv)⇒(i) shows thatthe c.e. martingale FS is universal.7.2.8 Theorem. The following are equivalent for a set Z.(i) Z is ML-random.(ii) CPS(Z) < ∞.(iii) JM(Z) < ∞.(iv) FS(Z) < ∞.Proof. The implications (iv)⇒ (iii) ⇒ (ii) hold since CPS(Z) ≤ JM(Z) ≤ FS(Z).The proof of (ii)⇒(i) coincides with the proof of (ii)⇒(i) in Theorem 3.2.9. Theimplication (i)⇒(iv) follows from 7.2.6 since FS is a c.e. martingale. ✷Of particular interest is the implication (i)⇒(iii), due to Miller and Yu (2008),that JM(Z) < ∞ for each ML-random set Z. They call this result the “AmpleExcess Lemma” because it says that K(Z ↾ n ) exceeds n considerably for large n.In the following let f,g: N → Z. To provide more detail, a function f suchthat ∑ n 2−f(n) < ∞ cannot grow too slowly. (For monotonic functions, theborderline is somewhere between λn. log n and λn.2 log n, as the sum divergesfor the former, and converges for the latter.) For ML-random Z, the functionf Z (n) =K(Z ↾ n ) − n has a positive value for almost all n, and does not grow tooslowly. Theorem 3.2.9 merely states that there is b such that f Z (n) ≥−b for all n.Proposition 3.2.21, that lim n f Z (n) =∞, is already a stronger condition on thegrowth rate of f Z . The condition JM(Z) < ∞ is even stronger and clearly impliesProposition 3.2.21. (If there is d such that f Z (n) =d for infinitely many n, thenJM(Z) =∞ since each such d contributes 2 −d to the sum JM(Z).)

268 7 Randomness and betting strategiesIn this context, Miller and Yu (20xx) have proved that if ∑ n 2−g(n) < ∞then there is a ML-random set Z such that ∀nf Z (n) ≤ + g(n). Thus, for a MLrandomset Z in general, we cannot prove a stronger growth condition on f Zthan JM(Z) < ∞.7.2.9 Exercise. (Miller and Yu, 20xx) (i) Show that if ∑ n 2−f(n) < ∞ then there isa function g ∈ ∆ 0 2(f) such that lim n(f(n) − g(n)) = ∞ and ∑ n 2−g(n) < ∞.(ii) Use the result of Miller and Yu (20xx) mentioned above and (i) to show that foreach ML-random set Y there is a ML-random set Z< K Y (see 5.6.1 for ≤ K).7.3 Computable supermartingalesWe will study a randomness notion where the tests are the computable bettingstrategies. This notion lies properly in between Martin-Löf randomness andSchnorr randomness defined in 3.5.8.7.3.1 Definition. (i) A supermartingale L is called computable if L(x) isa computable real number uniformly in x. Equivalently, the undergraph{〈x, q〉 : q ∈ Q 2 & q|x|+k B G i(x) for each k. Given a string x and k ∈ N, for eachi ≤ |x| + k we may compute q i,x ∈ Q 2 such that 0 ≤ B Gi (x) − q i,x < 2 −k−i−1 .Then 0 ≤ B(x) − ∑ i≤|x|+k q i,x < 2 −k+1 .✷

7.3 Computable supermartingales 269Below we will apply the fact that, if α i is a computable real number uniformlyin i ∈ N and 0 ≤ α i ≤ 2 −i for each i, then ∑ i α i is computable (Exercise 1.8.18).7.3.3 Theorem. The following are equivalent for a set Z.(i) Z ∈ ⋂ i G i for a Schnorr test (G i ) i∈N .(ii) There is a computable martingale B and an order function h such that∃ ∞ nB(Z ↾ n ) ≥ h(n).(iii) There is a computable martingale D and a strictly increasing computablefunction g such that ∃ ∞ kD(Z ↾ g(k) ) ≥ k.Proof. (iii)⇒(ii): If D and g are as in (iii), let h(n) = max{l : g(l) ≤ n}. Thenh(g(k)) = k for each k, so (ii) holds via D and h.(ii)⇒(i): We may assume that B(∅) ≤ 1. For each i ∈ N let S i be the computableset of minimal strings x (under the prefix relation) such that B(x) ≥ h(|x|) ≥ 2 i ,and let G i =[S i ] ≺ . Then Z ∈ ⋂ i G i, and λG i ≤ 2 −i by Proposition 7.1.9. Itremains to show that λG i is computable uniformly in i. Given r, we will computea rational q r ≤ λG i such that λG i − q r ≤ 2 −r . Compute the least p such thath(p) ≥ r. Then B(x) ≥ r for each x ∈ S i of length at least p, soλ[{x ∈ S i : |x| ≥p}] ≺ ≤ 2 −r by Proposition 7.1.9. Thus q r = λ[{x ∈ S i : |x|

270 7 Randomness and betting strategiesWe check that the martingale property holds for D k . First suppose that w ∈ U k .Since E k (wa) =E k (w)+2 |w| p w (a ∈ {0, 1}) and F k (w0) + F k (w1) = 2(F k (w) −2 |w| p w ), we have(E k (w0) + E k (w1))/2 =E k (w)+2 |w| p w ,(F k (w0) + F k (w1))/2 =F k (w) − 2 |w| p w ,so the martingale property holds. The case that w ∉ U k is simpler since we donot need the term 2 |w| p w in either equation.Since ∑ r≥k α r ≤ 2 −k+1 ,wehaveF k (w) ≤ 2 −k+1 2 |w| . Also E k (w) =0fork ≥ |w|, soD = ∑ k D k is a computable martingale.To show (iii), let g(k) = min{n: f(n) >k}. Given k 0 ∈ N, let m be least suchthat k = f(m) ≥ k 0 and Z ↾ m ∈ R. Then Z ↾ m ∈ U k .Ifx ∈ U k then D k (z) ≥ 2 kfor each z ≽ x. Asg(k) >m, we have in fact D k (Z ↾ g(k) ) ≥ D k (Z ↾ m ) ≥ 2 k .✸Note that the set R = ⋃ i S i is computable since x ∈ S i implies |x| ≥ i.Further, the real number ∑ x∈R 2−|x| is computable because λG i ≤ 2 −i and λG iis computable uniformly in i. IfZ ∈ ⋂ i G i then for each i there is m such thatZ ↾ m ∈ S i , and m ≥ i since λG i ≤ 2 −i .Thus∃ ∞ mZ↾ m ∈ R.✷Exercises.7.3.5. For a prefix-free machine M let FS M be the martingale ∑ y 2−K M (y) E y (seepage 266). Show that if M is a computable measure machine then FS M is computable.Use this to give yet another proof of Proposition 7.3.2.7.3.6. Use Lemma 7.3.4 to give another proof of Theorem 3.5.21 that each Schnorrrandom set satisfies the law of large numbers.7.3.7. ⋄ Show that Z is Schnorr random ⇔ lim inf n K M (Z ↾ n) − n = ∞ for each computablemeasure machine M. Hint. Adapt the proof of Proposition 3.2.21.Preliminaries on computable martingalesIf V is a computable martingale then by Fact 1.8.15(iv) one can approximate eachvalue V (x) ∈ R by a rational up to an arbitrary precision. Nonetheless, it wouldbe desirable to have complete information about these values. For this reason wewill mostly work with computable (super)martingales B taking values in Q 2 .Bythe following fact of Schnorr (1971), this restriction does not affect the notion ofcomputable randomness: if V is a computable martingale and V (Z) =∞ thenB(Z) =∞ for a computable Q 2 -valued martingale B.7.3.8 Proposition. For each computable martingale V there is a Q 2 -valuedcomputable martingale B such that V (x) ≤ B(x) ≤ V (x)+2 for each x.The constant 2 is chosen for notational simplicity, and could be replaced by anyrational ɛ > 0.Proof. Suppose that at a string y the martingale V bets r(y) =V (y0) − V (y)on the prediction 0. We allow a negative value r(y) here, which means that V

7.4 How to build a computably random set 271actually bets −r(y) on 1. Since V (x) is a computable real number uniformly in x,there is a computable function f : {0, 1} ∗ → Q 2 such that abs(r(y) − f(y)) ≤2 −|y|−2 for each y.We assign B a start capital of B(∅) ∈ [V (∅) +1/2,V(∅) + 1]. This implies0 ≤ V (∅)+1− B(∅) ≤ 1/2. In order to define B on strings other than ∅, wereplace r by f. Recursively we defineB(ya) =B(y)+(−1) a f(y)for each string y and each a ∈ {0, 1}. Then the martingale equality holds. Toshow that V (x) ≤ B(x) ≤ V (x) + 2 for each x, note that, where y ranges over{0, 1} ∗ , and a over {0, 1}, wehaveV (x) =V (∅)+ ∑ y,a (−1)a r(y) [[ya ≼ x] andsimilarly B(x) =B(∅)+ ∑ y,a (−1)a f(y) [[ya ≼ x], so thatabs(V (x)+1− B(x)) = abs(V (∅)+1− B(∅)Thus V (x) ≤ B(x) ≤ V (x) + 2 for each x.+ ∑ y,a(−1) a (r(y) − f(y)) [ya ≼ x]])≤ 1/2+ ∑2 −j−2 ≤ 1.j|z|, one may compute relative to S the leftmost w of length u suchthat S does not increase along w, namely,∀n [ |z| ≤ n

272 7 Randomness and betting strategies(I) We build a computably random Z that has a slowly growing initial segmentcomplexity, say K(Z ↾ n ) ≤ + 3 log n for each n.(II) Turing below each high set C we build a computably random set Z.The set Z in approach (I) is not ML-random because each ML-random set Zsatisfies ∀nK(Z ↾ n ) ≥ + n by Theorem 3.2.9. In approach (II), if the high setC is c.e. and C < T ∅ ′ , then Z is not ML-random because it is not even ofd.n.c. degree: otherwise C is of d.n.c. degree and hence Turing complete by thecompleteness criterion 4.1.11.By Theorem 3.5.13, a computably random set that is not high is already MLrandom.Theorem 7.5.9 is a result stronger than (II) that complements 3.5.13:each high degree contains a computably random set that is not ML-random.Besides computable randomness we will consider partial computable randomness,which lies strictly in between ML-randomness and computable randomness.The tests are now Q 2 -valued partial computable martingales B. The fact thatB(x) may be undefined for some strings x gives the betting strategy B a considerableadvantage. It can wait as long as it wishes before it decides on its bet at aposition. As before, B succeeds on Z if sup n B(Z ↾ n )=∞; in particular, B mustbe defined for all the initial segments of Z. In Section 7.5, as a preparation toTheorem 7.5.9, we give a direct construction of a computably random but notpartial computably random set.One can also separate all three randomness notions mentioned above by consideringthe initial segment complexity. This extends approach (I). It turns outthat a partial computably random set Z cannot satisfy ∀n K(Z ↾ n ) ≤ + c log nfor a constant c (Theorem 7.6.7), while a computably random set can do so. Aslightly larger bound, such as λn. log 2 n, is consistent with being partial computablyrandom, but still not with being ML-random.We summarize the implications between the randomness notions:ML-random ⇒ partial computably random⇒ computably random⇒ Schnorr random.The converse implications fail. A set that is Schnorr random but not computablyrandom is obtained in Theorem 7.5.10, a variant of Theorem 7.5.9.Three preliminary theorems: outlineThe results in this and the next section will be achieved in small steps. In each ofthree preliminary theorems, we prove that there is a computably random set Zwith certain additional properties. Each theorem introduces a new main idea.Eventually all the ideas are combined to prove Theorem 7.5.9 that each highdegree contains a computably random set that is not ML-random.The proofs share a template. In the simplest case, a supermartingale L isintroduced such that for each total B k , for an appropriate c>0wehave∀x [c ·L(x) ≥ B k (x)]. Then a set Z is built on which L does not succeed. Thus Z iscomputably random.

7.4 How to build a computably random set 273We describe the three preliminary theorems.• In Theorem 7.4.8 we carry out approach (I) on page 271. We introduce thetemplate and the idea of copying partial computable martingales.• In Theorem 7.5.2 we carry out approach (II). The idea is to use the highnessof C to build a supermartingale L ≤ T C that dominates each B k up to amultiplicative constant.• In Theorem 7.5.7 we build a computably random set that is not partialcomputably random. We introduce the idea of encoding the definition of Linto Z.The template will be used to separate all the randomness notions that can becharacterized via martingales, that is, all the notions between Schnorr randomnessand ML-randomness.Partial computable martingalesFirst we provide the formal definition of a concept already mentioned above.7.4.1 Definition. A partial computable martingale is a partial computable functionB : {0, 1} ∗ → {x ∈ Q 2 : x ≥ 0} such that dom(B) is closed under prefixes,and for each x ∈ dom(B), B(x0) is defined iff B(x1) is defined, in which casethe martingale equality B(x0) + B(x1) = 2B(x) holds.By this definition, partial computable martingales are always Q 2 -valued, whilewe allow computable supermartingales with values in R. This fine point rarelymatters because of Proposition 7.3.8.For a function f, we write f(x) ↓ to denote that f(x) is defined, and f(x) ↑otherwise. Definition 7.1.1 can be adapted to partial computable martingales.Thus, we let B(Z) = sup{B(Z ↾ n ): B(Z ↾ n )↓}, and B succeeds on Z if B(Z) =∞. This is only possible if B is defined along Z, that is, B(Z ↾ n )↓ for each n.7.4.2 Definition. Z is partial computably random if no partial computable martingalesucceeds on Z. The class of partial computably random sets is denotedby PCR.Each partial computable martingale B yields a c.e. supermartingale ̂B in thesense of Definition 7.2.1: let ̂B(x) =B(x) if the latter is defined, and otherwiseB(x) = 0. Note that B and ̂B succeed on the same sets. Hence each ML-randomset Z is partial computably random by Proposition 7.2.6.For the template and its applications we will need some more notation. Viathe effective identifications of {0, 1} ∗ and {q ∈ Q 2 : q ≥ 0} with N we may vieweach Φ k as a partial function {0, 1} ∗ → {q ∈ Q 2 : q ≥ 0}. Let B k be the partialcomputable function that copies Φ k as long as it looks like a martingale:B k (∅) ≃ Φ k (∅), andB k (x) =Φ k (x) ifx = ya for a ∈ {0, 1}, B k (y) has been defined already,and Φ k (y0) + Φ k (y1) = 2Φ k (y) (all defined).

274 7 Randomness and betting strategiesThen (B k ) k∈N is an effective listing of all the partial computable martingales.LetTMG = {k : B k is total}.We frequently need a partial computable martingale B k,n , n ∈ N, which copies B k ,but only for strings x such that |x| ≥ n, and with some scaling. For shorter stringsit returns the value 1.7.4.3 Fact. There is an effective listing (B k,n ) k,n∈N of partial computable martingalessuch that(i) B k,n (x) =1for |x| 0 such that, for each x, ifB k (x) ↓then B k,n (x)↓ and B k (x) ≤ c · B k,n (x).Proof. For |x| ≥ n, ifB k (x ↾ n )=0letB k,n (x) = 0. Otherwise let B k,n (x) ≃B k (x)/B k (x↾ n ). Then (ii) holds via c = max{B k (y): |y| = n}.✷7.4.4 Definition. For each n let G n be the supermartingale such thatG n (x) =1for|x|

7.4 How to build a computably random set 2751. Define the sequence 0 = n 0

276 7 Randomness and betting strategies7.4.9 Fact. If |x| ∈ I k we have V j (x) =1for all j ≥ k, so the total contributionof all such V j to L(x) is 2 −k+1 . In particular, L is Q 2 -valued.✸Step 3. Let Z be the leftmost non-ascending path of L, i.e., Z is leftmost suchthat ∀n L(Z ↾ n+1 ) ≤ L(Z ↾ n ). Then L(x) ≤ 2 for all x ≺ Z, that is, L(Z) ≤ 2. ByFact 7.4.3(ii), L multiplicatively dominates each total B k .SoZ is computablyrandom.It remains to show that K(Z ↾ n | n) ≤ h(n) for almost all n. To do so, whenn ∈ I k we compute Z ↾ n from n and TMG↾ k (recall that TMG = {e: B e total}).7.4.10 Compression Lemma. There is a partial computable function F suchthat ∀k∀n ∈ I k[Z ↾n = F (n, TMG↾ k ) ] .Subproof. We describe a procedure for F on inputs n and τ which in the relevantcase that τ = TMG↾ k simply provides more formal details in the definitionof Z. For other strings τ it may attempt to compute undefined values of partialmartingales, and thus may fail to terminate.(1) Compute k such that n ∈ I k .(2) Let z = ∅. For p =0ton − 1 do:(a) Let r be such that p ∈ I r . WriteS r (x) ≃ 2 −r+1 + ∑ i 2−i Bi ∗ (x)[i

7.4 How to build a computably random set 277Proof. Let Ord = {i: Φ i is an order function}. We modify the sequencen 0

278 7 Randomness and betting strategies7.4.15 Theorem. There is a partial computably random set Z such that∀ ∞ nK(Z ↾ n | n) ≤ h(n) log n for each order function h.Note that K(x | n) ≤ + K(x) ≤ + K(x | n) +K(n) ≤ + K(x | n) +2lognby Section 2.3, so we could as well state that ∀ ∞ nK(Z ↾ n ) ≤ h(n) log n foreach order function h. We merely keep conditional complexity to be notationallyconsistent with Theorem 7.4.11.Proof. We define the sequence (n k ) k∈N by (7.10). Let{Bk ∗ V k (x) =(x) if B∗ k (x)↓0 otherwise.As before, let L = ∑ k 2−k V k , and let Z be the leftmost non-ascending pathof L. Then Z is partial computably random: suppose Bk ∗(Z ↾ n)↓ for each n, thenV k (Z ↾ n )=Bk ∗(Z ↾ n) is bounded along Z.In the proof of the foregoing theorem, to determine Z(n) from z = Z ↾ n forn ∈ I k , we merely required a yes/no information for each i

7.5 Each high degree contains a computably random set 2797.4.17 Remark. A slight modification of the proof yields a left-c.e. set Z in Theorem7.4.15. The V k are uniformly c.e. supermartingales, hence L is computably enumerable.Recall from Remark 7.1.13 that T2L = {x: ∀y ≼ x [L(y) ≤ 2]}. Since L is c.e.and L(∅) ≤ 2, Paths(T2 L ) is a nonempty Π 0 1 class. The leftmost path Z of T2L is left-c.e.by Fact 1.8.36, and Z is partial computably random. To show the bound on the initialsegment complexity, in the procedure for F we replace step (2.b) by(2.b ′ ) Let z := z0 ifS r(z0) ≤ 2, and z := z1 otherwise.7.5 Each high degree contains a computably random setHaving studied the initial segment complexity of computably random sets, wenow turn to their computational complexity. We carry out the approach (II) onpage 271. Given a high set C, our first goal is to build a computably random setZ ≤ T C. Later on, we will achieve that Z ≡ T C, and we will also ensure that Zis not partial computably random. Each Schnorr random set of non-high degreeis ML-random by 3.5.13, so this characterizes the Turing degrees of sets that arecomputably random but not partial computably random (or ML-random).Martingales that dominateTheorem 1.5.19 states that C is high iff there is a function f ≤ T C dominatingevery computable function. From such an f we will obtain a Q 2 -valued supermartingaleL ≤ T C, L(∅) ≤ 2, that multiplicatively dominates each computablemartingale. If Z is the leftmost path on T2L = {x: ∀y ≼ x [L(y) ≤ 2]} then Z iscomputably random and Z ≤ T C.7.5.1 Domination Lemma. If C is high then there is a Q 2 -valued supermartingaleL ≤ T C that multiplicatively dominates each computable martingale.Proof. Only steps 2 and 3 of the template on page 274 are needed here (thinkof n e as being e). The partial computable martingales B k,n were defined in 7.4.3.Suppose f ≤ T C dominates each computable function. For each e, letV e (x) =g e (n) ≃ µt [∀x (|x| ≤ n → B e,e+1 (x)[t]↓)], and let{B e,e+1 (x) if |x| ≤ e or ∀m [ e ≤ m ≤ |x|) → g e (m) ≤ f(m)+e ]0 else.As usual let L(x) = ∑ e 2−e V e (x). Clearly L is a supermartingale. Further, L isQ 2 -valued by the standard argument: for e>|x| we have V e (x) = 1, so the tailof the sum due to the e>|x| equals 2 −|x| . Note that L ≤ T C since V e ≤ T Cuniformly in e.If B e is total then there is m such that g e (n) ≤ f(n) for all n ≥ m. BythePadding Lemma 1.1.3 we may choose e ′ such that the Turing program P e ′ doesexactly the same as P e and e ′ > max{g e (n): n

280 7 Randomness and betting strategiesThe following preliminary result is now immediate.7.5.2 Theorem. For each high set C there is a computably random set Z ≤ T C.Proof. Let L be as in Lemma 7.5.1. Note that L(∅) ≤ 2. As in Remark 7.1.13,let Z be the leftmost path on the tree T2L = {x: ∀y ≼ x [L(y) ≤ 2]}. ThenZ ≤ T L ≤ T C and Z is computably random.✷Each high c.e. degree contains a computably random left-c.e. setEventually we want to improve the conclusion in Theorem 7.5.2 to Z ≡ T C. Thisis easier when C is c.e., and in this case we even obtain a left-c.e. set Z. Theproof deviates from the template.7.5.3 Theorem. For each high c.e. set C, there is a computably random left-c.e.set Z ≡ T C.Proof. We need a version of the Domination Lemma 7.5.1 for the c.e. setting.7.5.4 Lemma. Suppose the set C is high and also c.e. Then there is a supermartingaleL as in 7.5.1 such that, in addition, L is computably enumerable.Subproof. We refine the proof Lemma 7.5.1 in order to obtain a supermartingaleapproximation (L s ) s∈N for L (see Definition 7.2.3). Suppose ∆ is a Turingfunctional such that ˜f = ∆ C dominates every computable function. We will replace˜f by an even larger function computable in C which can be approximatedin a non-decreasing way: letf s (x) = max t≤s ∆ C (x)[t], and f(x) = lim s f s (x).Note that f(x) < ∞ since ∆ C is total. To compute f(x) relative to C, let s beleast such that ∆ C (x)[s]↓ with C s correct up to u the use of this computation andoutput f s (x). Then f t (x) =f(x) for each stage t>s(here we have used that Cis computably enumerable). Now let L ≤ T C be as in the proof of Lemma 7.5.1.To obtain the supermartingale approximation (L s ) s∈N , let{B e (x) if |x|

7.5 Each high degree contains a computably random set 281Note that Z exists by (7.4). In our case, since L is Q 2 -valued and L ≤ T C,wehave L + N ≤ T C.ThusZ ≤ T C as well, which establishes Theorem 7.5.3.To prove the Lemma, we build N via a supermartingale approximation (N s) s∈N .Wemay assume that #(C s+1 − C s) = 1 for each stage s. At stage s + 1, let Z s be theleftmost path of length s +3 of T s = {x: ∀y ≼ x (L s + N s)(y) ≤ 4[s]}. Since L s(y)and N s(y) are nondecreasing in s, the approximation (Z s) s∈N only moves to the right,so Z is left-c.e.We devote 2 −n of the initial capital of N to code into Z whether n ∈ C. Recall that G nis the supermartingale given by G n(x) =1for|x| t, this causes (L + N)(Z s ↾ 2n+3) ≥ 4,so Z t ↾ 2n+3 is not on the leftmost path of T .✷A computably random set that is not partial computably randomCoding a set C into the leftmost path of a Π 0 1 class works when the set is c.e., butfor coding an arbitrary set a different idea is needed. Suppose that as before L isa supermartingale dominating all the partial computable supermartingales. Weintroduce a new method for coding information into a set Z such that L(Z) < ∞.Given a string x which already encodes k bits of information, one finds distinctstrings y 0 ,y 1 ≻ x such that along each y a , L increases by at most a factor of1+2 −k .IfZ extends x, one may code one further bit a into Z by letting Zextend y a . Such strings y 0 ,y 1 ≻ x exist by the following Lemma. It is an analogof Lemma 3.3.1 used for the proof of the Kučera–Gács Theorem 3.3.2.7.5.6 Lemma. Let L be a supermartingale. Then for each string x and eachnumber k, there are distinct strings y 0 ,y 1 ≻ x of length |x| + k +1 such that Ldoes not grow beyond (1 + 2 −k )L(x) along each y a , a =0, 1. That is,∀z [x ≼ z ≼ y a → L(z) ≤ (1 + 2 −k )L(x)]. (7.11)Proof. Because of the localization principle 7.1.7(iii) we may suppose x = ∅.Let y 0 be a string of length k+1 along which L does not increase. Let z 0 ,...,z n−1be the strings z of length k + 1 that are minimal under the prefix relation withL(z) > (1 + 2 −k )L(∅). Assume for a contradiction that for each y ≠ y 0 of

282 7 Randomness and betting strategieslength k + 1 there is iL(z) for z = Z ↾ u7 ).To show that Z is computably random, it suffices to verify that L does notsucceed on Z. We need the following fact from analysis.7.5.8 Fact. If (a k ) k∈N is a sequence of nonnegative real numbers ands = ∑ k a k < ∞, then ∏ k (1 + a k) < ∞.Proof. e s = sup k∏ ki=0 eai , and 1 + a i ≤ e ai .Thus ∏ k (1 + a k) ≤ e s < ∞.We apply this fact to a k =2 −k .AsL(∅) ≤ 2wehaveL(Z ↾ n ) ≤ 2 ∏ k

7.5 Each high degree contains a computably random set 283We define a partial computable martingale D that succeeds on Z. It can predicteach value Z(u k ), and hence bets all of its current capital on that value. Theprediction is based on TMG ↾ k , since TMG ↾ k determines the values L(x) for|x| ∈ I k . D merely “reads” Z at the bit positions j

284 7 Randomness and betting strategiesquestion whether B e is total is divided into infinitely many subquestions whetherB e (x) is defined for all x such that |x| ∈ I r . This is similar to the transition fromTheorem 7.4.8 to Remark 7.4.13.7.5.9 Theorem. For each high set C there is a computably random, but notpartial computably random set Z ≡ T C. Moreover, given an order function h,one can achieve that ∀ ∞ nK(Z ↾ n | n) ≤ h(n).Proof. Let n 0 =0.Fork ≥ 0, letn k+1 = µn > n k + k +1.h(n) > 4(k + 1).Let I k =[n k ,n k+1 ), u k = n k + k + 1, and u −1 = 0. As in the proof of theDomination Lemma 7.5.1, choose a function f ≤ T C which dominates eachcomputable function, and for each e let g e (n) ≃ µt. ∀x [|x| ≤ n → B e,t (x)↓]. LetQ = {〈0,r〉: r ∈ C} ∪{〈e, r〉: e>0&∀p ≤ r [g e (n 〈e+p+1,e+p+1〉 ) ≤ f(e + p)]}.Clearly Q ≡ T C.WesayB e is active in I k if k ≤ e +1, or 〈e, r〉 ∈Q for themaximal r such that r =0or〈e, r〉 n k . Hence B e (w)↓ for each w such that |w| 0, as always let Be ∗ = B e,ne+1 . Let{Be ∗ (x) if |x| ∈ I k & B e is active in I kV e (x) =0 else.As usual L(x) = ∑ e 2−e V e (x) isaQ 2 -valued supermartingale. Given x, k, thestrings yx,k+1,0 L and yL x,k+1,1are defined in (7.12). The set Z is given by thefollowing recursion.• If x = Z ↾ nk has been defined let Z ↾ uk = y L x,k+1,Q(k) .• If z = Z ↾ uk has been defined let Z ↾ nk+1 be the leftmost non-ascendingstring above z given by L (see 7.3.9).We show that Z is computably random. Firstly, we verify that L(Z) < ∞.Note that L(Z ↾ n ) ≤ 2 ∏ k≤n (1 + 2−k ) by the choice of the strings in (7.12) andsince L is non-ascending along Z on bit positions in the intervals [u k ,n k+1 ). ByFact 7.5.8 ∏ k (1 + 2−k ) < ∞. SoL(Z) = sup n L(Z ↾ n ) < ∞.Secondly, we show that L multiplicatively dominates each total B e . The functionv given by v(i) =g e (n 〈i+1,i+1〉 ) is computable, so that ∀i ≥ i 0 [v(i) ≤ f(i)]for some i 0 . By the Padding Lemma 1.1.3 we may choose e ′ such that theTuring program P e ′ behaves like P e and e ′ > i 0 . Hence, for all p, we havev(e ′ + p) =g e ′(n 〈e′ +p+1,e ′ +p+1〉) ≤ f(e ′ + p), so 〈e ′ ,r〉∈Q for e ′ and all r.Then B e ′ is active in all intervals I r , whence V e ′ = Be ∗ ′. Since B e = B e ′, thisshows that L multiplicatively dominates B e . (The Padding Lemma is more thana notational convenience here: it saves us from keeping track of the point fromwhich on f exceeds g e .)

7.5 Each high degree contains a computably random set 285Since C ≡ T Q, to show C ≡ T Z it suffices to verify that Q ≡ T Z.ForQ ≤ T Zwe introduce a Turing functional ∆ and verify inductively thatfor each k. Asalways∆ ∅ = ∅.∆ Z↾u k = Q↾k+1 (7.15)1. If |z| = u k and τ = ∆ z↾u k−1 has been defined where |τ| = k, letE(τ) ={e: 0

286 7 Randomness and betting strategiesOn the other hand we are able to build a computable martingale that uses theseguaranteed zeros to succeed on Z.Let θ e (m) ≃ µs. Φ e (m)[s]↓, and letψ(e, m) ≃ u 〈〈e,θe(m)〉,m〉+1.Note that ψ is one-one, ψ(e, m) ≥ u m+1 >mfor all pairs e, m in its domain,and its range is computable. We define a computable function p by the recursion{p(i)+1 ifi

7.5 Each high degree contains a computably random set 287(1) Let c =1/(2p(i)).(2) Let r := r +1.Ifr ∉ G i goto 2.(3) Let c := 2c. Bet the amount min(N,c) that the next bit is 0.(4) Request the next bit b. If b = 0 (and hence N is updated to N + c), leti := r and goto 1.(5) (Now N has been updated to N − c.) If N>0 goto 3, else end.Suppose now that the strategy is fed the bits of Z. Given i, let m be the numberof times S bets at (3) for this value of i. Since Z(r) = 0 for some r ∈ G i ,wehavem ≤ #G i ≤ log p(i) − 1. The maximum value of c is 2 m /p(i) ≤ 1/2, so there isalways some capital left for S to bet with. S loses 1/p(i), 2/p(i), 4/p(i) but thenwins 2 m /p(i). So it wins 1/p(i) onZ with parameter i.If S updates its parameter i to r in (4) then r = ψ(e, i) for some emfor infinitely many m. Since p takes eachvalue only finitely often, for almost all m we have h(log log m) >g(m).We claim that L(Z ↾ g(m) )=O(log m) for each m>1: in an interval of theform [n k ,u k ), L can increase its capital by a factor of at most 1 + 2 −k . Since∏k (1 + 2−k ) < ∞, overall the capital increases only by a constant factor. At aposition u k it can only increase its capital if u k ∈ ran(h). For almost all m, thereare at most log log(m) such u k below g(m). So even if L doubles the capital eachtime, it will only achieve an increase by a factor of O(log m). This proves theclaim.In the proof of Theorem 7.5.9 we showed that L multiplicatively dominateseach total B i .ThusB i (Z ↾ g(m) ) = O(log m) for each m > 1, contradiction.✷Nies, Stephan and Terwijn (2005) proved Theorem 7.5.3 by extending the methodsof Theorem 7.5.9. In a similar vein, they proved that in 7.5.10, if the high set C is c.e.,the Schnorr random set Z can be chosen left-c.e.Exercises.7.5.11. Show that if C is hyperimmune, there is a set Z ≡ T C such that Z is weaklyrandom but not Schnorr random. (This complements 3.6.4.)7.5.12. ⋄ Remark 7.4.17 contains an example of a left-c.e. partial computably randomset with a slowly growing initial segment complexity. As a further way to separate thisrandomness notion from ML-randomness, show that there is a Turing minimal pairZ 0,Z 1 of left-c.e. partial computably random sets.Hint. Modify the usual construction of a minimal pair of high c.e. sets via a tree ofstrategies already mentioned after 6.3.4; see for instance Soare (1987, XIV.3.1).7.5.13. ⋄ Problem. Decide whether for each high (c.e.) set C there is a (left-c.e.) setZ ≡ T C such that Z is partial computably random.

288 7 Randomness and betting strategies7.6 Varying the concept of a betting strategyWe consider two variants of betting strategies: selection rules and nonmonotonicbetting strategies. The former are weaker, the latter stronger than betting strategieswith the same type of effectivity condition.Instead of betting, a selection rule selects from a set Z (viewed as a sequenceof bits) a subsection of bits that “count”. It succeeds on the set if the law oflarge numbers fails for this selected subsequence. This notion can be traced backto von Mises (1919).A nonmonotonic betting strategy bets on the bits in an order it determines.Basics of selection rulesA selection rule ρ processes a set Z bit by bit. For each bit position n it makes adecision whether to select the bit in position n based on what it has seen so far.Thus, ρ maps a string x of length n (thought of as a possible initial segment Z ↾ n )to a bit.Recall from 1.7.1 that p E (n) denotes the n-th element of a set E ⊆ N.7.6.1 Definition.(i) A selection rule is a partial function ρ: {0, 1} ∗ → {0, 1} with a domainclosed under prefixes.(ii) Given a set Z, let E = {n: ρ(Z ↾ n )=1} (the set of positions selected fromZ by ρ). The set selected from Z by ρ is S = {k : Z(p E (k)) = 1}.For instance, suppose ρ selects a bit position if the last pair of bits it has readis 01. If Z = 110110110 ... then E = {3m +1: m>0} and S = N.StochasticityRecall from page 109 that a set Z satisfies the law of large numbers iflim n #{i

7.6 Varying the concept of a betting strategy 289Proof. Let b =1ifZ is infinite and b = 0 else. Define a computable selectionrule ρ by ρ(x) =1ifZ(|x|) =b. Then the set selected from Z by ρ is N. ✷If a computable selection rule ρ succeeds on a set Z then ρ(Z ↾ n ) is defined foreach n. By the main result of this section, Theorem 7.6.7 below, extra power isgained if we allow ρ to be undefined on strings that are not initial segments of Z.7.6.4 Definition. Z is partial computably stochastic (or Church–Mises–Waldstochastic) if no partial computable selection rule ρ succeeds on Z.The two notions just introduced correspond to computable randomness and partialcomputable randomness. Martingales as tests are strictly more powerful than the selectionrules with the same effectivity condition. For instance, the computably randomsets form a proper subclass of the computably stochastic sets. In fact, Merkle et al.(2006, Thm. 30) proved that some partial computably stochastic set is not even weaklyrandom.Anbos-Spies et al. (1996) called a (possibly partial) martingale B simple if for someq ∈ (0, 1) Q , the set of betting factors {B(x0)/B(x): x ∈ {0, 1} ∗ } is contained in{1,q,1 − q}. They showed that Z is computably stochastic iff no simple computablemartingale succeeds on Z, and Z is partial computably stochastic iff no simple partialcomputable martingale succeeds on Z.Stochasticity and initial segment complexityWe think of a K-trivial set A as far from random because K(A↾ n ) grows as slowlyas possible. If we relax the growth condition somewhat, we can still expect the setto have strong nonrandom features. Such a condition is being O(log) bounded.Recall that log n denotes the largest k ∈ N such that 2 k ≤ n.7.6.5 Definition. A string x is O(log) bounded via b ∈ N if C(x)

290 7 Randomness and betting strategiesWe need some preparations to prove 7.6.7. The following notation will be usedthroughout. For an interval I =[n, m) let Z ↾ I = Z(n) ...Z(m − 1). Moreover,(a) m 0

7.6 Varying the concept of a betting strategy 291Proof details. Suppose (i) holds for all p ≥ p 0 . Fix a ∈ {0, 1}. On input astring y, the following procedure attempts to compute ρ a (y) ∈ {0, 1}, whereρ a (y) = 1 means that the next bit is selected.(1) Let p be so that |y| ∈ I p . If p

292 7 Randomness and betting strategiesC M (Z ↾ I p ) 0,m p := 10ku p ,where as always u p = ∑ i

7.6 Varying the concept of a betting strategy 293(In particular, z p,1 = Z ↾ [u p ,u p +(k − 1)l p ) and z p,k = ∅.) Let us show thatfor some t ∈ {1,...,k}, for infinitely many p the set of advisors when ρ a,t isprocessing Z has a size of less than 0.2l p . Thereafter, we can argue as in theproof of Lemma 7.6.8. The idea is that otherwise, because there are k levels, foralmost every p the size of A p exceeds the bound (m p ) k−1 in (7.18).p pJ 1 J 2 . . .J kpZz p,1. . .T 1pz p,k –2T kp–2z p,k–1pT k –1z p,kT kpFig. 7.2. The sets T p s (z p,s ).7.6.10 Lemma. There is t ∈ {1,...,k} such that(i) t ∀ ∞ p Z ↾ J p̂t ∈ T p t (z p,t ).(ii) t ∃ ∞ p #T p t (z p,t ) < 0.2l p .Proof. (i) 1 states that Z ↾ I p ∈ A p for almost all p, that is, C(Z ↾ I p )

294 7 Randomness and betting strategiesintervals Jsp what ρ a did in the intervals I p , and ρ a,s uses the strings in Ts p (z)as sets of advisors for an appropriate z derived from the input.On input y the procedure attempts to compute ρ a,s (y) as follows.(1) Let p, t be so that |y| ∈ J p t . If p

7.6 Varying the concept of a betting strategy 295pair P =(π,B) consisting of a computable permutation and a partial computablemartingale. We say that P succeeds on Z if Z ◦ π ∈ Succ(B).The permutation random sets form the largest subclass of PCR that is closedunder computable permutations. Kastermans and Lempp (20xx) constructed apermutation random set that is not ML-random. It is not known whether sucha set can be left-c.e.; it is also open whether a permutation random set must beof d.n.c. degree.Muchnik’s splitting techniqueWe give two applications of a technique due to Andrej Muchnik (see Thm. 9.1.of Muchnik, Semenov and Uspensky 1998) to show that a set Z satisfies a propertyindicating non-randomness. Let S be an infinite co-infinite computable set.Split Z into sets Z 0 and Z 1 , the bits with positions in S and in N − S, respectively.In either application, the hypothesis states that for each r ∈ {0, 1}, foreach m ∈ N, the set Z r satisfies a “local” non-randomness condition P m,r . Theseconditions are uniformly Σ 1 (Z r ); let t m,r be the stage by which we discover thatP m,r holds for Z r . Choose p such that t m,p ≥ t m,1−p for infinitely many m. Wecan for instance use Z p as an oracle to build a test for Z 1−p . This is all we needthe first application of the technique given here, due to Merkle et al. (2006). Inour second application, for which the technique was introduced by Muchnik, therequired bits of the oracle Z p are read by a permutation betting strategy in orderto succeed on Z.7.6.13 Theorem. Suppose Z = Z 0 ⊕ Z 1 and neither Z 0 nor Z 1 is ML-random.Then Z 1−p is not Schnorr random relative to Z p for some p ∈ {0, 1}.Proof. Suppose that for each r ∈ {0, 1}, (G r m) m∈N is a ML-test such that Z r ∈⋂m Gr m. (The local conditon P m.r states that Z r ∈ G r m.)1. Let G r m,s be the approximation of G r m at stage s by strings of length atmost s as in (1.16) on page 54.2. Let t m,r = µt. Z r ∈ [G r m,t] ≺ . Note that λm. t m,r ≤ T Z r .3. Let p ∈ {0, 1} be least such that ∃ ∞ m [t m,p ≥ t m,1−p ].4. Let C p m be the clopen set [G 1−pm,t m,p] ≺ .5. Let V n = ⋃ m>n Cp m. Then Z 1−p ∈ ⋂ n V n by the choice of p.We show that (V n ) n∈N is a Schnorr test relative to Z p . Clearly, V n is uniformlyc.e. relative to Z p and λV n ≤ 2 −n . Also, λV n is uniformly computable relativeto Z p , because for k>nthe rational α k = λ ⋃ nk Cp m ≤ 2 −k ; now apply 1.8.15(iv) relative to Z p . ✷The second application shows that for a permutation random set the initialsegment complexity (in the sense of C, and hence also in the sense of K) hasto be large at infinitely many positions in a given infinite computable set. Fromthis we will conclude that PCR is not closed under computable permutations.

296 7 Randomness and betting strategies7.6.14 Theorem. (Andrej Muchnik) Suppose Z is a set such that, for somestrictly increasing computable function g with g(1) > 1, we have∀k [C(Z ↾ g(k) ) ≤ + g(k) − k]. Then Z is not permutation random.Proof. First we show that there is a strictly increasing computable h such that,where (I n ) n∈N are the consecutive intervals of N of length h(n), we have∀nC(Z ↾ I n ) ≤ + h(n) − n.Let h(0) = 1 and h(n) =g(n + ĥ(n)) − ĥ(n) where ĥ(n) =∑ i

7.6 Varying the concept of a betting strategy 297Kolmogorov–Loveland randomnessMost of the material in this subsection is due to Muchnik, Semenov and Uspensky(1998) or Merkle, Miller, Nies, Reimann and Stephan (2006).A nonmonotonic computable betting strategy determines the next position ofa set Y where a bet is placed from the previous positions and the values of Y atthese positions. For instance, initially the strategy bets at position 3; if Y (3) = 0it next bets at position 1, otherwise at 2, and so on.The history up to the n-th bet of betting positions and values is given bya finite assignment α = (〈d 0 ,r 0 〉,...,〈d n−1 ,r n−1 〉) as defined in Section 1.3.Thus all d i ∈ N are distinct and r i ∈ {0, 1}. We let dom(α) denote the set{d 0 ,...,d n−1 }, called the set of positions of α. We write α⊳Y if Y (d i )=r i foreach i

298 7 Randomness and betting strategiesWe say that Z is Kolmogorov-Loveland (KL-)random if no KL-betting strategysucceeds on it.Thus, Z is KL-random if S(Z) is partial computably random whenever it is aset. By Exercise 7.6.25 we may assume that S and B are total.7.6.20 Proposition. Each Martin-Löf random set Z is KL-random.Proof. We use ideas from the proof of Proposition 3.2.16. Given a partial computablescan rule S, for each string x let C x = {Y ∈ 2 N : x ≼ S(Y )}. Note thatC x is a c.e. open set uniformly in x. Moreover, either C x0 = C x1 = ∅ (this canonly happen if S is partial) or these two sets split C x into parts of the samemeasure. Then by induction on |x| we have λC x ≤ 2 −|x| for each x. IfG is c.e.open then so is G ∗ = ⋃ [x]⊆G C x, and λG ∗ ≤ λG.Now suppose the KL-betting strategy (S, B) succeeds on Z. We may assumethat B(∅) ≤ 1. By Proposition 7.1.9 the c.e. open set G d = {X : ∃rB(X ↾ r ) >2 −d } satisfies λG d ≤ 2 −d .ThusZ fails the Martin-Löf test (G ∗ d ) d∈N. ✷Each KL-betting strategy has an “Achilles heel”: it does not succeed on somec.e. set by Exercise 7.6.26. However, by Merkle et al. (2006) there are two KLbettingstrategies such that on each c.e. set one of them succeeds. In fact, it isan open question whether there are two KL-betting strategies such that on eachset that is not ML-random one of them succeeds.Although we do not know whether each KL-random set is already Martin-Löfrandom, various results of Merkle et al. (2006) show that KL-randomness is atleast closer to ML-randomness than the previously encountered notions (exceptfor, possibly, permutation randomness).7.6.21 Theorem. If Z = Z 0 ⊕Z 1 is KL random, then Z p is Martin-Löf randomfor some p ∈ {0, 1}.Sketch of proof. Suppose otherwise, then by Theorem 7.6.13 Z 1−p is notSchnorr random relative to Z p for some p ∈ {0, 1}, say p = 0. By Proposition7.3.2 relative to Z 0 we may choose a Turing functional Ψ such that Ψ Z0 isa Q 2 -valued martingale that succeeds on Z 1 . We use Ψ to build a KL-bettingstrategy that succeeds on Z. We may assume Ψ σ |σ| (y)↓ implies Ψσ |σ|(x)↓ for eachx ≺ y. The strategy reads more and more bits on the Z 0 -side in ascending orderwithout betting until a new computation q = Ψ σ |σ| (x0) appears, where σ ≺ Z 0has been scanned on the Z 0 -side, and x ≺ Z 1 on the Z 1 -side. Then it betsq − Ψ σ |σ| (x) on the prediction that the n-th bit on the Z 1-side is 0. Each bit onthe Z 1 -side is scanned eventually, so the strategy succeeds on Z.In more detail, the sequence of scanned bits has the form σ 0b 0σ 1 ...b n−1σ n, wherethe σ i are strings of consecutive oracle bits on the Z 0-side and the b j are consecutivebits on the Z 1-side. To formalize the above by a KL-betting strategy (S, B), for instancelet n = 1. The finite assigment corresponding to σ 0b 0σ 1, where k i = |σ i|, isα =(〈0, σ 0,0〉,...,〈2(k 0 − 1), σ 0,k0 −1〉, 〈1,b 0〉, 〈2k 0, σ 1,0〉,...,〈2(k 0 + k 1 − 1), σ 1,k1 −1),and we define S(α) = 3 since q = Ψ σ 0σ 1k 0 +k 1(b 00) newly converges. Also B(w) =1forw ≼ σ 0, B(w) =Ψ σ 0(b 0) for σ 0b 0 ≼ w ≼ σ 0b 0σ 1, and B(σ 0b 0σ 10) = q.✷

7.6 Varying the concept of a betting strategy 299The following is an effective version of the Hausdorff dimension of the class {Z}.See for instance Downey et al. (2006).7.6.22 Definition. Let dim(Z) = lim inf n K(Z ↾ n )/n.Note that by 2.4.2 we could as well take C instead of K. By Schnorr’s Theorem3.2.9 and Proposition 4.1.2 we have:7.6.23 Corollary. If Z is KL-random then dim(Z) ≥ 1/2, and Z has d.n.c.degree via a finite variant of the function λn. Z ↾ n .✷In particular, each left-c.e. KL-random set is wtt-complete by the completeness criterion4.1.11. It is not known whether such a set is already ML-random. An extensionof the argument in Theorem 7.6.21 shows that dim(Z) = 1 for each KL-random set Z.Further, if Z is ∆ 0 2 then both Z 0 and Z 1 are ML-random. See Merkle et al. (2006).We say that a set Z is KL-stochastic if, for each partial computable scan rule S, ifS(Z) is a set then S(Z) is partial computably stochastic. Merkle et al. (2006) showedthat actually dim(Z) = 1 for each KL-stochastic set. On the other hand, there is anonempty Π 0 1 class of KL-stochastic sets that are not even weakly random.The following implies that CR is closed under computable permutations.7.6.24 Theorem. Suppose that S is a partial computable scan rule that scansall positions for each set Y , namely, S(Y ) is a set and ⋃ n dom ΘY S (n) =N.Then, if Z is computably random, so is S(Z).Sketch of proof. S is total, so by 7.6.17 we may assume that S(α) is definedfor each α ∈ FinA. Suppose the computable martingale B succeeds on S(Z). Wemay assume that B(x) > 0 for each x and, by the Savings Lemma 7.1.14, thatlim n B(S(Z)↾ n )=∞. We define a computable sequence 0 = n 0 n i such that for each y of length n all the positionsin [0,n i ) have been scanned. For some p ∈ {0, 1} we may assume that the KLbettingstrategy ρ =(S, B) only bets on positions in [n 2k+p ,n 2k+p+1 ), becausethe product over all l of the betting factors B(S(Z)↾ l+1 )/B(S(Z)↾ l ) is infinite.Say p =0.We inductively define a computable martingale D that succeeds on Z. LetD(∅) =B(∅). Suppose |x| = n 2k and D(x) has been defined. Let m = n 2k+2 −n 2k . We will define D(xv) for each string v of length at most m.For each string u of length m, let r u be least such that [0,n 2k+1 ) ⊆ dom Θ xuS (r u).Note that n 2k+2 ≥ r u ≥ n 2k+1 . Letw u = S(xu) ↾ r u . (7.21)To proceed with the definition of D we need to verify the following.Claim. D(x) = ∑ |u|=m 2−m B(w u ).Then, recalling the martingales E u from 7.1.4, define for |v| ≤ mD(xv) = ∑2 −m B(w u )E u (v). (7.22)|u|=m

300 7 Randomness and betting strategiesFor v = ∅ the right hand side is D(x) by the claim. For |v| = m we have thevalue B(w v ). Since lim n B(S(Z)↾ n )=∞ this shows D(Z) =∞.To prove the claim, let G be the prefix-free set {w u : |u| = m}. Since S(α) isdefined for each α ∈ FinA, from a string w we can determine the finite assignmentα = α w , |α| = |w| that induces w in the sense that w(i) =S(α ↾ i ) for i0 it ignores thepositions less than n 2k . On these positions B does not bet anyway, since all thepositions less than n 2k−1 have been visited already after processing x, namely,[0,n 2k−1 ) ⊆ dom Θ x S . This leaves q w positions.For each w ∈ G there are 2 m−qw strings u of length m such that α w ⊳ xu,or equivalently w u = w. Hence D(x) = ∑ w∈GB(w) = ∑ 2−qw |u|=m 2−m B(w u ).This proves the claim.✷By the solution to Exercise 7.6.25, each KL-betting strategy can be replacedby a KL-betting strategy (S, B) with total S and B that on every set scans atleast as many positions as the given one. Thus, by the foregoing theorem, to bemore powerful than a monotonic betting strategy, a KL-betting strategy needsto avoid scanning positions on some sets (actually, on a class of sets of non-zeromeasure by Exercise 7.6.28).Exercises.7.6.25. (Merkle) Show that, if Z is not KL-random, some KL-betting strategy (S, B)with total scan rule S and total B suceeds on Z.7.6.26. (i) Show that for each computable set E and each partial computable scanrule S, there is a c.e. set A such that either S(A) is not a set or S(A) =E.(ii) Use 7.4.6 to infer that each KL-betting strategy (S, B) does not succeed on somec.e. set A.7.6.27. Extend Schnorr’s Theorem to finite assignments:Z is ML-random ⇔∃b ∀α⊳Z[K(α) ≥ |α| − b].7.6.28. Strengthen 7.6.24: if a KL-betting strategy (S, B) succeeds on a computablyrandom set Z, then the probability that S scans all places of a set Y is less than 1.

8Classes of computational complexityA lowness property of a set specifies a sense in which the set is computationallyweak. In this chapter we complete our work on lowness properties and how theyrelate to randomness relative to an oracle. Recall that Low(MLR) denotesthesets that are low for ML-randomness. In Chapter 5 we showed that this propertycoincides with a number of other lowness properties, such as being low for K.It also coincides with K-triviality, a property that expresses being far from MLrandom.We will carry out similar investigations for two classes that are variantsof Low(MLR). We give examples of incomputable sets in each class and provethe coincidence with classes that arise from a different context.(1) By Definition 3.6.17, Low(Ω) is the class of sets A such that Ω is ML-randomrelative to A. Clearly,Low(MLR) is contained in Low(Ω). Note that the classLow(Ω) is conull because each 2-random set is low for Ω. We think of being lowfor Ω as a weak lowness property. However, it is very restrictive within the ∆ 0 2sets: each ∆ 0 2 set that is low for Ω is a base for ML-randomness, and hence is inLow(MLR) by5.1.22.We prove that each nonempty Π 0 1 class contains a set that is low for Ω. Thereafterwe show that Low(Ω) coincides with the class of sets A that are weakly lowfor K, namely,∃b ∃ ∞ y [K A (y) ≥ K(y)−b]. This yields a further characterizationof 2-randomness via the initial segment complexity.(2) The second variant of Low(MLR)istheclassLow(SR), the sets that are low forSchnorr randomness. In Theorem 3.5.19 we characterized Schnorr randomness interms of the initial segment complexity given by computable measure machines.In this chapter we show that Low(SR) coincides with the class of sets that arelow for computable measure machines. This corresponds to the coincidence ofLow(MLR) andbeinglowforK.More importantly, the sets in Low(SR) canbecharacterizedbybeingcomputablytraceable, a restriction on the functions they compute. This characterizationis a further example of the interactions between randomness and computabilityin both directions. A computability theoretic property characterizeslowness for Schnorr randomness, but also, Schnorr randomness helps us to understandwhat this property means.The general idea of traceability of a set A is that the possible values of eachfunction computed by A are in finite sets which, together with a bound ontheir size, can be determined in some effective way from the argument. Theset A is computably traceable (Terwijn and Zambella, 2001) if for each functionf ≤ T A,eachvaluef(n) is in a finite set D g(n) ,whereg is a computable function

302 8 Classes of computational complexitydepending on f; furthermore,thereisacomputableboundonthesize#D g(n)independent of f (see Definition 8.2.15 below for the details). If we merely requirethat the trace be uniformly c.e., we obtain the weaker notion of c.e. traceability.Asetiscomputablytraceableiff it is c.e. traceable and computably dominated.Broadly speaking, lowness for Schnorr randomness and computable traceabilitycoincide because they are linked by the idea of a covering procedure (seepage 226). A is low for Schnorr randomness iff A is low for Schnorr null classes,that is, for each Schnorr test (G m ) m∈N relative to A there is a Schnorr test(H k ) k∈N such that ⋂ m G m is covered by ⋂ k H k. A is computably traceable iffor each f ≤ T A we can trace, or “cover”, the possible values f(n) byafiniteset obtained without the help of A. Theproofofthiscoincidenceshowshowacovering procedure of one type can be transformed into a covering procedure ofthe other type.How about being low for computable randomness? Surprisingly, the only setsof this kind are the computable ones. Recall from Definition 5.1.32 that for randomnessnotions C ⊆ D, we denote by Low(C, D) theclassofsetsA such thatC ⊆ D A . We determine the lowness notions for SR and CR by first characterizingLow(C, D) foreachpairC ⊂ D among the three randomness notions underdiscussion. We show that Low(MLR, SR) coincides with being c.e. traceable,Low(CR, SR) withbeingcomputablytraceable,andLow(MLR, CR) withbeinglow for K. IfA is low for computable randomness then A is computably dominatedand ∆ 0 2,soA is computable. For the same reason, although MLR ⊆ SR,the corresponding lowness properties exclude each other for incomputable sets.For the classes Low(MLR)orLow(Ω), no characterization in purely computabilitytheoretic terms is known. However, both classes are contained in interestingcomputability theoretic classes: each set in Low(MLR) issuperlowbyCorollary5.5.4, and each incomputable set that is low for Ω is of hyperimmune degreeby Theorem 8.1.18 below. In particular, the classes Low(SR) andLow(Ω) excludeeach other for incomputable sets, strengthening the afore-mentioned similar factabout Low(SR) andLow(MLR).A further example of a traceability property of a set A is jump traceability withan order function g as a bound (Nies, 2002). As for c.e. traceability, the tracefor J A is a uniformly c.e. sequence (T n ) n∈N such that #T n ≤ g(n) foreachn.We ask that J A (n) ∈ T n whenever J A (n) isdefined.Ifg grows suffciently fastthen there is a perfect class of jump traceable sets with bound g. On the c.e.sets this property coincides with superlowness. Unlike the case of computabletraceability, or c.e. traceability, it now matters which order function we chooseas a bound. We will show that if the order function h grows much slower than g,the class for the bound h is properly included in the class for the bound g.We say that A is strongly jump traceable if for each order function g thereis a c.e. trace for J A with bound g. Figueira,NiesandStephan(2008)builtapromptlysimplestronglyjumptraceableset.Theirhopewasthat,atleastfor the c.e. sets, this property might be a computability theoretic characterizationof Low(MLR). However, Cholak, Downey and Greenberg (2008) proved that

8.1 The class Low(Ω) 303Table 8.1. Properties in the same row coincide (except for the facile sets).Low(MLR) Low for K K-trivialLow for ΩWeakly low for KLow(SR) Low for computable (facile) Computablymeasure machinestraceableLowly for Cstrongly jump tr.the strongly jump traceable c.e. sets form a proper subclass of the c.e. sets inLow(MLR).Table 8.1 summarizes the classes studied in this chapter. The first row illustratessome of the coincidences of Chapter 5, the second row relates to (1) above,the third to (2) above, and the fourth to strong jump traceability. Classes in thesame row coincide, disregarding the class of facile sets. Properties in the samecolumn are analogous.The analog of the coincidence of lowness for ML-randomness and being a basefor ML-randomness fails in the domain of Schnorr randomness: Hirschfeldt, Niesand Stephan (2007) proved that the class of bases for Schnorr randomness islarger than Low(SR). Similarly, the analog of the coincidence of lowness forSchnorr randomness and computable traceability fails in the domain of MLrandomness,because the strongly jump traceable c.e. sets form a proper subclassof the c.e. sets in Low(MLR).8.1 The class Low(Ω)In 3.6.17 we introduced the weak lowness property Low(Ω). In Fact 3.6.18 weobserved that Low(Ω) ⊆ GL 1 ,andthatLow(Ω) iscloseddownwardunderTuringreducibility (actually, it is easily seen to be closed downward under ≤ LR ). Wenow study the class Low(Ω) inmoredetail.Wewillfrequentlyusethefactthatin the definition of Low(Ω) one can replace Ω by any left-c.e. ML-random set.8.1.1 Proposition. The following are equivalent for a set A.(i) Some left-c.e. set is ML-random relative to A.(ii) A is low for Ω.(iii) Every left-c.e. ML-random set is ML-random relative to A.Proof. The implications (iii)⇒(ii) and (ii)⇒(i) are trivial. For the implication(i)⇒(iii), suppose that Z is left-c.e. and ML-random relative to A. IfY is leftc.e.and ML-random then by Theorem 3.2.29 we have 0.Z ≤ S 0.Y ,whichimpliesthat 0.Z ≤ S A 0.Y ,where≤ S A denotes Solovay reducibility relativized to A.Thus β ≤ S A 0.Z ≤ S A 0.Y for any real number β that is left-c.e. in A. HenceYis ML-random relative to A by Theorem 3.2.29 relativized to A.✷The Low(Ω) basis theoremWe give some examples of sets in Low(Ω). By 5.1.27 the only ∆ 0 2 sets in Low(Ω)are the ones in Low(MLR). By 3.6.19 each 2-random set is low for Ω.

304 8 Classes of computational complexityThe Low(Ω) basis theorem states that each nonempty Π 0 1 class P contains a setin Low(Ω). For instance, there is a set A ∈ Low(Ω) in the Π 0 1 class of two-valuedd.n.c. functions (see Fact 1.8.31). This yields a further type of sets in Low(Ω): theTuring degree of A does not contain a ML-random set, for otherwise A ≥ T ∅ ′by Theorem 4.3.8, contrary to the fact that Low(Ω) ⊆ GL 1 . Also, there is aML-random set Z ≤ T A by Theorem 4.3.2, so A is not in Low(MLR).The Low(Ω) basistheoremcanbederivedeasilyfromthefollowingresultofDowney, Hirschfeldt, Miller and Nies (2005). Recall from Definition 3.4.2 thatΩ A M = λ(dom M A )foraprefix-freeoraclemachineM.8.1.2 Theorem. For each prefix-free oracle machine M and each nonempty Π 0 1class P there is a left-Σ 0 2 set Z ∈ P such that Ω Z M = inf{ΩX M : X ∈ P }, whichisaleft-c.e.realnumber.Proof. Recall from 1.8.3 that T P = {x ∈ {0, 1} ∗ : P ∩ [x] ≠ ∅} is the treecorresponding to P .Welabeleachx ∈ T P with the real numberr x = inf{Ω X M : X ∈ P ∩ [x]}.If x ∉ T P we let r x = ∞. Clearly,r x = min(r x0 ,r x1 ). For q ∈ Q 2 , the classS x,q = P ∩ {X ≻ x: Ω X M ≤ q}is Π 0 1 uniformly in x, q. ThenL(r x )={q ∈ Q 2 : qr x1 ↔ L(r x0 ) ⊃ L(r x1 ) ↔ ∃q ∈ Q 2 [S x0,q = ∅ & S x1,q ≠ ∅],even in the case that P ∩ [x0] = ∅ (namely r x0 = ∞). This is a Σ 0 2 propertyof x, andhenceΣ 0 1 in ∅ ′ .ThereforetheclassP ∩ {X : ∀n [r X↾ (n+1) ≤ r X ↾n ]} isΠ 0 1(∅ ′ ). Let Z be its leftmost path, then Z is left-c.e. in ∅ ′ by the Kreisel BasisTheorem 1.8.36, that is, Z is left-Σ 0 2.We verify that Ω Z M = r ∅.BydefinitionofZ we have r ∅ = r x for each x ≺ Z.Let Ω x M denote λ(dom M x ), then Ω x M ≤ r x = r ∅ for each x ≺ Z. Butclearlylim n Ω Z↾nM= ΩZ M , as ΩZ M − ΩZ↾n Mis the measure of the M Z -descriptions with usegreater than n. ThusΩ Z M = r ∅ as required.✷If M is the optimal oracle prefix-free machine U, weobtainasetZ that is lowfor Ω, because the left-c.e. real number Ω Z is ML-random relative to Z. Thus:8.1.3 Corollary. Each nonempty Π 0 1 class contains a left-Σ 0 2 set Z ∈ Low(Ω).We mention a further consequence of Theorem 8.1.2: there is a left-c.e. realnumber r with a preimage under the operator Ω of positive measure. (Thus theoperator Ω is rather different from the jump operator in that for many sets X,Ω X has no information at all about X.) First we observe that such an r isnecessarily left-c.e. For r ∈ [0, 1] R let G r = {X : Ω X = r}.

8.1.4 Proposition. If λG r > 0 then r is left-c.e.8.1 The class Low(Ω) 305Proof. By Theorem 1.9.4 there is σ ∈ 2 2 −|σ|−1 .For each s ∈ N, sinceΩ X s ∈ Q 2 for each X, we can compute the largest q s ∈ Q 2such that λ{X ∈ [σ]: Ω X s ≥ q s } > 2 −|σ|−1 .Thenr =lim s q s . Since the sequence(q s ) s∈N is non-descending, this shows that r is left-c.e. ✷8.1.5 Lemma. Let r ∈ [0, 1] R be left-c.e. ThenλG r > 0 ⇔ G r contains a ML-random set.Proof. The implication “⇒” is trivial because the class MLR is conull.For the implication “⇐”, suppose that X is ML-random and Ω X = r. Let R bethe set such that r =0.R. NowR is ML-random in X, sobythevanLambalgenTheorem 3.4.6, X is ML-random in R.ButR ≡ T ∅ ′ because R is ML-random andleft-c.e., so X is 2-random. By Exercise 8.1.8 below, G r is a Π 0 2 class containing X,so we may conclude that λG r > 0.✷8.1.6 Corollary. There is a real number r such that λG r > 0: ifP is a Π 0 1 classsuch that ∅≠ P ⊆ MLR, then r = min{Ω X : X ∈ P } is such a number.Proof. Immediate from Theorem 8.1.2 and Lemma 8.1.5.✷Exercises.8.1.7. Show that not every nonempty Π 0 1 class P contains a Σ 0 2 set in Low(Ω).8.1.8. (Bienvenue) Show that for each ∆ 0 2 real number r ∈ [0, 1),the class G r = {X : Ω X = r} is Π 0 2.Being weakly low for KBy Theorem 5.1.22, A is low for K ⇔ A ∈ Low(MLR). Here we prove an analogousresult of Miller (20xx). Both sides of the equivalence are weakened.We say that A is weakly low for K if ∃b ∃ ∞ y [K A (y) ≥ K(y) − b].8.1.9 Theorem. A is weakly low for K ⇔ A is low for Ω.Proof. ⇒: Suppose that A is not low for Ω, thatis,foreachb there is m suchthat K A (Ω↾ m ) ≤ m − b. Wewanttoshowthatforeachb, foralmostally, thereis a U A -description that is by b + O(1) shorter than a U-description σ of y.If for some sufficiently large m, A “knew” a stage t such that Ω − Ω t ≤ 2 −m ,it could ensure this by starting a bounded request set L at t, andenumeratingarequest〈|σ| − m, y〉 whenever a computation U(σ) =y converges after t.The set A actually does not know such a stage t, butforeachb there is an msuch that Ω↾ m has a U A -description τ of length at most m − b. Thisinformationabout Ω is sufficient: at stage s, ifthereisanewU A -description τ of Ω s ↾ m ,we start a bounded request set L τ , which is enumerated similar to the set Ldescribed above, but only at stages t ≥ s, andonlyaslongasΩ t ↾ m = Ω s ↾ m .The c.e. index for L τ can be computed from A, hence we obtain prefix-free

306 8 Classes of computational complexitydescriptions relative to A by concatenating τ and the descriptions for the prefixfreemachine for L τ .IfU A (τ) =Ω ↾ m and |τ| ≤ m − b, thenthenewprefix-freedescription of y has a length of at most (m − b)+(|σ| − m) =|σ| − b.Construction relative to A of bounded request sets L τ .Initially, let L τ = ∅ for all τ ∈ {0, 1} ∗ .If U A s (τ) newlyconvergesatstages with output Ω s ↾ m ,startP (τ,s).Procedure P (τ,s): at stage t ≥ s, ifΩ t ↾ m = Ω s ↾ m and a computation U t (σ) =ynewly converges (necessarily |σ| ≥ m), then put the request 〈|σ| − m, y〉 into L τ .The set L τ is a bounded request set because we only enumerate into it as longas Ω has not increased by more than 2 −m since stage s. Bytheuniformityofthe Machine Existence Theorem 2.2.17, we have a prefix-free machine M τ foreach L τ ,andanindexforM τ as a partial computable function can be computedby A. Thus, letting M(τσ) ≃ M τ (σ) wedefineaprefix-freemachineM relativeto A.To show that A is not weakly low for K, givenb choose m, ands least for m,such that Ω s ↾ m = Ω ↾ m and U A s (τ) =Ω ↾ m for some τ of length at most m − b.Then the procedure P (τ,s) is started (at stage s). If a computation U(σ) =yconverges after s, the enumeration of L τ ensures that M(τρ)=y for some ρsuch that |ρ| ≤ |σ| − m. ThusK A (y) ≤ + (m − b)+(|σ| − m) =|σ| − b.⇐: Suppose that A is not weakly low for K. LetR ≠2 N be a c.e. open set suchthat 2 N − R ⊆ MLR (for instance, let R be the component R 1 of the universalML-test in Definition 3.2.8). Let Z be the leftmost member of the Π 0 1 class 2 N −R.Then Z is left-c.e. and ML-random. We show that Z is not ML-random relativeto A, whence A is not low for Ω by 8.1.1.The set Z will fail a ML-test (Ĝe) e∈N relative to A. We first build a ML-test(G e ) e∈N relative to A such that G e ⊆ R and λG e ≤ 2 −e−1 for each e. WhileZpasses this test, for each e we copy sufficiently much of R into G e to ensure that(◦)[p, 0.Z) ⊆ G e for some p ∈ Q 2 such that p 0&p +2ɛ ≤ 1&[p, p + ɛ) ⊆ G e }.Then λĜe ≤ 2λG e and Ĝe is c.e. relative to A uniformly in e. Thus(Ĝe) e∈N isaML-testrelativetoA and Z ∈ ⋂ e G e by (◦).By Fact 1.8.26 there is a computable prefix-free set B such that [B] ≺ = R. Wewill define sets B e ⊆ B that are uniformly c.e. relative to A. ThereafterweletG e =[B e ] ≺ . For each m, anystringoflengthm in B is enumerated into B e aslong as the number of such strings does not exceed 2 m−KA (m)−e−1 .IfK A (m)decreases we are allowed to enumerate more strings into B e .Moreformally,wedefine an A-computable enumeration (B e,s ) s∈N of B e .ThefinitesetB e,s is givenby requiring that, for each m,B e,s ∩ {0, 1} m = B t ∩ {0, 1} m ,

8.1 The class Low(Ω) 307where t ≤ s is the greatest number such that #(B t ∩ {0, 1} m ) ≤ 2 m−KA s (m)−e−1 .Note that λG e = ∑ m 2−m #(B e ∩ {0, 1} m ) ≤ 2 −e−1 ∑ m 2−KA (m) ≤ 2 −e−1 .Claim. For each e, forsufficiently large m we have B ∩ {0, 1} m = B e ∩ {0, 1} m .To see this let b m = #(B ∩ {0, 1} m ). By Fact 2.2.20 there is c ∈ N such thatK(m) − c ≤ m − log b mfor each m. SinceA is not weakly low for K, thereism 0 ∈ N such that for eachm>m 0 ,K A (m) ≤ K(m) − c − e − 2.Thus m − K A (m) − e − 2 ≥ log b m for all m>m 0 ,whichimpliesthat2 m−KA s (m)−e−1 ≥ b m (since 2 log bm ≥ b m /2byourdefinitionofthelogarithm),and hence B ∩ {0, 1} m = B e ∩ {0, 1} m .We use the claim to verify the property (◦) forp =0.Z ↾ m0 .Thebasicopencylinder [Z ↾ m0 ] is identified with the interval [p, p +2 −m0 ), which contains[p, 0.Z). Since Z ∉ R we have [Z ↾ m0 ] ∩ [x] = ∅ for each x ∈ B such that|x| ≤ m 0 .Now[p, 0.Z) ⊆ R by the definition of Z. Since strings in B of lengthno more than m 0 do not help to cover [p, 0.Z) andalllongerstringsinB areenumerated into B e ,thisimpliesthat[p, 0.Z) ⊆ G e .✷The foregoing theorem has interesting consequences. The first is presented here,the second in Theorem 8.1.14. For background on the first, see the discussionafter Theorem 5.6.13.8.1.10 Corollary. (J. Miller) If A ≤ LR B and B is low for Ω then A ≤ T B ′ .In particular, if B is low for Ω then {X : X ≤ LR B} is countable.Proof. Note that A ≤ LK B by Theorem 5.6.5. Hence, for each n we haveK B (A↾ n ) ≤ + K A (A↾ n )= + K A (n) ≤ + K(n).The set B is weakly low for K by Theorem 8.1.9, so S = {n: K(n) ≤ K B (n)+d}is infinite for some d ∈ N. Then,forsomeb ∈ N, thesetA is a path of the treeV = {z : ∀n ≤ |z| [n ∈ S → K B (z ↾ n ) ≤ K B (n)+b]}.Note that S ≤ T B ′ ,andthereforeV ≤ T B ′ . By Theorem 2.2.26(ii) relativizedto B, foreachn ∈ S there are only O(2 b )stringsoflengthn on V .HenceA isan isolated path of V ,whichimpliesthatA ≤ T V ≤ T B ′ .✷Note that the set A in 8.1.10 is low for Ω, andhenceinGL 1 by Fact 3.6.18(ii).Therefore, from A ≤ T B ′ we can conclude that A ′ ≤ T A ⊕∅ ′ ≤ T B ′ .Thus,forsetsAand B in the conull class Low(Ω), the weak reducibility ≤ LR is well-behaved in thatA ≤ T B → A ≤ LR B → A ′ ≤ T B ′ .In Theorem 5.6.9, we obtained statements equivalent to the condition that A ≤ LR Band A ≤ T B ′ .HencealltheseholdunderthehypothesesofCorollary8.1.10.Thus,wecan also, more generally, invoke 5.6.9 via Exercise 5.6.10 to show that A ≤ LR B andA ≤ T B ′ implies A ′ ≤ T B ′ for each A and B.

308 8 Classes of computational complexityExercises.8.1.11. If A ∈ ∆ 0 2 is weakly low for K then A is already low for K.8.1.12. (Miller, Yu) If X ⊕ Y is 2-random then X and Y form a minimal pair withrespect to ≤ LR. Namely,C ≤ LR X, Y implies C ∈ Low(MLR) for each set C.8.1.13. ⋄ Problem. If {X : X ≤ LR B} is countable, is B low for Ω?2-randomness and strong incompressibility KIn Theorem 3.6.10 we proved that Z is 2-random iff for some b, infinitelymanyx ≺ Z are b-incompressible C . In our second application of Theorem 8.1.9, weshow that one can equivalently require that infinitely many initial segments arestrongly incompressible in the sense of K (Definition 2.5.2). This also yields anew proof of the implication “⇒” inTheorem3.6.10.TheresultisduetoMiller(20xx).8.1.14 Theorem. Z is 2-random ⇔∃b ∃ ∞ n [ K(Z ↾ n ) ≥ n + K(n) − b ] .Proof. ⇐: Each strongly incompressible K string is incompressible C in a sensemade precise in Proposition 2.5.5. So the right hand side implies that for some b ′ ,infinitely many strings x ≺ Z are b ′ -incompressible C .ThereforeZ is 2-randomby Theorem 3.6.10.⇒: Unlike the proof of 3.6.10, this proof is based on lowness properties. A 2-random set is low for Ω and hence weakly low for K. Wewillshowthat,ifZ isML-random and weakly low for K, thenZ satisfies the right hand side.8.1.15 Remark. A c.e. operator L is called a request operator if L X is a boundedrequest set relative to X for each set X. TheproofoftheMachineExistenceTheorem 2.2.17 can be viewed relative to an oracle: from a request operator Lone can effectively obtain a prefix-free oracle machine M = M d (d>1) suchthat ∀X ∀r, y [ 〈r, y〉 ∈ L X ↔ ∃w [|w| = r & M X (w) = y] ] .Inparticular,if 〈r, y〉 ∈ L X then K X (y) ≤ r + d. Asbefore,wecalld acodingconstantfor L. Remark 2.2.21 remains valid for request operators: suppose we build ac.e. operator L based on a given parameter d ∈ N in such a way that for eachchoice of d, L is a request operator. Then, by the oracle version of the RecursionTheorem we may assume that the coding constant d for L is given in advance.The following lemma says that, for ML-random sets, the slower the initialsegment complexity grows, the stronger is the set computationally.8.1.16 Lemma. If B is ML-random then K B (n) ≤ + K(B ↾ n ) − n for each n.Subproof. By Theorem 7.2.8 there is c ∈ N such that ∑ n 2n−K(B↾n) ≤ c.Define a request operator L as follows: for each X, n, at each stage s>0suchthat K s (X ↾ n )

8.1.17. ⋄ Exercise. (Miller and Yu, 2008) (i) Show that8.1 The class Low(Ω) 309K B (m) = + min{K(B ↾ 〈m,i〉 ) −〈m, i〉: i ∈ N}for each ML-random set B. (ii)Deducethat≤ K implies ≥ LK on MLR. (iii)Further,for each n>1, within MLR the class of n-random sets is closed downwards under ≤ LR,and closed upwards under ≤ K.Hint. (i) For the inequality “≤ + ”generalize8.1.16.Theconverseinequality“≥ + ”holdsfor every set X: defineanappropriateprefix-freemachinethatsimulatesacomputationU X (ρ) =m by processing both input bits and oracle bits. By modifying U so that itreads further oracle bits if necessary, you may assume that the use of such a computationis of the form 〈m, i〉 for some i.Each computably dominated set in Low(Ω) is computableBy Proposition 1.5.12, each low (and in fact each ∆ 0 2)computablydominatedset is computable. We say that the two lowness properties are orthogonal. Wewill prove that Low(Ω) andbeingcomputablydominatedareorthogonal.ByExercise 5.1.27 the only ∆ 0 2 sets in Low(Ω) aretheonesinLow(MLR), so thisresult is rather different from Proposition 1.5.12. Since each 2-random set is inLow(Ω), we obtain an alternative proof of Corollary 3.6.15 that no 2-random setis computably dominated.8.1.18 Theorem. (Miller and Nies) Suppose that A is low for Ω and computablydominated. Then A is computable.Proof. We will construct a computable binary tree T such that for some fixed k,for each n the level T n = {σ ∈ T : |σ| = n} has at most k elements. The set Awill be a path of T .AnypathofT is isolated, and hence computable.Fix b ∈ N such that Ω ∉ R A b.Thenthefunctiong defined byg(r) =µt ≥ r. ∀m m− b]is total and g ≤ T A.SinceA is computably dominated, there is a computablefunction f such that f(r) ≥ g(r) foreachr. WedefineT level by level. Letn>0, and suppose T n−1 has been defined. In the beginning we let T n be theset of extensions of length n of strings in T n−1 .WeobtainthefinalT n by athinning-out procedure consisting of at most 2 n rounds. Given σ of length n,we use f in an attempt to rule out that σ ≺ A. Ifwesucceed,weremoveσfrom T n .Wechallengeσ at stage r by causing K σ (Ω r ↾ m ) ≤ m − b for some mwhich exceeds n by a constant to be determined later. Then, by the definitionof f, wehaveσ ⊀ A unless Ω ↾ m changes before stage f(r). In that case weplan to account our investment to ensure that K σ (Ω r ↾ m ) is small against Otto’sinvestment to increase Ω. WebuildarequestoperatorL (see 8.1.15) to makeK σ (Ω r ↾ m ) small. As usual, our investment into a single L σ is c times his, for aconstant c ∈ N known to us in advance. In each round, if c strings of length nare challenged (in a sequential fashion) and survive, then Otto has to increase Ωeach time, so his investment is as large as ours.We actually ask that k =2c strings survive on T n for the procedure to continue,for we want at our disposal an extra weight of 1/2 ineachsetL X .Weneedthis

310 8 Classes of computational complexityextra weight, for if σ is challenged and Ω↾ m does not change then σ is removedfrom T n ,buttherequestduetoσ already is in L X . Given an oracle X, thiscan happen at most once for each length n, forσ = X ↾ n :eitherithappenswhen σ is removed from T n ,orwhenwearedonewithleveln. Theroundsare necessary because we are only allowed to contribute 2 −n−1 to L X in thatway. (This an example of a controlled risk strategy. Such strategies were alreadyused in Section 5.4.) The thinning-out procedure can have at most 2 n rounds forlevel n because each time Ω increases by 2 −n .Construction of the levels T n and the request operator L, givend ∈ N.Let k =2 b+d+1 .LetT 0 = {∅}.Suppose n>0andwehavecompletedthedefinitionofT n−1 by stage s. Letm = n + b + d +1.BeginwithT n,s = { ρa: ρ ∈ T n−1,s & a ∈ {0, 1} } .Carryoutthe thinning-out procedure, starting at stage s.(1) If #T n,s ≤ k, letT n = T n,s and return.Else start a new round by declaring all strings σ ∈ T n,s available.(2) If no string σ ∈ T n,s is available goto (1).(3) Let σ be the leftmost available string. Put the request 〈n+1, Ω s ↾ m 〉 into L σand declare σ unavailable.(4) Wait for the first stage r>ssuch that K σ (Ω s ↾ m ) ≤ n + d +1=m − b.(If d is indeed a coding constant for L then r exists.)If Ω f(r) ↾ m = Ω s ↾ m ,removeσ at stage r. Thatis,letT n,r = T n,s − {σ}.(We will show that this cannot happen if σ ≺ A.)(5) Let s := r and goto (2).Verification. At first we do not assume that d ∈ N is a coding constant for L.Claim 1. Suppose skthen Ω t − Ω s ≥ 2 −n .If σ is declared unavailable at (3) during stage u ∈ [s, t], and σ is not removedwhen the procedure reaches (4) at stage v>u,thenΩ v ↾ m ≠ Ω u ↾ m .Atleastk +1 strings are declared unavailable at some stage v, s ≤ v ≤ t. ThereforeΩ t − Ω s ≥ k2 −m =2 −n . (We need here that Ω↾ m change more than k times, forwhen the procedure for n begins at stage r, Ω r could already be very close to2 −m +0.Ω r ↾ m .InthebinaryrepresentationofΩ r there could be a long sequenceof ones from the position m on.)Claim 2. L is a request operator.Fix X ⊆ N. Arequest〈n +1, Ω s ↾ m 〉 is put into L X at (3) during a stage sbecause of some σ = X ↾ n for n>0. If σ is next declared available at stage t>sthen Ω has increased by 2 −n ,twicetheweightofourrequest.Thecontributionof such requests to the total weight of L X is therefore bounded by 1/2. If σ isnot declared available again (possibly because we have finished level n), we haveaone-timecontributionof2 −n−1 for this length n. ThusthetotalweightofL Xis at most 1/2+ ∑ n>0 2−n−1 ≤ 1, and Claim 2 is proved.

8.1 The class Low(Ω) 311From now one we may assume d is a coding constant for L. Thenastager asin (4) always exists. As a consequence, by Claim 1, the thinning-out procedurefor T n returns (after passing (1) for at most 2 n times).Claim 3. A is computable.It suffices to show that A is a path of T since each path of T is isolated andhence computable. We must verify that for each n>0thestringσ = A↾ n is notremoved from T n .Assumeσ is chosen at stage s and removed at r>s.ThenΩ f(r) ↾ m = Ω s ↾ m .Bythedefinitionofg and since g(r) ≤ f(r), there is a stage t,r ≤ t ≤ f(r), such that Kt A (Ω s ↾ m ) >m− b. Thiscontradictsthefactthatalready Kr σ (Ω s ↾ m ) ≤ n + d +1=m − b.✷A related result on computably dominated sets in GL 1Each 2-random set is generalized low (i.e., in GL 1 ) by Corollary 3.6.20. Lewis,Montalban and Nies (2007) were interested in the question whether this is stilltrue for weakly 2-random sets. As a counterexample, they found a ML-randomcomputably dominated set that is not in GL 1 .LateritturnedoutthatactuallynoML-random computably dominated set is in GL 1 !(Thiswasalreadymentionedafter Proposition 3.6.4.)To see this, let us replace in Theorem 8.1.18 being in Low(Ω) by the weakerproperty of being in GL 1 . Not all computably dominated sets in GL 1 are computable.For, by Exercise 8.2.20 below, there is a perfect Π 0 1 class of incomputablesets that are in GL 1 , and such a class has a perfect subclass of computably dominatedsets by Theorem 1.8.44.8.1.19 Theorem. (Miller and Nies) Suppose A is in GL 1 and computably dominated.Then A is not of d.n.c. degree.Proof idea. Let h ≤ T A.Wewanttoshowthath(x) =J(x) forsomex. LetΨbe a Turing functional such that A ′ = Ψ(A ⊕∅ ′ ). The function g defined byg(r) =µt ≥ r. ∀k

312 8 Classes of computational complexity∀σ ∀e [Γ σ (e) ≃ J σ (p(e))] and ∀j [∆(j) ≃ J(q(j))].For Γ we only need the input 0: we define Γ σ (0) for lots of strings σ. Wewrite0for p(0). Since A ′ (0) = Ψ(A ⊕∅ ′ ; 0) we can wait for a computation Ψ(α ⊕∅ ′ ; 0)at a stage t>0. Let j = 〈α,t〉. Weattempttoruleoutanystringσ ≽ α suchthat y = Θ σ (q(j)) converges by defining Γ σ (0) at a stage s>t.Wewaitfortheresponse J σ (0) ↓ at stage r>s, and then look ahead till stage f(r): if ∅ ′ haschanged below the use of Ψ(α ⊕∅ ′ ; 0), then because this is a one-time event forj = 〈α,t〉, weareallowedtouseupj as an input for ∆. Wedefine∆(j) =y.Then x = q(j) isasrequired.If∅ ′ has not changed then σ ⊀ A.Construction of a Turing functional Γ and a partial computable function ∆, givencomputable functions p and q.At stage t>0=p(0), if there is a new convergence Ψ(α ⊕∅ ′ , 0)[t] = 0, where αis shortest for this computation, let m be its use on the ∅ ′ -side, let j = 〈α,t〉,and start the procedure P (α,t)whichrunsatstagess>t. Several proceduresmay run in parallel.Procedure P (α,t):(1) If at stage s>tthere is a string σ ≽ α such that Γ σ (0)↑ andy = Θ σ s (q(j)) ↓, letσ be the least such and define Γ σ (0) = 1(the output is irrelevant).(2) Wait for the first stage r>ssuch that J σ (0)↓. If ∅ ′ f(r) ↾ m≠ ∅ ′ t ↾ m ,define∆(j) =y and end the procedure P (α,t), else goto (1).Verification. We claim that there is x such that J(x) =h(x). First we showthat Γ A (0)↓. Otherwise,lett be the least stage > 0suchthatthecomputationΨ(A ⊕∅ ′ , 0)[t] =0isstable.Letα be the initial segment of the oracle A used bythis computation, and let m be its use on the ∅ ′ side. We start procedure P (α,t)at stage t. Let j = 〈α,t〉, and let σ ≺ A be least such that Θ σ (q(j))↓. Wheneverwe enter (2) for some σ ′ ,wegetbackto(1)because∅ ′ t ↾ m is already stable. So,eventually it is the turn of σ, andwedefineΓ σ (0) = 1, contradiction.We may conclude that we define Γ σ (0) = 1 for some least σ ≺ A in (1) of someprocedure P (α,t). We use the notation of the construction for this procedure: letj = 〈α,t〉, x = q(j) andsupposer>sis least such that Jr σ (0)↓. Bythedefinitionof g and since 0

8.2 Traceability 313states that each value f(n) is in a finite set T n that can be determined from nin an effective way (see the chapter introduction for more background). We willstudy four traceability properties. The first two are covered in this section: c.e.traceability and computable traceability, where only total functions computedby A are traced. Each c.e. traceable set is array computable, a weak lownessproperty introduced in 8.2.7 below. Each computably traceable set is computablydominated.In Section 8.4 we study jump traceability and strong jump traceability, wherealso functions that are partial computable in A are traced.In this section and Section 8.4 we consider traceability for its own sake. InSections 8.3 and 8.5 we will see that traceability is closely related to lowness forrandomness notions.Recall that an order function is a nondecreasing unbounded computable function.From now on we will assume that order functions only have positive values.8.2.1 Definition. Let h be an order function. A trace with bound h is a sequence(T n ) n∈N of nonempty sets such that #T n ≤ h(n) foreachn. Wesaythat(T n ) n∈Nis a trace for the function f : N → N if f(n) ∈ T n for each n. Wecall(T n ) n∈N ac.e. trace if (T n ) n∈N is uniformly computably enumerable.The following notion was first studied by Ishmukametov (1999).8.2.2 Definition. We say that a set A is c.e. traceable if there is an orderfunction h such that each function f ≤ T A has a c.e. trace with bound h.The point is that all the functions f ≤ T A can be traced via the same bound.It suffices to require that there is n 0 such that f(n) ∈ T n for all n ≥ n 0 ,forwe can modify the trace by letting T n = {f(n)} for n

314 8 Classes of computational complexityc.e. traces, no matter how slowly it grows. This result of Terwijn and Zambella(2001) relies on the fact that one traces only total functions.8.2.3 Theorem. Ac.e.traceablesetisc.e.traceableviaeveryorderfunction.Proof. Recall that on page 67 we defined a computable injection D : N ∗ → N byD(m 0 ,...,m k−1 )= ∏ k−1i=0 pmi+1 i = α where p i is the i-th prime number. We willwrite α(i) form i . If f ∈ N N and k ∈ N, wewillwritef ↾ k for D(m 0 ,...,m k−1 ),where m j = f(j) forj

8.2 Traceability 3158.2.6 Proposition. If A has d.n.c. degree then A is not c.e. traceable.Proof. Assume for a contradiction that A is c.e. traceable. By Theorem 4.1.9there is a nondecreasing unbounded function g ≤ T A such that ∀n [g(n) ≤K(A ↾ n )]. Let f ≤ T A be the function given by f(r) =A ↾ h(r) where h(r) =min{n: g(n) ≥ r}. Ifn = h(r) thenf(r) =A ↾ n ,soK(f(r)) ≥ g(n) ≥ r foreach r. By Theorem 8.2.3 there is a trace (T r ) r∈N for f such that #T r ≤ r +1for each r. Butf(r) ∈ T r implies K(f(r)) ≤ + 4logr, contradiction. ✷Kjos-Hanssen, Merkle and Stephan (20xx) proved a stronger result: A is c.e. traceable ⇔there is an order function q showing that each f ≤ T A strongly fails to be a d.n.c.function, in the sense that for each k we have k ≤ #{e

316 8 Classes of computational complexity8.2.12. Some low c.e. set A is not c.e. traceable. Explain why your construction doesnot make A superlow.8.2.13. (Nies, 2002) By 1.5.8, a set A is in Low 2 iff Tot A = {e: Φ A e total} is Σ 0 3.Show that a c.e. traceable ∆ 0 2 set A is uniformly in Low 2 in that, from a computableapproximation (A s) s∈N of A, one can effectively obtain a Σ 0 3-index for Tot A .8.2.14. ⋄ Problem. Decide whether each weakly 2-random set is array computable.Computably traceable setsTerwijn and Zambella (2001) introduced a further notion of traceability. One hascomplete information about the components of the traces. Recall from 1.1.14 that(D n ) n∈N is an effective listing of the finite subsets of N.8.2.15 Definition. A trace (T n ) n∈N is called computable if there is a computablefunction g such that T n = D g(n) for each n. WesaythatA is computablytraceable if there is an order function h such that each function f ≤ T A has acomputable trace with bound h.The following fact of Kjos-Hanssen, Nies and Stephan (2005) characterizes thecomputably traceable sets within the class of c.e. traceable sets.8.2.16 Proposition. A is computably traceable ⇔ A is c.e. traceable and computablydominated.Proof. ⇒: Each computable trace (D g(n) ) n∈N is a c.e. trace. If it is a trace forafunctionf then f is dominated by the computable function λn.max D g(n) .⇐: Suppose A is c.e. traceable via the bound h. Letf ≤ T A,andlet(T n ) n∈Nbe a c.e. trace for f with bound h. Letp(n) =µs.f(n) ∈ T n,s ,thenp ≤ T A.Since A is computably dominated, we may choose a computable function r suchthat r(n) ≥ p(n) foreachn. Then(T n,r(n) ) n∈N is a computable trace for f withbound h.✷Recall from page 56 that a nonempty closed class C ⊆ 2 N is called perfect ifthe corresponding tree T C = {x: [x] ∩ C ≠ ∅} has no isolated paths. We willbuild a perfect class C of computably traceable sets. Since a perfect class hascardinality 2 ℵ0 ,thisimpliesthatthereisacomputablytraceablesetthatisnotcomputable. In fact we will obtain such a set in ∆ 0 3.8.2.17 Theorem. There is a perfect class C of computably traceable sets. Moreover,the corresponding tree {x: [x] ∩ C ≠ ∅} is ∆ 0 3.The proof is similar to the proof of Theorem 1.8.44, yet we cannot expect tomake C asubclassofanarbitrarygivenΠ 0 1 class without computable members.For instance, the class 2 N − R 1 of ML-random sets does not even contain a c.e.traceable set by Proposition 8.2.6. However, we can ensure that C is a subclassof a given perfect class P such that T P is computable.Proof. We need an alternative notation for the tree representation of perfectclasses given by 1.8.3. A function tree is a map F : {0, 1} ∗ → {0, 1} ∗ such thatfor each σ, both F (σ0) and F (σ1) properly extend F (σ), and F (σ0) < L F (σ1).

8.2 Traceability 317Function trees correspond to binary trees without dead branches or isolatedpaths; if F is a function tree, the corresponding binary tree is B F = {x: ∃σ [x ≼F (σ)]}. NotethatB F ≡ T F .LetPaths(F )={Z : ∀n∃σ [Z ↾ n ≼ F (σ)]}. Theperfect classes are precisely the ones of the form Paths(F ) for a function tree F .We define a ∅ ′′ -computable sequence of computable function trees (F e ) e∈Nsuch that Paths(F e+1 ) ⊆ Paths(F e )foreache. ForF 0 we may choose an arbitrarycomputable function tree, for instance the identity function on {0, 1} ∗ .The class C will be a perfect subclass of ⋂ e Paths(F e ).For each e, weletF e+1 copy F e on strings of length up to e. For each σ oflength e+1, all paths Y of F e+1 extending F e+1 (σ) behave in the same way withregard to Φ e :eitherΦ Y e is partial for each such Y ,orthereisasinglecomputabletrace for each Φ Y e with bound h(n) =2 n .Inductive definition of the computable function trees F e for e ∈ N.Step e +1. Suppose that F e has been determined. For each ν such that |ν| ≤ e,let F e+1 (ν) =F e (ν).Let S e be the set of strings σ of length e +1suchthat∀η ≽ σ ∀n ∃ρ ≽ η [ Φ F e (ρ)e,|ρ|(n)↓ ]. (8.1)For each σ of length e +1wedefineF e+1 on all strings extending σ:If σ ∉ S e ,letη ≽ σ and n ∈ N be a pair of witnesses for the failure of (8.1),and define F e+1 (σα) =F e (ηα) foreachstringα. ThisensuresthatΦ Y (n)↑ foreach path Y of F e+1 extending F e+1 (σ).If σ ∈ S e ,defineF e+1 (σα) byinductiononn = |α|: letF e+1 (σ) =F e (σ). IfF e+1 (σα) =F e (η) has been defined, then for each r ∈ {0, 1}, letF e+1 (σαr) =F e (ρ), where ρ ≽ ηr is least such that Φ F e (ρ)e,|ρ|(n) ↓. Notethatρ exists sinceσ ∈ S e . This completes the inductive definition.For each e, F e+1 is computable. Moreover, using ∅ ′′ as an oracle, we can computeS e ,andhencefindaTuringprogramcomputingF e+1 .WedefineafunctionG ≤ T ∅ ′′ by G(σ) =F |σ| (σ). Then G is a function tree since F e+1 (ν) =F e (ν)for |ν| = e, andF e+1 (ν0) < L F e+1 (ν1). Thus C = Paths(G) isperfect.We show that each Y ∈ C is computably traceable with bound λn.2 n .Givene,let σ be the string of length e +1suchthatF e+1 (σ) ≺ Y .Ifσ /∈ S e then Φ Y e ispartial. Otherwise let g be the computable function such thatD g(n) = {Φ F e+1 (σα)e,|F e+1 (σα)|(n): |α| = n}.Then #D g(n) ≤ 2 n for each n, and(D g(n) ) n∈N is a computable trace for Φ Y e .Note that G ≡ T ran G ≡ T {x: [x] ∩ C ≠ ∅} by Exercise 8.2.19, so the set{x: [x] ∩ C ≠ ∅} is ∆ 0 3.✷8.2.18 Corollary. There is a computably traceable set A in ∆ 0 3 that is not computable.Proof. Let the function tree G be as above. We define inductively a ∆ 0 3 sequenceof strings σ 0 ≺ σ 1 ≺ ... such that |σ e | = e, andletA = ⋃ e G(σ e). If σ e has been

318 8 Classes of computational complexitydetermined, using ∅ ′′ as an oracle, check whether G(σ e 0)(k) =1 ↔ Φ e (k) =1for each k

8.2 Traceability 319Suppose M is an oracle prefix-free machine such that Ω A M is A-computable.We actually build a c.m.m. N such that ∀y [ KM A (y) ≥ K N(y) − 1 ] .Fortn 2m 2 −2m =2 −n−1 .Thus w is a computable real number by Fact 1.8.15. Also, w 0 ≤ 1/2 since#D g(0) =1,sow ≤ 1. If N is the prefix-free machine obtained from L throughthe Machine Existence Theorem 2.2.17, then Ω N = w, soN is a computablemeasure machine. Clearly N is as required.⇐: Suppose that f ≤ T A.LetM be the prefix-free machine defined byM(0 |x| 1x) =f(x).Then K M (f(x)) ≤ + 2logx for each x. Sincethestringsoflengthn contribute2 n 2 (−2n+1) = 2 −n−1 to the measure of dom M, thismeasureiscomputable.In particular, M is a c.m.m. relative to A. Hencethereisac.m.m.N suchthat ∀yK N (y) ≤ + K M (y). By Exercise 3.5.17 a strong index for the finite set{y : K N (y) ≤ r} can be computed from r. Letg be the computable functionsuch that D g(x) = {y : K N (y) ≤ 3logx}. Notethat#D g(x) ≤ 2x 3 +1. Moreover,f(x) ∈ D g(x) for almost all x. Thus,afinitevariantof(D g(x) ) x∈N is a computabletrace for f with bound λx.2x 3 +1.✷Exercises.8.2.24. Show that a variation of the preceding result yields a characterization of thec.e. traceable sets in terms of oracle prefix-free machines: A is c.e. traceable ⇔ foreach c.m.m. M relative to A, thereisb such that ∀y [KM A (y) ≥ K(y) − b].8.2.25. We say that A is Schnorr trivial (Downey, Griffiths and LaForte, 2004) if foreach c.m.m. M there is a c.m.m. N such that ∀nK N (A↾ n) ≤ + K M (n). Show that if Ais low for c.m.m. then A is Schnorr trivial.Facile sets as an analog of the K-trivial sets ⋆We will characterize the computably traceable sets within the class of computablydominated sets by a property defined in terms of a growth condition onthe initial segment complexity.

320 8 Classes of computational complexity8.2.26 Definition. Z is facile if for each order function h,∀n [ K(Z ↾ n | n) ≤ + h(n)].By Theorem 7.4.11 a facile set can be computably random. By the next twofacts, similar to the K-trivial sets, the facile sets determine an ideal in the weaktruth-table degrees (see Definition 1.2.27 for ideals).8.2.27 Proposition. If A is facile and B ≤ wtt A then B is facile as well.Proof. Suppose that B = Φ A and ∀n [ use Φ A (n) ≤ f(n)] for a computable f.We may assume that f is strictly increasing. A prefix-free description of B ↾ ngiven n may be obtained from a prefix-free description σ of A ↾ f(n) given n viathe binary machine M defined by M(σ,n) ≃ Φ U2 (σ,n) ↾ n .Thus,K(B ↾ n | n) ≤ + K(A↾ f(n) | n) ≤ + K(A↾ f(n) | f(n))(for the second inequality, we have used that f is computable and strictly increasing).If h is an order function, then ĥ(r) =h(min{n: f(n) ≥ r})isanorderfunctionas well (without the extra condition that ĥ(0) > 0). Also, ĥ(f(n)) = h(n) foreach n. SinceK(A↾ r | r) ≤ + ĥ(r) foreachr, weobtainthatK(B ↾ n| n) ≤ + h(n)for each n.✷8.2.28 Proposition. If A and B are facile, then A ⊕ B is facile.Proof. It suffices to observe that, for each n, wehaveK(A ⊕ B ↾ 2n | 2n) = + K(A ⊕ B ↾ 2n | n) ≤ + K(A↾ n | n)+K(B ↾ n | n).We will show that each c.e. traceable set A is facile. In fact, a weaker propertysuffices. We say that set A is weakly c.e. traceable if Definition 8.2.2 holds restrictedto the functions f ≤ T A that are bounded from above by a computablefunction. As before, the choice of the bound for the trace does not matter as longas it is an order function, because in the proof of Theorem 8.2.3, if f boundedby a computable function then so is ˜f.8.2.29 Theorem. A is weakly c.e. traceable ⇔ each set Z ≤ T A is facile.Proof. ⇒: Suppose Z ≤ T A, and an order function h is given. The function λn.Z ↾ nis computable in A, and dominated by the function λn.2 n .Thereforeithasac.e.trace (T n) n∈N with bound h. Givenn, wemaydescribez = Z ↾ n by the numberi ≤ h(n) such that z is the i-th string enumerated into T n.SinceK(i) ≤ + 2logi wehave K(z | n) ≤ + h(n).⇐: Suppose that f ≤ T A and ∀nf(n) ≤ b(n) where b is an order function. For r ∈ N,let n r = ∑ i

8.3 Lowness for randomness notions 321obtain ∀ ∞ rK(Z ↾ nr | r) ≤ r, and hence ∀ ∞ r [ K(Z ↾ nr ) ≤ r + 2 log r ]. This leads tothe desired trace (T r) r∈N for f: given r, for each σ such that |σ| ≤ r + 1 + 2 log(r +1),if U(σ) =z where |z| = n r+1, then enumerate #{i ∈ [n r,n r+1): z(i) =1} into T r.Clearly f(r) ∈ T r for almost all r. While(T r) r∈N depends on the sequence (n r) andhence on the computable bound b for f, thesize #T r is bounded by 2 cr for a fixed c,independent of such a computable bound for f.✷We are now able to characterize computable traceability within the class ofcomputably dominated sets.8.2.30 Corollary. Let A be computably dominated. ThenA is computably traceable ⇔ A is facile.Proof. By 1.5.11, Z ≤ T A ⇔ Z ≤ wtt A for each Z. By8.2.27,beingfacileisclosed downward under ≤ wtt , so by Theorem 8.2.29 A is weakly c.e. traceableiff A is facile. Finally, by 8.2.16, for computably dominated sets A we have:A is computably traceable ⇔ A c.e. traceable ⇔ A weakly c.e. traceable. ✷Franklin and Stephan (20xx) have given an alternative characterization of thiskind: let A be computably dominated. Then A is computably traceable ⇔ Ais Schnorr trivial (as defined in Exercise 8.2.25). Note that “⇒” holdsforeachset by 8.2.23 and 8.2.25. On the other hand, a Schnorr trivial set can be Turingcomplete by Downey, Griffiths and LaForte (2004).Each K-trivial set is c.e. traceable by Exercise 8.2.11, and hence facile. The conversefails. For instance, no K-trivial set is Schnorr random, otherwise, being nonhigh, itwould already be ML-random (5.5.10). So the facile computably random set of Theorem7.4.11 is not K-trivial. Alternatively, there are 2 ℵ 0many computably traceable andhence facile sets, while each K-trivial set is ∆ 0 2.Exercises. Show the following.8.2.31. Each facile set Z is O(log) bounded (as defined in 7.6.5, page 289).8.2.32. If Z is facile then ∃ ∞ nK(Z ↾ n)

322 8 Classes of computational complexity(1) Low(MLR, SR) ⇔ c.e. traceable (Thm. 8.3.3)(2) Low(CR, SR) ⇔ comp. traceable (Thm. 8.3.7)(3) Low(MLR, CR) ⇔ low for K (Thm. 8.3.10).The implications “⇐” are the easier ones. For (3), we have already seen inFact 5.1.8 that each set that is low for K is low for ML-randomness. For (1) and(2), we use that the relevant traceability notion can be expressed in terms of computablemeasure machines (Exercise 8.2.24 and Theorem 8.2.23, respectively).Then we apply Theorem 3.5.19 that Schnorr randomness can be described bysuch machines. In the case of (2), this argument actually establishes that eachcomputably traceable set is in Low(SR); see Proposition 8.3.2.The implications “⇒” areharder.Wederive(2)from(1)andLemma8.3.8thateach set in Low(CR, SR) iscomputablydominated.Toprove(1)and(3),webuild a test in the sense of D A ,andusethefactthatnosetZ on which thetest succeeds is in C. Wefirstderivefromthisfactaconditiononfiniteobjects(we call such a condition combinatorial). This enables us to obtain the desiredlowness property; see Lemmas 8.3.4 and 8.3.13, respectively. For instance, in (3),to show that each set in A ∈ Low(MLR, CR) islowforK, webuildaQ 2 -valuedmartingale L A ≤ T A. By the hypothesis, Succ(L A ) ⊆ non-MLR. Fromthiswederive in Lemma 8.3.13 a combinatorial condition stating, roughly, that [x] ⊆ R 1for any string x such that N(x) exceeds a certain value 2 d .The technique to derive a combinatorial condition from the hypothesis that A islow for some randomness notion(s) was first used in (i)⇒(iii) of Theorem 5.1.9.The proof is always by contraposition: if the condition fails, one can build aset Z that shows A does not satisfy the lowness property. The condition iscombinatorial because at some point the attempted definition of Z by finiteextensions must stop.The characterizations of Low(SR)andLow(CR)arenoweasilyobtained.By(2),A ∈ Low(SR)impliesthatA is computably traceable, and the converse also holds.By (2) and (3), A ∈ Low(CR) iff A is computable, because computably traceablesets are computably dominated while sets that are low for K are ∆ 0 2.Lowness for C-null classesA randomness notion C is usually introduced by specifying a notion of a C-test.We say that S ⊆ 2 N is a C-null class if there is a C-test such that all the sets inS fail the test. For instance, S is a Schnorr null class if there is a Schnorr test(G m ) m∈N such that S ⊆ ⋂ m G m.A C-test is called universal if the sets that fail the test form the largest C-nullclass. If there is a universal C-test, then S is a C-null class iff S ∩ C = ∅.The following is a lowness property defined in terms of a covering procedure(see page 226).8.3.1 Definition. We say that A is low for C-null classes if each C A -null classalready is a C-null class.Since a set Z ∉ C A is in some C A -null class, this implies C ⊆ C A ,namely,Ais low for C. BeinglowforC-null classes appears to be stronger than being low

8.3 Lowness for randomness notions 323for C: itcouldbethatC ⊆ C A because each Z ∉ C A fails some C-test thatdepends on Z. IfthereisauniversalC-test then the two are clearly equivalent.For deeper reasons, lowness for C-null classes and lowness for C also coincidewhen C is SR, CR or W2R. ForW2R this was shown already in Theorem 5.5.17.In this section we provide the proof for SR, andofcourseforCR it follows fromthe result that each set in Low(CR) is computable. No randomness notion C isknown for which being low for C-null classes is stronger than being low for C.The following uses computable measure machines (c.m.m.) via Theorem 8.2.23.8.3.2 Proposition. Each computably traceable set A is low for Schnorr nullclasses.Proof. As in Definition 3.2.6, for an oracle prefix-free machine M and an oracleX, weletR M,Xb=[{x ∈ {0, 1} ∗ : KM X (x) ≤ |x| − b}]≺ . Let (G m ) m∈N be aSchnorr test relative to A. ByLemma3.5.18relativizedtoA, thereisac.m.m.M relative to A such that ⋂ m G m ⊆ ⋂ d RM,A d. By Theorem 8.2.23, there is ac.m.m. N and b ∈ N such that ∀y [KM A (y) ≥ K N(y) − b]. Then (Rk N) k∈N is aSchnorr test such that ⋂ m G m ⊆ ⋂ k RN k .✷The class Low(MLR, SR)The main result of Terwijn and Zambella (2001) states that lowness for Schnorrnull classes is equivalent to computable traceability. (The easier implication “⇐”has already been proved in Proposition 8.3.2.) This result represented an importantadvance, characterizing for the first time lowness for tests by a computabilitytheoretic property.Kjos-Hanssen, Nies and Stephan (2005) extended this, answering a questionof Ambos-Spies and Kučera (2000): actually, lowness for Schnorr randomness isequivalent to computable traceability. To get around the problem that there isno universal Schnorr test, they proceeded indirectly.Firstly, they proved that A ∈ Low(MLR, SR) ⇔ A is c.e. traceable, by adaptingthe measure-theoretic combinatorics of Terwijn and Zambella for their implication“⇒”.Secondly, they used a lemma of Bedregal and Nies (2003) that each set A inLow(CR, SR) iscomputablydominated.Eachc.e.traceablecomputablydominatedset is already computably traceable by Proposition 8.2.16. Therefore eachset in Low(SR) is computably traceable.8.3.3 Theorem. A ∈ Low(MLR, SR) ⇔ A is c.e. traceable.Proof. Recall from page 71 that for a string v and a measurable class C, weletC | v = {X : vX ∈ C}. Furthermore,λ(C | v) =2 |v| λ(C ∩ [v]) as in Definition1.9.3. We will also use the notation λ v (C) asashortformofλ(C |v).⇐: If Z is not Schnorr random in A, then by Theorem 3.5.19 there is a computablemeasure machine M relative to A such that ∀b ∃n K A M (Z ↾ n) ≤ n − b.

324 8 Classes of computational complexitySince A is c.e. traceable, by Exercise 8.2.24 K(y) ≤ + KM A (y) foreachy. HenceZis not ML-random.⇒: Given g ≤ T A,wewilldefineac.e.trace(T n ) n∈N for g with bound λn.2 n .We may assume that g(n) > 0andn | g(n) (that is, n divides g(n)) for each n.Otherwise, we replace g by the function g = λn.ng(n)+n, and a c.e. trace for gcan be turned into a c.e. trace for g with the same bound.The basic idea is to use a Schnorr test (U g d ) d∈N relative to A which contains asufficient amount of information about g. Foreachd, theopensetU g dconsistsof the sets Z such that for some n>d,thereisarunofn zeros starting atposition g(n). By the hypothesis on A, wehave ⋂ d U g d⊆ Non-MLR, andinparticular ⋂ d U g d ⊆ R where R = R 2 =[{x ∈ {0, 1} ∗ : K(x) ≤ |x| − 2}] ≺ .Recall that λR ≤ 1/4. We first need a lemma saying that, when we restrictourselves to an appropriate basic open cylinder [v], then λ v (R) issmallbutλ v (U g d − R) = 0 for some d. Thetraceforg will be obtained from U g dand v. Thelemma holds in more generality.8.3.4 Lemma. Let ɛ =1/4, andlet(U d ) d∈N be a sequence of open sets suchthat ⋂ d U d ⊆ R. Then there is a string v and d ∈ N such that λ v (R) ≤ ɛ andλ v (U d − R) =0.Subproof. Assume the Lemma fails. We define inductively a sequence of strings(v d ) d∈N such that v 0 ≺ v 1 ≺ ...and ∀d λ(R | v d ) ≤ ɛ. Letv 0 be the empty string.Suppose v d has been defined and λ(R | v d ) ≤ ɛ. Thenλ((U d − R) | v d ) > 0sincethe Lemma fails, so we can choose y such that [y] ⊆ U d and λ([y] − R | v d ) > 0;in particular, y ≽ v d .ByTheorem1.9.4wemaychoosev d+1 ≻ y such thatλ(R | v d+1 ) ≤ ɛ. NowletZ = ⋃ d v d,thenZ ∈ ⋂ d U d and Z ∉ R.✸Next we provide the details on how to code information about g into a Schnorrtest relative to A. Foreachn, k ∈ N, letB n,k =[{x0 n : |x| = k}] ≺ .Thus, B n,k is the clopen class consisting of the sets that have a run of n consecutivezeros from position k on. LetU g d = ⋃ n>d B n,g(n).Claim 1. (U g d ) d∈N is a Schnorr test relative to A.Since g ≤ T A and λB n,k =2 −n ,(U g d ) d∈N is a ML-test relative to A. Toshowitis a Schnorr test relative to A, wehavetoverifythatλU g dis computable in Auniformly in d. Let q d,t = λ ⋃ t≥n>d B n,g(n), then q d,t ∈ Q 2 ,thefunctionλt. q d,tis uniformly computable in A, andλU g d − q d,t ≤ λ ⋃ n>t B n,g(n) ≤ 2 −t . ✸We think of T n as the set of k such that B n,k − R has small measure, in asense to be specified (which depends on n). Since B n,k − R generally tends tohave large measure, there can only be very few k for which B n,k − R has smallmeasure. This leads to the bound on #T n .What does this have to do with the function g? By the hypothesis, ⋂ d U g d ⊆ R,so choose v, d for this test as in Lemma 8.3.4. We will actually carry out the idea

8.3 Lowness for randomness notions 325sketched above within [v]. By Lemma 8.3.4, for n>d, B n,g(n) − R is null in [v],so g(n) is traced.For the formal details, let p = |v|. SinceU g d ⊇ U g d+1for each d, wemayassumethat d ≥ p, whichimpliesthatg(n) ≥ n ≥ p for each n ≥ d.Let ˜R = R | v. Forn ≤ d, let˜g(n) = 0 and ˜T n = {0}; forn > d,let˜g(n) =g(n) − p and˜T n = {k : n|k + p & λ(B n,k − ˜R) < 2 −(k+4) }.In the remaining claims we show that ( ˜T n ) n∈N is a c.e. trace for ˜g with boundλn.2 n .Then(T n ) n∈N is a c.e. trace for g with the same bound, where T n = {g(n)}for n ≤ d, andT n = {k + p: k ∈ ˜T n } for n>d.Claim 2. ∀n ˜g(n) ∈ ˜T n .Firstly, by hypothesis n | g(n) =˜g(n)+p. Secondly, since B n,k | v = B n,k−p foreach k ≥ p,U g d | v = ⋃ n>d B n,g(n) | v = ⋃ n>d B n,˜g(n) = U ˜g d .By the choice of v, thisimpliesthatλ(U ˜g d − ˜R) =λ v (U g d− R) =0,whence˜g(n) ∈ ˜T n for each n>d.✸Claim 3. ( ˜T n ) n∈N is a c.e. trace with bound λn.2 n .For a Π 0 1 class P and m ∈ N, λP

326 8 Classes of computational complexity(i) (Kjos-Hanssen) (a) If A does not have d.n.c. degree then A ∈ Low(MLR, WR).(b) Conclude that if Z is ML-random and D ⊆ Z is infinite then D has d.n.c. degree.(ii) (Greenberg and Miller) If A has d.n.c. degree then A computes an infinite subset Dof some ML-random set. In particular, A ∉ Low(MLR, WR).Classes that coincide with Low(SR)We give a number of characterizations for the class of sets that are low for Schnorrrandomness. However, the main technical result of this subsection is on lownessfor two randomness notions:8.3.7 Theorem. A ∈ Low(CR, SR) ⇔ A is computably traceable.Proof. ⇐: A is low for Schnorr null classes by 8.3.2, so A ∈ Low(CR, SR).⇒: A is in Low(MLR, SR), and hence c.e. traceable by Theorem 8.3.3. By Proposition8.2.16, each c.e. traceable computably dominated set is computably traceable.So the following lemma of Bedregal and Nies (2003) suffices.8.3.8 Lemma. If A ∈ Low(CR, SR) then A is computably dominated.To prove the lemma, suppose that A is not computably dominated. We will buildasetZ that is computably random and not Schnorr random relative to A. Forthe former, we ensure that sup n M(Z ↾ n ) < ∞ whenever M is a computableQ 2 -valued martingale. For the latter, we define a bounded request set relativeto A with total weight (a real number) computable in A, insuchawaythat,if N is the associated c.m.m. relative to A, then∀b ∃x ≺ Z [ K A N (z) ≤ z − b] .The uniform listing (M e ) e∈N of the Q 2 -valued partial computable martingaleswas introduced on page 273. Recall that TMG = {e: M e total}.The basic strategy to make Z computably random but not Schnorr randomrelative to A is not unlike the one used in the proof of Theorem 7.4.8. We ensurethat M e (Z) < ∞ whenever M e is total. In the following α, β, γ denote finitesubsets of N. Givenα, in order to beat all the martingales M e for e ∈ α together,we consider a linear combination M α with positive rational coefficients of thosemartingales. For all α ⊂ TMG (and inevitably some others) we will define stringsx α in such a way that x α ≺ x α∪{e} when e>max(α) andx α∪{e} is defined. Weensure that, for each total M e , each γ containing e and each x ≺ x γ , M e (x) isbounded by a constant only depending on e. ThenthesetZ = ⋃ {x TMG∩{0,...,e} : e ∈ N}is computably random.Fix a function g ≤ T A not dominated by any computable function. If α ⊂ TMG,there are infinitely many m such that, for all x of length at most m, M α (x) hasconverged by stage g(m). If α = β ∪ {e} where e>max(β), we define x α to bethe leftmost non-ascending path of M α of length m above x β .ThesetA canrecognize whether a string of length m is x α .Therearefewx α of each length,so a computable measure machine relative to A can assign short descriptions toeach of them.

8.3 Lowness for randomness notions 327Let ˜M e = M e,0 where M e,0 was defined in Fact 7.4.3; thus ˜M e is M e scaledin such a way that the start capital is 1 (unless M e (∅) = 0). For each α letn α = ∑ e∈α 2e .Inductive definition of partial computable martingales M α and strings x α .We maintain the condition that if x α is defined, thenM α (x α )convergesing(|x α |)stepsandM α (x α ) < 2. (8.2)Let x ∅ = ∅, andletM ∅ be the constant zero function. We may assume thatM ∅ (∅) convergesing(0) steps by increasing g(0) if necessary, so (8.2) holds forα = ∅. Nowsupposeα = β ∪ {e} where e>max(β) ifβ ≠ ∅, andinductivelysuppose that (8.2) holds for β. LetM α = M β + p α ˜Me where the binary rationalp α > 0 is chosen in such a way that M α (x β ) < 2: since ˜M e (x) ≤ 2 |x| for each x,it is sufficient to let p α =2 −|x β|−1 (2 − M β (x β )).To define x α ,letm>|x β |, m ≥ 2 nα +20beleastsuchthatM j (z) convergesin g(m) stepsforeachj ∈ α and |z| ≤ m. Choosex α ≻ x β of length m in suchawaythatM α (yr) ≤ M α (y) foranystringy ≽ x β and r ∈ {0, 1} such thatyr ≼ x α . If m fails to exist then leave x α undefined. This completes the inductivedefinition.Claim 1. If α ⊂ TMG then x α is defined.This is clear for α = ∅.Supposenowtheclaimholdsforβ,andthatα = β∪{e} ⊂TMG where e>max(β) ifβ ≠ ∅. Sincethefunctionf defined byf(m) =µs. ∀e ∈ α∀x [ |x| ≤ m → M e (x)↓ [s] ]is computable, there is a least m>|x β |, m ≥ 2 nα +20suchthatg(m) ≥ f(m).Since M α is a martingale, we have M α (y0) ≤ M α (y) orM α (y1) ≤ M α (y) forany y. Sowecanchoosex α as required.✸Claim 2. Z is computably random.Suppose M e is total, and let α = TMG ∩ {0,...,e}. Weneedtoshowthatsup{M e (x): x ≺ Z} < ∞. Givenx, letk ∈ TMG be largest such that x γ ≼ xwhere γ = TMG ∩ {0,...,k}. Leti>kbe least such that i ∈ TMG. Thusx ≺ x γ∪{i} ≺ Z. Wemayassumethatx α ≼ x γ .Thenp α ˜Me (x) ≤ M γ (x) ≤ M γ (x γ ) < 2by (8.2) for γ and the definition of x γ∪{i} .SoM e (x) is bounded by a constant.✸Claim 3. Z is not Schnorr random relative to A.We have x α ≺ Z for infinitely many α, soby3.5.19itissufficient to show that{〈|x α | − n α ,x α 〉: x α is defined}is a bounded request set relative to A with total weight w computable in A.Let r m = ∑ α 2nα−m [[|x α | = m]], then w = ∑ m≥20 r m.Sinceg ≤ T A we have{〈y, α〉: y = x α } ≤ T A.Thusthefunctionλm. r m is computable in A. Since

328 8 Classes of computational complexity|x α | ≥ 2 nα for each α, thesumdefiningr m contains at most m terms (far fewer,in fact). Then, since m ≥ 20, and since log m ≥ n α for each index α in the sum,r m ≤ m2 log m−m ≤ m 2 2 −m ≤ 2 −m/2 ,sothat ∑ m≥i+1 r m ≤ 2 −i/2 for each i.This shows that the real number w is computable in A.✷We summarize the characterizations of lowness for Schnorr randomness.8.3.9 Theorem. The following are equivalent for a set A.(i) A is computably traceable (Definition 8.2.15).(ii) A is low for computable measure machines (8.2.22).(iii) A is low for Schnorr null classes (8.3.1).(iv) A is low for Schnorr randomness.(v) A ∈ Low(CR, SR).Note that (i)⇔(ii) is Theorem 8.2.23, (i)⇒(iii) is Proposition 8.3.2, the implications(iii)⇒(iv)⇒(v) are trivial, and (v)⇒(i) holds by Theorem 8.3.7.By way of analogy, property (i) corresponds to K-triviality, (ii) to being lowfor K, (iv) to Low(MLR) and(v)toLow(W2R, MLR).Low(MLR, CR) coincides with being low for KThe following theorem of Nies (2005b) concludes a series of preliminary results.The technique to wait till some clopen set is contained in a fixed c.e. open set Rsuch that λR

8.3 Lowness for randomness notions 329⇒: First we recall some notation and facts. For the remainder of this proof, anopen set S ⊆ 2 N is identified with the set of strings y such that [y] ⊆ S. Thuswewrite y ∈ S instead of [y] ⊆ S. ForR ⊆ {0, 1} ∗ we write λR instead of λ[R] ≺ .We let R = R 1 =[{x ∈ {0, 1} ∗ : K(x) ≤ |x| − 1}] ≺ .Thereisacomputableenumeration (R s ) s∈N of R such that R s contains only strings of a length up to s,and R s is closed under extensions within those strings (see (1.16) on page 54).We may also assume that x0,x1 ∈ R s → x ∈ R s for each string x. Forastringvand a measurable class C, we use the notation λ v (C) insteadofλ(C | v). ByExercise 3.3.4 there is a computable function g such that for each string v,v ∉ R → λ v (R) < 1 − 2 −g(v) . (8.3)8.3.13 Lemma. Let N be a martingale such that Succ(N) ⊆ Non-MLR. Thenthere is v ∈ {0, 1} ∗ and d ∈ N such that v ∉ R and∀x ≽ v [ N(x) ≥ 2 d → x ∈ R ] . (8.4)This is an analog of Lemma 8.3.4. Once again, a condition on subsets of N (thecontainment Succ(N) ⊆ Non-MLR) implies a condition on finite objects, whichis easier to work with.Subproof. Suppose the Lemma fails. We build a set Z ∉ R on which N succeeds.Define a sequence of strings (v d ) d∈N as follows: let v 0 = ∅, and let v d+1 besome proper extension y of v d such that N(y) ≥ 2 d but y ∉ R. ThenN succeedson Z = ⋃ d v d.OntheotherhandZ ∉ R, soZ ∈ MLR.✸We build a Turing functional L such that L X is a (total) Q 2 -valued martingalefor each oracle X. (WesaythatL is a martingale functional.) If A is inLow(MLR, CR) thenSucc(L A ) ⊆ Non-MLR, so we can apply the Lemma withN = L A .Atfirstwepretendtoknowawitness〈v, d〉 as in the Lemma.We will define an effective sequence (T s ) s∈N of finite subtrees of 2

330 8 Classes of computational complexitymeasure of new enumeration into R (see 5.1.21). If the clopen sets belonging todifferent procedures are disjoint, then W is a bounded request set.Roughly, the procedure α chooses a clopen set ˜C = ˜C(α) suchthatλ v ( ˜C) =2 −(r+c) ,and ˜C is disjoint from R s and the sets chosen by other procedures.Then α causes (in a way also to be specified) L X (z) ≥ 2 d for each X ≽ γ andeach string z ∈ ˜C of minimal length. If at a stage t>sonce again we haveγ ∈ T t ,then ˜C ⊆ R t ,andα now has the permission to put 〈r + c, y〉 into W .This approach has to be refined in order to guarantee the disjointness of clopensets belonging to different procedures. Suppose β ≠ α is a procedure that wantsto choose its set ˜C(β) atastages ′ >s.Ifγ =(α) 2 is in some T q with s

8.3 Lowness for randomness notions 331all the capital on 0, and reaches an increase of 2 d at z. Everystringextendingxis called used by α.The procedure α has to obey the following restrictions at a stage s.1. Choose the extension z not in [Ẽs−1] ≺ where Ẽs−1 is the set of stringspreviously appointed by other procedures β. (Theopensetsgeneratedbythe strings appointed by different procedures need to be disjoint.)2. Let C t (α) denote the set of strings x used by α up to stage t. Ensurethatx ∉ [C s−1 (α)] ≺ so that the capital of α is still available at x. Suchachoiceis possible for sufficiently many x, becauseforallt we haveλ v [ ˜C t (α)] ≺ ≤ 2 −(r+c) ,sothatλ v [C t (α)] ≺ ≤ 2 −(r+c) 2 1+r+d+m =2 −(u+2) .There is no conflict between α = 〈σ,y,γ〉 and any other procedure β = 〈σ ′ ,y ′ , γ ′ 〉as far as the capital is concerned: if γ ′ is incomparable with γ then γ and γ ′ canonly be extended by different oracles X. Otherwise,α and β own different partsof the initial capital of L X m for every set X extending their third components.We now proceed to the formal definition of the martingale functionals L m .Therelevant objects are summarized in Table 8.2. For a procedure α = 〈σ,y,γ〉, letn α > max(|σ| + m + d +1, |γ|, |v|) beanaturalnumberassignedtoα in someeffective one-one way. Each procedure α defines an auxiliary function F α :2

332 8 Classes of computational complexityTable 8.2. Summary of the objects occuring in the proof.R R 1 =[{x ∈ {0, 1} ∗ : K(x) ≤ |x| − 1}] ≺g computable function as in (8.3)δ m witness for Lemma 8.3.13, of the form 〈v, d〉u g(v)c m+ d + u +3L m martingale functional for witness δ mαn αF αC t (α)˜C t (α)T sWẼ tprocedure of form 〈σ,y,γ〉 where U γ (σ) =y; letr = |σ|code number of α, chosen> max(|σ| + m + d +1, |γ|, |v|)auxiliary function defined by αset of strings x used by α up to (the end of) stage tset of strings appointed by α up to stage t, offormx0 r+m+d+1tree for m at the end of stage sbounded request set for mset of strings appointed by procedures up to stage tConstruction for parameter m. Let δ m = 〈v, d〉 and u = g(v). The constructionproceeds at stages which are powers of 2; the variables s and t denote suchstages. At stage s, wedefineT s and extend the functions F α (w) toallw suchthat s ≤ |w| < 2s. Foreachw such that s ≤ |w| < 2s and each string η (whichmay be shorter than w), by the end of stage s we can calculateL m (η,w)=2 −m + ∑ αF α (w)[[(α) 2 ≼ η]]. (8.6)Stage 1. Let T 1 = {∅}, andF α (w) = 0 for each α and each w, |w| ≤ 1.Let Ẽ1 = ∅.Stage s>1. Suppose T t has been determined for t

8.3 Lowness for randomness notions 333where G = ⋃ {D β : β has so far acted at stage s}. (InClaim2wewillverify that D exists.) Let ˜D = {x0 m+d+r+1 : x ∈ D}, putD into C s (α),and put ˜D into ˜C s (α) andẼs. (Note that |w| < 2s for all strings w ∈ ˜D,since m + d + r +1 |γ|. (B2) and (B3) are satisfied since each x chosen in (3b)goes into C(α). So by the choice of D in (3a), no future definition of F α onextensions of x is made except for by (4).✸Claim 2. Each procedure α is able to choose a set D α in (3a).– By the definition of T s ,foreachβ = 〈σ ′ ,y ′ , γ ′ 〉 and each t ≥ 2, if γ ′ ∈T t then ˜C t/2 (β) ⊆ R t . Thus for each procedure β, λ v ( ˜C t (β) − R t ) ≤2 −(nβ+u+2) because ˜C t (β) − R t consists of a single set ˜D β .Then,lettingt = s/2, we have λ v (Ẽs/2 − R s ) ≤ 2 −(u+2) .–EachsetD β chosen during stage s satisfies λ v (D β ) ≤ 2 −(n β+u+2) ,henceλ v G never exceeds 2 −(u+2) .– For each s, λ v ˜Cs (α) ≤ 2 −(r+c) ,andhenceλ v C s (α) ≤ 2 r+d+m+1 2 −(r+c) =2 −(u+2) .Since the query in (1) was answered negatively, λ v R s < 1−2 −u , so relative to [v]a measure of 2 −(u+2) is available outside [R s ∪ Ẽs/2 ∪ G ∪ C s/2 ]forchoosingD α .All strings in R s ∪ Ẽs/2 ∪ G ∪ C s/2 are shorter than s (for strings in Ẽs/2 thisholds by the comment at the end of (3a)), so the strings in D α can be chosen oflength s.✸Claim 3. Each procedure α acts only finitely often.Each time α acts at s and s ′ >sis least such that γ ∈ T s ′ we have increasedλ v ˜C(α) bythefixedamountof2−(n α+c+r) .Soeitherγ ∉ T s for almost all s, orα ends.✸In (8.6) we defined L m (η,w).Claim 4. For each string η, there is a stage s η such that no procedure α with(α) 2 ≼ η acts at any stage ≥ s η .Further,foreachw ≽ v, if|w| ≥ s η thenL m (η,w)=L m (η,w ′ ) for some w ′ such that v ≼ w ′ ≼ w and |w ′ |

334 8 Classes of computational complexityThis holds because there are only finitely many procedures α with (α) 2 ≼ η. ByClaim 3 there is a stage s η by which these procedures have stopped acting, andfurther definitions F α (w) areonlymadein(4).✸Claim 5. T (η) =lim s T s (η) exists.Suppose s ≥ s η is least such that η ∈ T s .Weshowη ∈ T t for each t ≥ s. Supposev ≼ w, |w| ≤ t and L m (η,t) ≥ 2 d .ByClaim4,L m (η,w)=L m (η,w ′ ) for somew ′ ≼ w of length

8.3 Lowness for randomness notions 335Recall that W2R denotes the class of weakly 2-random sets. A modification ofthe proof of 8.3.10 yields a stronger result, which also entails Theorem 5.1.33that Low(W2R, MLR) = Low(MLR).8.3.14 Theorem. (Extends 8.3.10)A ∈ Low(W2R, CR) ⇔ A is low for K.Proof. We only need to prove the implication “⇒”. The construction of L mis now based on potential witnesses of the form δ m = 〈v, d, V, q, ɛ〉, wherev ∈{0, 1} ∗ , d ∈ N, V is a c.e. open set and q, ɛ ∈ Q 2 ,0< ɛ,q,andq + ɛ ≤ 1. Givensuch a potential witness δ m ,weletR =[{x ≽ v : λ x (V ) ≥ q + ɛ}] ≺ .Letu ∈ Nbe least such that q/(q +ɛ) < 1−2 −u .IntheconstructionofL m ,eachprocedureα is specified as before but with the new definitions of R and u. Inparticular,aprocedure based on δ m is active at stages s only as long as λ v (R s ) < 1−2 −u .Thefunction x → λ x (V ) is a martingale, so if λ v (V ) ≤ q, thenbyProposition7.1.9we have λ v (R) < 1 − 2 −u .Lemma 8.3.13 is modified: for an appropriate witness such that λ v (V ) ≤ q, ifaprocedureα causes L A (x) ≥ 2 d for some x ≽ v, thenλ x (V ) ≥ q + ɛ.8.3.15 Lemma. Let N be a martingale such that Succ(N) ⊆ non-W2R.Then there is a δ m = 〈v, d, V, q, ɛ〉 as above such that λ v (V ) ≤ q and∀x ≽ v [ N(x) ≥ 2 d → λ x (V ) ≥ q + ɛ ] . (8.7)Subproof. The argument is similar to the one in the proof of Theorem 5.1.33.(The negation of the lemma now plays the role of Claim 5.1.34.) As before, let{G e n} e,n∈ω be a listing of all the generalized ML-tests (Definition 3.6.1) with noassumption on the uniformity in e. Ifthelemmafailswemayinductivelydefineasequencew 0 ≺ w 1 ≺ ... and numbers n d (d ∈ N) suchthatN(w d ) ≥ 2 d & λ(V d | w d ) ≤ γ d , (8.8)where γ d =1− 2 −d and V d = ⋃ i

336 8 Classes of computational complexity8.4 Jump traceabilityOur basic notion is c.e. traceability of a set, introduced in 8.2.2. In Definition8.2.15 we strengthened this to computable traceability, where the tracesare effective sequence of strong indices for finite sets. We can also stay with c.e.traces, but strengthen the basic notion by tracing functions partial computablein A. We show in Fact 8.4.2 that it suffices to require a c.e. trace for J A .8.4.1 Definition. Let g be an order function. The set A is jump traceable withbound g (Nies, 2002) if there is a c.e. trace (T e ) e∈N with bound g such that∀e [ J A (e)↓→ J A (e) ∈ T e].We say that (T e ) e∈N is a jump trace for A. ThesetA is called jump traceableif A is jump traceable with some bound.For c.e. traceability and computable traceability, by the argument in Theorem8.2.3, the growth rate of the bounds of the traces is irrelevant as long asthey are order functions. The argument relies on the fact that only total functionsare traced, so it is not surprising that the case of jump traceability isdifferent: there is a proper hierarchy depending on the growth of the bounds byTheorem 8.5.2.Later on in this section we will study sets that are jump traceable for everybound. We will see in Section 8.5 that this lowness property, called strong jumptraceability, is much more restrictive than jump traceability. For instance, for c.e.sets, strong jump traceability strictly implies K-triviality, while jump traceabilityis equivalent to superlowness.Just like the class Low(Ω), the class of jump traceable sets is closed downwardunder ≤ T ,andcontainedinGL 1 .Infact,buildingajumptraceablesetisagoodway to obtain a set in GL 1 .Friedberg(1957a) showed that for each set C ≥ T ∅ ′there is a set A such that C ≡ T A ⊕∅ ′ ≡ T A ′ .WeprovethisinCorollary8.4.5via a jump traceable set A.Unlike sets that are low for Ω, byCorollary8.4.7thereisanincomputablejumptraceable set that is computably dominated. No characterization via relativerandomness is known for jump traceability. However, we will give one using C-complexity relative to an oracle.Basics of jump traceability, and existence theoremsFirst we show that it suffices to require that the jump is traced.8.4.2 Fact. If A is jump traceable, there is a c.e. trace (S m ) m∈N such that foreach Turing functional Φ we have ∀ ∞ m [ Φ A (m)↓→ Φ A (m) ∈ S m].In particular, A is c.e. traceable.Proof. There is an effective listing (p i ) i∈N of all the reduction functions in thesense of Fact 1.2.15. (This relies on the proof of the Parameter Theorem 1.1.2which is applied to obtain Fact 1.2.15.) Suppose that A is jump traceable viathe c.e. trace (T e ) e∈N with bound g. LetS m = ⋃ i≤m T p i(m). ThenS m is c.e.uniformly in m, andλm. ∑ i≤m g(p i(m)) is a bound for (S m ) m∈N .

8.4 Jump traceability 337Given a Turing functional Φ, chooseareductionfunctionp i for Φ. Then,foreach m ≥ i, Φ A (m)↓ implies Φ A (m) ∈ S m .✷If B ≤ T A then J B is partial computable in A. Thereforetheclassofjumptraceable sets is closed downward under Turing reducibility.8.4.3 Proposition. Each jump traceable set A is in GL 1 .Moreover,areductionprocedure for A ′ ≤ T A ⊕∅ ′ can be obtained effectively from an index of a jumptrace for A.Proof. Consider the Turing functional Ψ defined by Ψ X (n) ≃ µs. J X (n) ↓ [s].Choose a reduction function p for Ψ by 1.2.15. To determine whether e ∈ A ′ ,first compute t =maxT p(e) using ∅ ′ as an oracle. Next, using A check whetherJ A (p(e))↓ in at most t steps. If so, answer “yes”, otherwise answer “no”. ✷Function trees were introduced in the proof of Theorem 8.2.17 to build a perfectclass of computably traceable sets. They can also be used to obtain a perfectclass of jump traceable sets. Before, the purpose of the e-th function tree was toprovide a trace for the Turing functional Φ e .Nowweonlytracethejump.Thes-th function tree is used to trace computations that converge by stage s.8.4.4 Theorem. There is a perfect Π 0 1 class P such that each set in P is jumptraceable via a fixed c.e. trace (T e ) e∈N with bound λe.2 · 4 e .Proof. We build a uniformly computable sequence of function trees (F s ) s∈N .Wethink of the F s (σ) asmovablemarkersonbinarystrings.Theactivemovementofa marker is to an extensions of its present value. Thus, for each s, σ, ifF s+1 (σ) ≠F s (σ) thenwehaveF s+1 (σ) ≻ F s (σ) unless the change is caused by F s+1 (ρ) ≠F s (ρ) forsomeρ ≺ σ. WeensurethatF (σ) =lim s F s (σ) exists for each σ, thatis, the markers stabilize eventually. Then F is a function tree. Note that the classP = Paths(F )= { Z : ∀n ∀s ∃η [ |η| ≤ n & Z ↾ n ≼ F s (η) ]}is a perfect Π 0 1 class.The idea is to move F s (σ), |σ| = e, in order to make a computation JsF (σ) (e)converge whenever possible. Once achieved, we maintain this unless the movementof a marker F s (ρ) forρ ≺ σ interferes. Note the similarity to meeting thelowness requirements in the proof of Theorem 1.6.4. The interference of a markerF s (ρ) forρ ≺ σ corresponds to the injury of a lowness requirement L e .Construction of the function trees F s . Let F 0 (σ) =σ for each σ.Stage s +1. Look for the least e

338 8 Classes of computational complexitysuch that F t (σ) has stabilized for all σ, |σ| ≤ e +1. Let σ be the string such thatF t (σ) ≺ Y ,thenJ Y (e) ≃ J Ft(σ)t (e).For the full result, define a c.e. trace by T e = {JsFs(σ) (e): s ∈ N, |σ| = e}.For |σ| = e, avalueF s (σ) canchangeatmost2 e+1 − 1timesbecausethemarker F s (σ) movesatmostonceafterF s (ρ) isstableforallρ ≺ σ. Therefore#T e ≤ 2 e (2 e+1 − 1) < 2 · 4 e . ✷8.4.5 Corollary. For each set C ≥ T ∅ ′ there is a jump traceable set A such thatC ≡ T A ⊕∅ ′ ≡ T A ′ .Inparticular,thereisahighjumptraceableset.Proof. Let A = ⋃ σ≺CF (σ) whereF is the function tree introduced above.Since F ≤ T ∅ ′ ,wehaveC ≤ T A ⊕ F ≤ T A ⊕∅ ′ .AlsoA ≤ T C ⊕ F ≡ T C. ✷The construction in the proof of Theorem 8.4.4 can be modified in order to obtain abound for the trace closer to λe. 2 e .However,iftheboundg grows slowly enough, thenasetthatisjumptraceableforg is ∆ 0 2 by Downey and Greenberg (20xx).8.4.6 Exercise. Extend Theorem 8.4.4 by also ensuring that P contains no computablesets (at the cost of a somewhat larger bound for the jump trace).8.4.7 Corollary. There is a perfect class of sets that are both computably traceableand jump traceable.Proof. Let P ≠ ∅ be the Π 0 1 class of Exercise 8.4.6. By Theorem 1.8.44, there is aperfect subclass S of P such that each element of S is computably dominated. Eachmember of S is c.e. traceable and hence computably traceable.✷The foregoing result implies that a computably traceable set can be in GL 1.Notall computably traceable sets are in GL 1:Lerman(1983,Thm.V.3.12)constructedaset A of minimal Turing degree such that A ′′ ≡ T ∅ ′′ and A ∉ GL 1.Examininghisconstruction reveals that A is computably traceable.8.4.8 Exercise. Show in a direct way that if A is low for K, then A is jump traceablewith a bound in O(n log 2 n). (See 5.5.12 for a proof via K-triviality.)8.4.9. ⋄ Problem. Characterize the sets that are low for Ω and jump traceable (each K-trivial set is). Characterize the sets that are computably traceable and jump traceable.Jump traceability and descriptive string complexityFigueira, Nies and Stephan (2008) proved that A is jump traceable iff C A (x) isnot much smaller than C(x) foreachx. We need some notation to make thisprecise. For any function f, let̂f(y) =y + f(y).Let α be a reduction function for the plain optimal machine V, namely,∀X ∀σV X (σ) ≃ J X (α(σ)). Let c be a constant such that J A (|x|) ↓ impliesC A (x, J A (|x|)) ≤ |x| + c for each x. Suchac exists because given x we cancompute J A (|x|) relative to A.8.4.10 Theorem. A is jump traceable ⇔∀x [ C(x) ≤ + C A (x)+h(C A (x)) ] forsome order function h. Inmoredetail:

8.4 Jump traceability 339(i) Let A be jump traceable with bound g. Then ∀x [ C(x) ≤ + ĥ(CA (x)) ] , whereh(n) = max{2g(α(σ)) + 1: |σ| = n}.(ii) Suppose h is an order function such that ∀x [C(x) ≤ + ĥ(CA (x))]. Then Ais jump traceable with bound λe. 3 h(e+c) .Proof. (i) Let (T e ) e∈N be a jump trace for A with bound g. WedefineamachineM by letting M(0 |r| 1rσ) bether-th element in the enumeration of T α(σ)if such an element exists, and leaving it undefined otherwise.Suppose σ is a shortest V A -description of a string x. Thenx ∈ T α(σ) .Since2#T α(σ) +1≤ h(|σ|), we have M(0 |r| 1rσ) =x for some r such that 2r +1≤+h(|σ|). Thus C(x) ≤ ĥ(|σ|) asrequired.(ii) We need an auxiliary fact stating that, independently of x, wecanboundthe number of y such that C(x, y) exceedsC(x) bynomorethanb ∈ N.8.4.11 Lemma. #{y : C(x, y) ≤ C(x)+b} = O(2 b+2 log b ).Subproof. Let M be the machine given by M(σ) ≃ (V(σ)) 1 ,namely,M(σ)is the first component of V(σ) viewed as an ordered pair. By Lemma 5.2.21,#{σ : M(σ) =x & |σ| ≤ C(x)+b} = O(2 b+2 log b ). If C(x, y) ≤ C(x)+b thenV(σ) =〈x, y〉 for some σ such that |σ| ≤ C(x) +b. ThenM(σ) =x, sotherequired bound on #{y : C(x, y) ≤ C(x)+b} follows.✸By the hypothesis of (ii) and the definition of c, foralle, ifJ A (e) ↓ and |x| = ethenC(x, J A (|x|)) ≤ + ĥ(CA (x, J A (|x|))) ≤ ĥ(|x| + c).Now let T e = { y : ∀x [ |x| = e → C(x, y) ≤ ĥ(e + c) +d]} ,whered is theimplicit constant in the inequality above. Thus, if J A (e) ↓ then J A (e) ∈ T e .Clearly (T e ) e∈N is uniformly c.e. For the bound on #T e ,givene we choose xsuch that |x| = e and C(x) ≥ e. Ify ∈ T e then C(x, y) ≤ ĥ(e + c) +d ≤C(x) +c + h(e + c) +d. Thus,foralmostalle we have #T e ≤ 3 h(e+c) byLemma 8.4.11 where b = c + h(e + c) +d. Then a modification of (T e ) e∈N atfinitely many components is a jump trace for A with the required bound. ✷8.4.12 Exercise. Recall from 1.4.5 that V e is the e-th ω-c.e. set.Show that {e: V e is jump traceable} is Σ 0 3-complete.Similarly, show that {e: W e is jump traceable} is Σ 0 3-complete.The weak reducibility associated with jump traceabilityIn Section 5.6 we studied the weak reducibility ≤ LR associated with the lownessproperty Low(MLR). Simpson (2007) defined, in a different language, aweak reducibility ≤ JT associated with jump traceability. Also see Table 8.3 onpage 363.Note that a sequence of sets (T e ) e∈N is a c.e. trace relative to B iff there isfunction f ≤ T B such that T e = Wf(e) B for each e, andaboundh ≤ T B suchthat #T e ≤ h(e)foreache.Wemayequivalentlyrequirethatf is computable, for

340 8 Classes of computational complexitylet f = Γ B ,andletΘ be a Turing functional such that Θ B (e, x)↓ iff x ∈ W B Γ B (e) .By the Parameter Theorem for oracles (mentioned before 1.2.10), there is acomputable function r such that Φ B r(e) (x) ≃ ΘB (e, x) foreachB,e, soWf(e) B =Wr(e) B for each e. However,torequireacomputableboundh is a restriction.8.4.13 Definition. A is jump traceable by B, written A ≤ JT B,ifthereisac.e.trace (T e ) e∈N relative to B for J A ,andanorderfunctionh such that #T e ≤ h(e)for each e.Being jump traceable by B is somewhat different from being jump traceable relativeto B because we only require the existence of a c.e. trace for the function J A ,not for J A⊕B ;ontheotherhand,theboundforthistracemustbecomputable,not merely computable in B.8.4.14 Fact. The relation ≤ JT is transitive.Proof. Suppose A is jump traceable by B via a trace (S n ) n∈N with bound anorder function g, andB is jump traceable by C via a trace (T i ) i∈N with boundan order function h. Thereisacomputablefunctionβ such thatJ B (β(〈n, k〉)) ≃ the k-th element enumerated into S n .Let V n = ⋃ k

8.4 Jump traceability 341Ψ X (e) ≃ min{σ ≺ X : J σ |σ| (e)↓}.Choose a reduction function p for Ψ. Thus,Ψ X (e) ≃ J X (p(e)) for each X and e.We write ê for p(e).Let (A s ) s∈N be a B-computable enumeration of A. SupposethatA ≤ JT B viaa B-c.e. trace (T e ) e∈N with computable bound h. WemayassumethatJ σ |σ| (ê)↓for each σ ∈ Tê by only admitting such strings into Tê. Letσ e,k,s be the k-thstring enumerated into Tê,s ,ifsuchastringexists,andundefinedotherwise.Notethat B can compute the value of σ e,k,s ,andifσ e,k,s is defined then k

342 8 Classes of computational complexity8.4.23 Theorem. Let A be c.e. Then A is jump traceable ⇔ A is superlow.Proof. ⇒: (Range to domain) In Theorem 8.4.16 let B = ∅.⇐: (Domain to range) We will write j(X, e) ≃ use J X (e) andj(X, e, s) ≃use J X s (e). By Proposition 1.4.4, A is superlow iff there is a binary {0, 1}-valuedcomputable function q and an order function g such that, for each e,A ′ (e) =lim s q(e, s) &g(e) ≥ #{s >0: q(e, s) ≠ q(e, s − 1)}. (8.10)To obtain a jump trace (T e ) e∈N for A, wewilldefineanauxiliaryTuringfunctionalΨ which copies computations of J with some delay. We assume a Turingfunctional ˜Ψ is given. Let β be the reduction function effectively obtained from ˜Ψaccording to Fact 1.2.15. We build Ψ effectively from β. Then,bytheRecursionTheorem we may assume that ˜Ψ = Ψ, so β is a reduction function for Ψ as well.For each e let ê = β(e). At stage 0, Ψ is undefined for all inputs. At stage s>0we distinguish two cases.(a) q(ê, s) = 0. If Ψ A (e)[s − 1]↑ and J A (e)[s]↓= v, letΨ A (e)[s] =vwith use j(A s ,e,s).(b) q(ê, s) = 1. If Ψ A (e)[s]↓ then enumerate y = J A (e)[s] intoT e .Note that Ψ merely copies computations of J at a later stage, so when a newcomputation J A (e)[s] appears,nocomputationΨ A (e)[t] whichwasdefinedattt, and at stage r we enumerate z into T e .Next, we show that (T e ) e∈N is a c.e. trace with bound h(e) = ⌊ 1 2 g(β(e))⌋.Suppose that v

8.4 Jump traceability 343By Theorem 5.5.7, each K-trivial set A is Turing below a c.e. K-trivial set, which issuperlow, hence jump traceable, and hence c.e. traceable. Thus A is c.e. traceable. ByProposition 8.2.29 this implies the following.8.4.25 Corollary. Each K trivial set is facile. ✷See the solution to Exercise 8.2.11 for an alternative proof that each K-trivial set isc.e. traceable.8.4.26 Exercise. (Ishmukhametov, 1999) Recall Corollary 8.2.8. Show that each arraycomputable c.e. set A is c.e. traceable.More on weak reducibilitiesThe following extension of the implication from right to left in the foregoingTheorem 8.4.23 was proved by Simpson and Cole (20xx). It is a converse toTheorem 8.4.16 under the extra hypothesis that B ≤ T A.8.4.27 Corollary. Suppose A is c.e. in B and B ≤ T A. Then A ′ ≤ tt B ′ impliesA ≤ JT B.Inparticular,ifC is ∆ 0 2 and superhigh then ∅ ′ ≤ JT C.Proof. At first we merely relativize “⇐” of Theorem 8.4.23 to B: ifA is c.e.in B and (A ⊕ B) ′ ≤ tt(B) B ′ then there is a trace (T e ) e∈N for J A⊕B that is c.e.relative to B. Here≤ tt(B) denotes the reducibility where B can be used as anoracle to compute the truth table.Since B ≤ T A and A ′ ≤ tt B ′ ,weactuallyhavethestrongerhypothesisthat(A ⊕ B) ′ ≤ tt B ′ .Thus,in(8.10)wecanchoosethebinaryfunctionq ≤ T B,which now approximates (A ⊕ B) ′ ,insuchawaythatthenumberofitschangesis bounded by a computable function g (not merely one computable in B).When we view the proof of 8.4.23 relative to B, thereductionfunctionβ for thegiven functional ˜Ψ is still computable. Note that #T e ≤⌊ 1 2g(β(e))⌋ as before.This shows that A ≤ JT B.✷On the other hand, a superhigh set C can be jump traceable by Exercise 8.6.2.Thus C ≤ JT ∅ and hence ∅ ′ ≰ JT C. Together with Theorem 6.3.14, this showsthat all the highness properties in the first five rows of Table 8.3 on page 363can be separated.Strong jump traceabilityFigueira, Nies and Stephan (2008) introduced the following lowness property.8.4.28 Definition. A is strongly jump traceable if A is jump traceable withbound g (as defined in 8.4.1) for each order function g.For such a set A, ifΨ is a Turing functional and h an order function, there isac.e.traceforΨ A with bound h. To see this choose an increasing reductionfunction p, sothatΨ A (x) ≃ J A (p(x)) for each x. IfB ≤ T A then J B = Ψ A forsome Turing functional Ψ, soB is strongly jump traceable as well.We will see in Section 8.5 that for the c.e. sets being strongly jump traceableimplies most other lowness properties. For instance, by Corollary 8.5.5, a strongly

344 8 Classes of computational complexityjump traceable c.e. set is Turing below each ω-c.e. ML-random set, and hencelow for K.Figueira, Nies and Stephan (2008) built a promptly simple strongly jump traceableset. Their construction can be viewed as an adaptive cost function construction.Hence it does not qualify as being injury-free. In fact, in the proof we willmodify the construction with injury of Theorem 1.7.10, which in itself extendsTheorem 1.6.4 that there is a low simple set.8.4.29 Theorem. There is a promptly simple strongly jump traceable set A.Proof. As in Theorem 1.7.10 we meet the prompt simplicity requirementsPS e : #W e = ∞ ⇒ ∃s ∃x [x ∈ W e,at s & x ∈ A s ](where W e,at s = W e,s −W e,s−1 ), and the lowness requirements L k which attemptto stabilize a computation J A (k). However, now the priority ordering is dynamic.Recall from 2.1.22 that the function C given by C(x) =min{C(y): y ≥ x} isdominated by each order function g. NotethatC s (x) = min{C s (y): y ≥ x}defines a nonincreasing computable approximation of C. WestipulatethatPS ecan only injure a requirement L k at stage s if e0. For each euse J A (k)[s − 1] ] , (8.11)then put x into A s and declare PS e satisfied.Verification. The first claim is immediate from the construction.Claim 1. Given a number k and a stage t such that J A (k)[t − 1] ↓, we haveJ A (k)[t] =J A (k)[t − 1] unless some PS e such that e

8.4 Jump traceability 345x>use J A (k)[s 0 ] whenever e ≥ C(k) andJ A (k)[s 0 ] is defined. We enumerate xinto A at stage s if PS e has not been met yet.✷To turn the proof into a cost function construction similar to the one in the proof ofTheorem 5.3.35, one uses the adaptive cost function c(x, s) =max{2 −e : (8.11) holds}.In the following variant of the theorem we apply the Robinson guessing method alreadyused in Theorem 5.3.22.8.4.30 Theorem. For each low c.e. set B, there is a strongly jump traceable c.e. set Asuch that A ≰ T B.Proof. We make the necessary changes to the proof of Theorem 5.3.22. In the construction,instead of (5.8) we now ask that Φ B e (x) =0[s] and∀k [ (〈e, n〉 ≥C s(k) &J s−1(A s−1; k)↓) → x>use J s−1(A s−1; k) ] .Informally, if P e has acted n times before stage s, it has to obey the restraint of all L ksuch that 〈e, n〉 ≥C s(k). The same condition, but with the given partial enumeration(Ãs) s∈N instead of (A s) s∈N ,replacescondition(ii)intheΣ 0 1(B) questionforrequirementP e.OneshowsthatA is total as in Claim 5.3.23 in the proof of Theorem 5.3.22.Claim 1 and Claim 2 in the proof of Theorem 8.4.29 are verified as before. Claim 3 isreplaced by Claim 5.3.25, with the obvious changes.✷8.4.31 Remark. For each order function g there is a (much slower growing)order function h and a c.e. set A that is jump traceable with bound g but not withbound h. WegiveanewproofofthisresultofNg(2008a) in Theorem 8.5.2. Ng(2008a) used similar methods to show that the class of strongly jump traceablec.e. sets has a Π 0 4-complete index set.In Section 8.5 we will show that each strongly jump traceable c.e. set is lowfor K. Being strongly jump traceable is also related to relative C-complexity.A set that is low for C is computable by Exercise 5.2.22, but a weaker conditionthan being low for C yields a characterization of strongly jump traceability. Thisis an easy consequence of Theorem 8.4.10:8.4.32 Corollary. The following are equivalent.(i) A is strongly jump traceable.(ii) A is lowly for C, namely,foreachorderfunctionh,∀x [ C(x) ≤ + C A (x)+h(C A (x)) ] . (8.12)Proof. The function α and the constant c were introduced before Theorem 8.4.10.(i) ⇒ (ii): Let h be an order function. There is an order function g such that˜h(n) =max{2g(α(σ)) + 1: |σ| = n} ≤ h(n) foralmostalln. SinceA is jumptraceable with bound g, (i) of Theorem 8.4.10 implies (8.12).(ii) ⇒ (i): Let g be an order function. There is an order function h such that3 h(e+c) ≤ g(e) foralmostalle. Since(8.12)holdsforh, (ii) of Theorem 8.4.10implies that A is jump traceable with bound g.✷Downey and Greenberg (20xx) have announced that each strongly jump traceable setis ω-c.e. (even with the number of changes dominated by any order function). Being inGL 1,thisimpliesthateachstronglyjumptraceablesetislow.

346 8 Classes of computational complexityStrong superlowness ⋆This subsection mostly follows Figueira, Nies and Stephan (2008). We strengthenthe property of superlowness by universally quantifying over all order functions,the same way we passed from jump traceability to strong jump traceability.8.4.33 Definition. Let g be an order function. We say that A is superlow withbound g if there is a {0, 1}-valued computable binary function q such that, foreach x,A ′ (x) =lim s q(x, s) &#{s >0: q(x, s) ≠ q(x, s − 1)} ≤ g(x). (8.13)A is strongly superlow if A is superlow with bound g for each order function g.The construction in the proof of Theorem 8.4.29 yields a strongly superlow c.e.set (see Exercise 8.4.36). Actually, for c.e. sets, strong superlowness and strongjump traceability are equivalent by the techniques in the proof of Theorem 8.4.23.The implication from left to right holds for all sets. No direct proof of the latterresult is known. Rather, we use Corollary 8.4.32.8.4.34 Theorem. Each strongly superlow set A is strongly jump traceable.In fact, for each order function g there is an order function b such thatA is superlow with bound b ⇒ A is jump traceable with bound g.Proof idea. By Theorem 8.4.10(ii) it is sufficient to show that for each order function h,there is an order function b such thatA is superlow with bound b →∀x [C(x) ≤ + ĥ(C A (x))](where ĥ(y) =y + h(y) as before). For each pair of strings σ,x we haveC(x) ≤ + |σ| + K(x | σ) ≤ + |σ| +2C(x | σ).Now suppose that σ x is a shortest V A -description of x. SinceA is computationallyweak, very little extra information is needed to obtain x from σ x without the help of A,namely, 2C(x | σ x) ≤ + h(|σ x|). To show this, for each n we code into A ′ the bits ofthe first τ of length n we find such that x = V 2 (τ, σ x). If n = n x := C(x | σ x), anappropriate approximation to A ′ changes a sufficiently small number of times on thesebits (compared to h). Then a description of the corresponding τ from n x, σ x and thisnumber of changes is used to show that 2C(x | σ x) ≤ + h(|σ x|).Proof details. Let Ψ X be the Turing functional given by the following procedure (ifit gets stuck on input y we leave Ψ X (y) undefined). On input y = 〈i, n, σ〉:(1) Attempt to compute x = V X (σ).(2) Let s be least such that V 2 s(τ, σ) =x for some τ of length n.(3) If i

nh(|σ|) foralln>0andallσ. To obtain b, for each k let8.4 Jump traceability 347m k =max{α(n, n, σ): n>0&nh(|σ|) ≤ k}.Note that m k exists since h is an order function. Let b(j) = 1 for j ≤ m 1,andifk>1and m k−1 0: τ x,s ≠ τ x,s−1} ≤ n xb(α(n x,n x, σ x)) ≤ n 2 xh(|σ x|).There is a machine describing τ x in terms of n x, σ x, and d x for each x. SinceV 2 (τ x, σ x)=x, afurthermachinedescribesx in terms of n x, σ x, and d x. Note that foru, v ∈ N we have |bin(u · v)| = + |bin(u)| + |bin(v)|, where bin(u) ∈ {0, 1} ∗ is the binaryrepresentation of u. Wetemporarilywrite|n| for the length of bin(n). Note thatn x = C(x | σ x) ≤ + 2|n x| + |n 2 x · h(|σ x|)| ≤ + 4|n x| + |h(|σ x|)|. (8.14)Claim. We have n x ≤ + 2|h(|σ x|)| for all x.There is a constant N such that 8|n| ≤ n for all n ≥ N. Since h is an order function,|h(|σ x|)| ≥ N for almost all x, hence for almost every x either n x ≤ |h(|σ x|)| or4|n x| ≤ n x/2. In the latter case n x − 4|n x| ≥ n x/2 and by (8.14), n x/2 ≤ + |h(|σ x|)|.This proves the claim. To conclude the proof, note thatC(x) ≤ + |σ x| +2C(x | σ x) ≤ + |σ x| +4|h(|σ x|)| ≤ + |σ x| + h(|σ x|).The rightmost expression equals C A (x)+h(C A (x)) = ĥ(CA (x)).8.4.35 Corollary. Let A be computably enumerable. ThenA is strongly jump traceable ⇔ A is strongly superlow.Proof. ⇐: This implication holds for every set A.⇒: We show that A is superlow with bound g for any order function g by lookingat the proof of Theorem 8.4.16 for B = ∅. Foreachk there are two queries to ∅ ′in (8.9). Let h be an order function such that 2h(ê) ≤ g(e) foralmostalle.Then the approximation to A ′ obtained by evaluating the truth-table reductionin (8.9) on ∅ ′ s at stage s changes at most g(e) timesforalmostalle. ✷Exercises.8.4.36. Show directly from the construction that the set A we build in the proof ofTheorem 8.4.29 is strongly superlow.8.4.37. Define weak reducibilities ≤ SSL and ≤ SJT corresponding to strong superlownessand strong jump traceability. Show that if B ≤ T A, then A ≤ SSL B impliesA ≤ SJT B. (For work related to ≤ SJT see Ng 20xx.)8.4.38. ⋄ Problem. Is each strongly jump traceable set strongly superlow?✷

348 8 Classes of computational complexity8.5 Subclasses of the K-trivial setsNote that the term “subclass” does not imply being proper.By Theorem 8.3.9, lowness for Schnorr randomness can be characterized bythe computability theoretic property of computable traceability. In Chapter 5 weshowed that several properties coincide with lowness for Martin-Löf randomness,but none of them is purely computability theoretic. A characterization of thiskind would be desirable for lowness for Martin-Löf randomness, for instancebecause it might lead to shorter proofs of some of those coincidences.When the properties of strong jump traceability and strong superlowness wereintroduced in Figueira, Nies and Stephan (2008), the hope was that one of themis equivalent to Low(MLR), leading to such a characterization. Cholak, Downeyand Greenberg (2008) showed that the strongly jump traceable c.e. sets formasubclassofthec.e.setsinLow(MLR), introducing the so-called box promotionmethod. However, they also proved that this subclass is proper, therebydestroying the hope that either property characterizes Low(MLR).Following work of Greenberg and Nies, we give proofs using cost functions ofthese results in Cholak, Downey and Greenberg (2008). In 8.5.3 we will introducebenign cost functions, monotonic cost functions c with the property that thenumber of pairwise disjoint intervals [x, s) suchthatc(x, s) ≥ 2 −n is boundedcomputably in n. Thestandardcostfunctionisbenign(Remark5.3.14).Ifcis benign then by the box promotion method each strongly jump traceable c.e.set A has a computable enumeration obeying c. Ontheotherhand,thereisac.e. set obeying c which is not strongly jump traceable.We study further subclasses of the c.e. K-trivial sets that contain the stronglyjump traceable c.e. sets, such as the class of c.e. sets that are Turing below eachML-random set C ≥ LR ∅ ′ . None of these subclasses is known to be proper.The results in this section are rather incomplete and leave plenty of opportunityfor future research. For some updates see the end of this chapter.Some K-trivial c.e. set is not strongly jump traceableIn Theorem 5.3.5 we built a promptly simple set obeying a given cost functionwith the limit condition. For the standard cost function c K ,thisconstructionis more flexible than one might expect: there is a K-trivial c.e. set A that isnot strongly jump traceable. There are strategies R e (e ∈ N) to make A notstrongly jump traceable. Each R e is eventually able to enumerate e +1 numbersinto A at stages of its choice. The construction has to cope with a computablybounded number of failed attempts of R e .Failuremeansthatthesequenceofenumerations is terminated before the e + 1-th number is reached.In Exercise 8.5.8 the result is extended to an arbitrary benign cost function.An alternative proof of the result can be obtained from Corollary 8.5.5 andExercise 8.5.25: each strongly jump traceable c.e. set, but not each K-trivial c.e.set, is below each ω-c.e. ML-random set.8.5.1 Theorem. There exists a c.e. K-trivial set A that is not strongly jumptraceable. Indeed, for some Turing functional Ψ, thereisnoc.e.traceforΨ Awith bound λz.max(1, ⌊1/2loglogz⌋).

8.5 Subclasses of the K-trivial sets 349Proof. We build a computable enumeration (A s ) s∈N of a set A that obeys thestandard cost function c K (see 5.3.2 and 5.3.4 for the definitions). Then A is K-trivial.Let I 1 ,I 2 ,... be the consecutive intervals in N such that #I e =2 e2e .Leth bethe order function such that h(z) =e for z ∈ I e .Notethat1/2loglogz ≤ h(z)for each z because ∑ 1≤i0. Let (S e z) z∈N bethe e-th c.e. trace with bound h in some uniformly c.e. listing of all such traces.We build a Turing functional Ψ in such a way that for each e>0wehaveΨ A (z) ∉ S e z for some z ∈ I e .Since#S e z ≤ e it is sufficient to have e +1 numbersavailable for enumeration into A at stages of our choice before defining Ψ A (z)for the first time with large use (say the stage number). From now on, whenevery = Ψ A (z) isdefinedandy has appeared in S e z, we enumerate the next amongthose numbers into A and redefine Ψ A (z) withalargervaluethanbeforeandthe stage number as the use.The challenge is to implement this strategy while obeying the cost function c K .As a preparation, we will rephrase the construction of an incomputable K-trivialset in terms of procedures. We meet for all e the requirement that A ≠ Φ e .Theprocedure P e (b) forthisrequirementisallowedtoincuracostof1/b. It wants toenumerate a number w into A at a stage s when Φ e,s (w) = 0 in order to ensurethat Φ e (w) =A(w) fails.Ittriestofindanumberw such that c K (w, s) ≤ 1/b ata stage s when it wants to enumerate w.Procedure P e (b) (e >0,b∈ N)(1) Let w be the current stage number.(2) Wait for a stage s such that Φ e,s (w) = 0. If s is found put w into A s .If c K (w, t) > 1/b at a stage t during this waiting goto (1). We say that P e isreset.Since ∑ n 2−K(n) ≤ 1theprocedureP e (b)isresetatmostb times, so eventually wstabilizes. (This particular feature of c K was already discussed in Remark 5.3.14.)At stage e of the construction we start P e (2 e ). Since each procedure enumeratesinto A at most once, the total cost of all A-changes is finite.Now suppose that for each e we have to meet the requirement that (Sz) e z∈Nis not a trace for Ψ A . The strategy for this requirement needs e +1 numbersavailable for enumeration into A, so it now involves e + 1 levels of proceduresPj e, e ≥ j ≥ 0. The strategy begins with P e e ,whichcallsPe−1, e and so on downto P0e which picks z ∈ I e and defines Ψ A (z) for the first time. Each time thecurrent value Ψ A (z) appears in the trace set Sz,aprocedurereturnscontroltoethe procedure that called it, which now changes A in order to redefine Ψ A (z) toalargervalue.TheallowanceofeachrunofPj e is the same amount 1/b e,j, wherethe numbers b e,j ∈ N can be computed in advance.It may now happen that for j>0, a procedure Pj e calls P j−1 e but is reset (in thesame way as above) before Pj−1 e returns. In this case P j−1 e is cancelled. Typicallyit has incurred some cost already. To bound this type of garbage, Pje with theallowance of 1/b calls Pj−1 e with the smaller allowance of 1/b2 .TherunPj e (b) is

350 8 Classes of computational complexityreset at most b times, and each time the cost at level j − 1isatmostb −2 .Thetotal cost at this level is therefore at most 1/b. Wewillverifythatwecanaffordthis cost.If a run of P e j−1 is cancelled this also wastes inputs for ΨA ,becausetherunmay define computations Ψ A (z) whenA is already stable up to the use. However,the size of I e is at least the largest possible number of runs of P e 0 ,sothateachsuch run can choose a new number z ∈ I e .In the following let e>0. In the construction of A we call P e e (2 e ) at stage e.Procedure P e 0 (b) (at level 0, the parameter b merely simplifies the notation later):(1) At the current stage s, letz be the least unused number in I e .Define y = Ψ A (z)[s] =s with use s.(2) Wait for a stage r > s such that y ∈ S e z,r. Meanwhile,ifA ↾ y changes,redefine Ψ A (z) =y with the same use y. Ifr is found return z.Procedure Pj e (b) (b, j ∈ N, e>0, j>0).(1) Let w be the current stage number. Call Pj−1 e (b2 ).(2) Wait for a stage s at which this run of the procedure Pj−1 e returns a number z.If s is found put w into A s .Sincew < use Ψ A (z)[s − 1] this makes Ψ A (z)undefined. Let y = s and redefine Ψ A (z)[s] =y with use y.If c K (w, t) > 1/b at a stage t during this waiting, then at the beginning of tcancel the run of Pj−1 e and all its subruns, and goto (1). We say that the runof Pje is reset (note that it is not cancelled, and has not incurred any cost yet).(3) Wait for a stage r>ssuch that y ∈ Sz,r. e Meanwhile,ifA t−1 ↾ y ≠ A t ↾ y forthe current stage t, thenredefineΨ A (z)[t] =y with the same use y as before.If r is found return z.Claim 1. ArunPj e (b) is reset fewer than b times.Suppose the run is started at w 0 ,andresetatstagesw 1k/b < ∑ 0≤i

8.5 Subclasses of the K-trivial sets 351y 0

352 8 Classes of computational complexityx 0

8.5 Subclasses of the K-trivial sets 353T d z = T αd (z).By the definition of α d ,(Tz d ) z∈N is a c.e. trace for Φ A with bound h d .We are given A with a computable enumeration, and define the new enumeration(Âr) r∈N .Roughly,toensurethat(Âr) r∈N obeys c, thelargerc(y, r), the lessfrequent is the event that y ∈ Âr − Âr−1. TothisendwealsodefineΦ A (z) withuse greater than y for an appropriate z such that Tz d is small. If Φ A (z) entersTzdthen Otto has spent one trace element. If y enters A after that, we can redefineΦ A (z) withalargervalue.Wenowsaythatz is promoted. Thisprocesstakesplace at most #Tzd times.AprocedureR n wants to ensure that there are at most n + d stages r atwhich a number y enters Âr such that c(y, r) ≥ 2 −n . This suffices to make thetotal cost of changes bounded, for R n actually only has to consider the y such∑that also 2 −n+1 >c(y, r), else some R m for m

354 8 Classes of computational complexityProof details. In the following we fix d>0. Let (I n ) n∈N + be the consecutiveintervals of N such that #I n = g(n) (n+d+1) .Givenn, thevariablesk, l rangeover {1,...,g(n)}. IfB is an interval of N such that g(n) | #B, thenletB k bethe k-th subinterval of B of size #B/g(n).The procedure R n controls Φ A (z) foreachz ∈ I n .Wedefinetheorderfunctionh d byh d (z) =n + d if z ∈ I n . (8.15)To deal with the waiting for tracing, the construction (for parameter d) proceedsin stages s(i), similar to the proof of Theorem 5.3.27. We will only consider stagesof this sort, and use italics to indicate this. Let s(0) = 0. If s(i) hasbeendefinedand the construction has been carried out up to stage s(i), lets(i +1)≃ µs > s(i) [ 2 | s & ∀z 0 is a stage, thent denotes the largeststage less than t. WesaythatA t ↾ x is certified ifA t ↾ x = A t ↾ x .Construction of the Turing functional Φ, given the parameter d ∈ N. The proceduresR n act independently. We describe their actions at each stage. Toensuretracing of the relevant computations by the next stage, R n is only active fromstage s ∗ n = s(i +1)on,wherei is least such that s(i) > max I n .Action of R n . At stage s ∗ n let B = I n , m =0andx 0 = 0.Stage s>s ∗ n.(1) Check whether there is k ≤ m such that(◦) x = x k is defined by the end of s, A s ↾ x is certified and A s ↾ x ≠ A s ↾ x .If so, let k be the least such number. Note that Φ A (z)[s] ↑ for all z ∈ B k [s],because such a computation is only defined in (3) and has use x k .For each y ∈ B[s] suchthatg(n) | (y − min B[s]), lety ′ = min B k [s]+(y − min B[s])/g(n).Then y ′ ∈ B k [s]. If Φ A (y)[s] ↓, makeacopyofthiscomputationwithinputy ′ :define Φ A (y ′ )[s + 1] = Φ A (y)[s] withthesameuse.Let B[s +1]:=B k [s]. We say that R n shrinks its box at s.(2) If c(x m ,s) ≥ 2 −n ,thenincrementm and let x m = s.(3) For each l ≤ m and z ∈ B l [s +1],ifΦ A (z)[s] ↑, defineΦ A (z)[s + 1] = s +1with use x l .Verification. For the first two claims fix n.Claim 1. For each stage s>s ∗ n,ifx = x k is defined at s and A s ↾ x is certified,then Φ A (z)[s]↓∈ Tz,s d for each z ∈ B k [s +1].By the action in (3) at some stage ≤ s and the faithful copying, w = Φ A (z)[s+1]is defined with use x. SinceA s ↾ x = A s ↾ x ,wehavew ∈ Tx,s d by the definition ofstages and since s>s ∗ n.✸Claim 2. The procedure R n shrinks its box no more than n + d times.

8.5 Subclasses of the K-trivial sets 355Assume for a contradiction that t 0 0 and Âr(y) ≠ Âr−1(y). Let n be such that2 −n+1 >c(y, r) ≥ 2 −n and consider the procedure R n at stage t = q(r), whenx 0 ,...,x m have been defined. By the monotonicity of c in the second argumentwe have c(y, t) ≥ 2 −n .Alsoc(x m ,t) < 2 −n ,soy

356 8 Classes of computational complexityplete (page 202). However, the number of levels is fixed for the decanter method whilein the case of box promotion, the requirement R n needs n + d +1levels.Decanter method Box promotion methoda number put into F 0 an input x for Φ Aj-seta box of numbers that have been promoted j timesFact 5.4.2 about k -sets Claim 2a garbage number a number that is not promoted because it is inthe wrong box when R n shrinks its boxThe golden run method (which builds on the decanter method) is less relevant, becausein the proof of Theorem 8.5.4 we did not need a tree of runs, while parallel runsare necessary in the proof of Theorem 5.4.1. However, the present construction bearssome resemblance to the calling of Q-type procedures in that proof. When x = x m isdefined in (2) of the construction, one can think of calling such a procedure. It returnswhen Φ A (z)[s +1]istraced.Afterthat,achangeofA↾ xm releases the procedure andcauses R n to shrink its box.The proof of Theorem 8.5.1 can be viewed as a failed decanter type construction foreach interval I e.ThisisplausiblesincewewanttoensurethatA is not strongly jumptraceable, so we are now on the side that Otto occupies in the proof of 8.5.4. If (Sz) e z∈Nis a c.e. trace for Ψ A with the indicated bound, then we can promote some numberz ∈ I e for e +1times.ThismeansthattheanalogofFact5.4.2fails.Cholak, Downey and Greenberg (2008) proved that for any strongly jump traceablec.e. sets A 0,A 1,thesetA 0 ⊕ A 1 is strongly jump traceable. Thus, the degrees of thec.e. strongly jump traceable sets form a proper subideal of the ideal of c.e. K-trivialdegrees. The proof uses the box promotion method as well.Exercises.8.5.8. Extend 8.5.1: if c is a benign cost function with bound g, then there exists ac.e. set A with an enumeration obeying c that is not strongly jump traceable.8.5.9. Show that for each incomputable ∆ 0 2 set A there is a monotonic cost function cwith the limit condition such that no computable approximation of A obeys c.8.5.10. ⋄ Prove the converse of Theorem 8.5.4: if A is a c.e. set which for each benigncost function c has a computable enumeration obeying c, then A is strongly jumptraceable.The diamond operatorFor a class H ⊆ 2 N let H ✸ = {A: A is c.e. & ∀Y ∈ H ∩ MLR [A ≤ T Y ]}.Note that H ✸ determines an ideal in the c.e. Turing degrees. By Theorem 5.1.12,the class {Y : A ≤ T Y } is null for each incomputable set A. Thus,ifH ✸ containsan incomputable set then H is null. On the other hand, for each null Σ 0 3 class Hthere is a promptly simple set in H ✸ obtained via an injury-free construction byTheorem 5.3.15.We will study several subclasses of the c.e. K-trivial sets that are of theform H ✸ .OftenH is a highness property applying to some ML-random setZ ≱ T ∅ ′ .Theweakreducibilities≤ LR and ≤ JT were defined in 5.6.1 and 8.4.13,respectively. The main highness properties relevant here are being LR-hard,being JT-hard, and being superhigh:

8.5 Subclasses of the K-trivial sets 357LRH = {C : ∅ ′ ≤ LR C},JTH = {C : ∅ ′ ≤ JT C},Shigh = {C : ∅ ′′ ≤ tt C ′ }In Theorem 5.6.30 we proved that LRH coincides with the class of uniformly a.e.dominating sets. By 8.4.16 and 8.6.2, JTH is properly contained in the class ofsuperhigh sets introduced in 6.3.13. The two latter classes coincide on the ∆ 0 2sets by 8.4.27. Hence, by Theorem 6.3.14, LRH is a proper subclass of JTH.We provide a basic fact. For a stronger result see Exercise 8.5.21.8.5.11 Proposition. The sets that are high 1 form a null class.Proof. Otherwise high 1 is conull by the zero-one law 1.9.12, so high 1 ∩ GL 1 ={C : ∅ ′′ ≤ T C ⊕∅ ′ } is conull. By 5.1.12 relativized to ∅ ′ , {C : A ≤ T C ⊕∅ ′ } isnull for each set A ≰ T ∅ ′ .LettingA = ∅ ′′ this yields a contradiction. ✷8.5.12 Proposition. (i) The classes LRH and JTH are null Σ 0 3 classes.(ii) (Simpson) Shigh is contained in a null Σ 0 3 class.Proof. (i) Since LRH ⊆ JTH ⊆ Shigh the classes are null. By 8.4.20 JTH is Σ 0 3.To prove that LRH is Σ 0 3, recall from Theorem 5.6.5 that ≤ LR is equivalent to≤ LK ,andnotethat∅ ′ ≤ LK C is equivalent to ∃b ∀y, σ,s∃t ≥ sU ∅′ (σ) =y[s] → (U ∅′ (σ)[t]↑ ∨ K C↾tt (y) ≤ |σ| + b).(ii) Note that a function f is d.n.c. relative to ∅ ′ if ∀x ¬f(x) =J ∅′ (x). Let Pbe the Π 0 1(∅ ′ )classof{0, 1}-valued functions that are d.n.c. relative to ∅ ′ .ByExercise 5.1.15 relative to ∅ ′ , the class {Z : ∃f ≤ T Z ⊕∅ ′ [f ∈ P ]} is null. Then,since GL 1 is conull, the class H = {Z : ∃f ≤ tt Z ′ [f ∈ P ]} is also null. This classcontains Shigh by 1.8.30 relative to ∅ ′ (note that q there remains computable).To show that H is Σ 0 3,fixaΠ 0 2 relation R ⊆ N 3 such that a string σ is extendedby a member of P iff ∀u ∃vR(σ,u,v). Let (Ψ e ) e∈N be an effective listing of truthtablereduction procedures. It suffices to show that {Z : Ψ e (Z ′ ) ∈ P } is a Π 0 2class. To this end, note that, as use Ψ X e (j) isindependentoftheoracleX,Ψ e (Z ′ ) ∈ P ↔ ∀x ∀t ∀u ∃s >t∃vR(Ψ Z′e ↾ x [s],u,v). ✷From Theorem 5.3.15 we conclude:8.5.13 Corollary. There is a promptly simple set in Shigh ✸ ⊆ JTH ✸ ⊆ LRH ✸ .✷In Proposition 3.4.17 we showed that the 2-random set Ω ∅′ is high. This contrasts withthe following fact, which is immediate from 8.5.12(ii).8.5.14 Corollary. No weakly 2-random set is superhigh. ✷A further results suggests that high 1 ∩ MLR is much larger than Shigh ∩ MLR. Byunpublished work announced in Kučera (1990, Remark 8), for each incomputable ∆ 0 2set A there is a high ∆ 0 2 ML-random set Z ≱ T A (while Shigh ✸ contains a promptlysimple set A). In particular, if A ∈ high ✸ 1 then A is computable. For the latter fact onecan also argue that Ω ∅′ is high and 2-random (3.4.17), and therefore forms a minimalpair with the high ML-random set Ω.

358 8 Classes of computational complexityWe prove some inclusion relations among subclasses of the c.e. K-trivial sets.8.5.15 Theorem. Consider the following properties of a c.e. set A.(i) A ∈ Shigh ✸ ;(ii)A ∈ JTH ✸ ;(iii)A ∈ LRH ✸ ;(iv) A is ML-coverable, namely, there is a ML-random set Z ≥ T A such that∅ ′ ≰ T Z (see 5.1.23);(v) for each ML-random set Z, if∅ ′ ≤ T A ⊕ Z then ∅ ′ ≤ T Z;(vi) A is K-trivial.The following implications hold: (i)⇒(ii)⇒(iii); (iii)⇒(iv)⇒(vi); (iii)⇒(v)⇒(vi).All these classes are closed downward under ≤ T within the c.e. sets. The classesgiven by (i), (ii), (iii) and (vi) are even known to be ideals.Proof. (i)⇒(ii)⇒(iii): Immediate since Shigh ⊇ JTH ⊇ LRH.(iii)⇒(iv): By Theorem 6.3.14 there is a ML-random set C < T ∅ ′ such that∅ ′ ≤ LR C.IfA ∈ LRH ✸ then A is ML-coverable via C.(iv)⇒(vi): See Corollary 5.1.23.(iii)⇒(v): By the previous implications, A is K-trivial and hence low for MLrandomness.We show that for each set A that is low for ML-randomness, if Z isML-random and ∅ ′ ≤ T A ⊕ Z then ∅ ′ ≤ LR Z. (Intuitively, since A is computablyweak, Z must be computationally strong.) The proof, due to Hirschfeldt, involvesrelativizing the van Lambalgen Theorem 3.4.6 to a set D: foranysetsZ and Rsuch that Z ∈ MLR D ,wehaveR ⊕ Z ∈ MLR D iff R ∈ MLR Z⊕D .Then,foreach set R, wehaveR ∈ MLR Z → R ⊕ Z ∈ MLR → R ⊕ Z ∈ MLR A → R ∈MLR Z⊕A → R ∈ MLR ∅′ .Thus∅ ′ ≤ LR Z.(v)⇒(vi): If A is not K-trivial then A is not a base for ML-randomness byTheorem 5.1.22. Hence A ≰ T Ω A .Thus∅ ′ ≰ T Ω A while ∅ ′ ≤ T A ⊕ Ω A byFact 3.4.16.✷8.5.16 Remark. A is called ML-cuppable if A ⊕ Z ≡ T ∅ ′ for some ML-random setZ< T ∅ ′ (such a Z is called a cupping partner for A). The class of c.e. non-ML-cuppablesets clearly contains the class given by (v); we show that it is still contained in (vi).Suppose that A is not K-trivial. First assume that A is low, then Ω A < T ∅ ′ is a cuppingpartner. If A is not low, by 1.6.10 let A ≡ wtt A 0 ⊕A 1 for low c.e. sets A 0,A 1.Oneofthesets A 0,A 1 is not K-trivial, say A 0. Then Ω A 0is a cupping partner for A 0 and hencefor A.Nies (2007) proved by a direct construction that there is a promptly simple MLnoncuppableset. An easier proof is obtained by noting that the class LRH is Σ 0 3, andhence LRH ✸ contains a promptly simple set by Theorem 5.3.15.It is unknown whether LRH ✸ coincides with the c.e. K-trivial sets, that is, whethereach c.e. K-trivial set is Turing below each LR-hard ML-random set. If so, then allthe classes (iii)–(vi) in 8.5.15 would coincide. The same applies to JTH ✸ .Kjos-Hanssenand Nies (20xx) have shown that Shigh ✸ is a proper subclass of the c.e. K-trivial sets.The following strengthens 8.5.13.8.5.17 Theorem. Each strongly jump traceable c.e. set A is in JTH ✸ .

8.5 Subclasses of the K-trivial sets 359Proof. Our plan is to define a class H ⊇ JTH and a benign cost function csuch that H ✸ contains every ∆ 0 2 set with a computable approximation obeying c(this is similar to Fact 5.3.13). Then we apply Theorem 8.5.4: some computableenumeration of A obeys c, soA ∈ H ✸ ⊆ JTH ✸ .Thisisliketheproofof8.5.5(ii).For the present proof we develop some theory of interest by itself. We actuallydefine first a class G ⊇ JTH, and then in a second step the class H ⊇ G. For anorder function h, we consider uniform sequences of c.e. operators (T n ) n∈N suchthat (T Y n ) n∈N is a Y -c.e. trace with bound h, for each oracle Y .LetG = {Y : ∃ order function h ∀f ≤ wtt ∅ ′∃ uniform sequence of c.e. operators (T n ) n∈N(T Y n ) n∈N is a Y -c.e. trace with bound h for f}.To see that JTH ⊆ G, notethat,bydefinition,Y ∈ JTH iff ∅ ′ is jump traceableby Y ,whichimpliesthat∅ ′ is c.e. traceable by Y (defined similar to 8.4.13), andhence Y ∈ G.By the usual argument in Theorem 8.2.3 we may fix an order function h (saythe identity) in the definition of G. Then we may as well let the sequence oftrace operators be the universal sequence V =(VeY ) e∈N for this order function,obtained as in the proof of Corollary 8.2.4 but for c.e. operators. This yields thesimpler expression G = {Y : ∀f ≤ wtt ∅ ′ ∀ ∞ nf(n) ∈ Vn Y }.For the sake of simplicity we formulate the next two lemmas for this universalsequence V =(VeY ) e∈N with this fixed bound h, eveniftheywouldactuallyworkfor any uniform sequence of c.e. operators (T n ) n∈N with bound h as above. Thefirst lemma is based on an argument due to Hirschfeldt.8.5.18 Lemma. There is a function f ≤ wtt ∅ ′ such that2 −n ≥ λ{Y : f(n) ∈ V Y n } for each n.Subproof. We define a computable approximation for f in the sense of Definition1.4.6. Let f 0 (n) =1foreachn. Lets>0. If 2 −n < λ{Y : f s−1 (n) ∈ V Y n,s}then we let f s (n) =f s−1 (n) + 1, otherwise f s (n) =f s−1 (n).Claim. For each n, s we have f s (n) ≤ 2 n h(n).We apply Exercise 1.9.15 for ɛ =2 −n .Supposef s (n) is incremented N>2 n h(n)times. Let C i = {Y : i ∈ VeY }. Foreachi ≤ N, sincei is not the final value off s (n), we have λC i ≥ ɛ. AlsoNɛ >h(n), so by 1.9.15 there is F ⊆ {1,...,N}such that #F = h(n) +1 and ⋂ i∈F C i ≠ ∅. IfY ∈ ⋂ i∈F C i then F ⊆ Vn Y ,contradiction.✸For a function f let H(f) ={Y : ∀ ∞ nf(n) ∈ V Y n }. The class H ⊇ G will be ofthe form H(f) forf as in the foregoing lemma.8.5.19 Lemma. Suppose f is a function as in Lemma 8.5.18. Then there is abenign cost function c f such that, if (B s ) s∈N is a computable approximation of aset B obeying c f , then B ≤ T Y for each ML-random set Y ∈ H(f).Subproof. We modify the argument in Fact 5.3.13. Let λn, s. f s (n) beacomputableapproximation of f. Wemaysupposethat2 −n ≥ λ{Y : f s (n) ∈ V Y n,s} for

360 8 Classes of computational complexityeach n, s. The cost function c f is defined as in (5.7) on page 189: let c f (x, s) =2 −xfor each x ≥ s. Ifxn 0such that f(m) isstableats 0 for m ≤ n 0 and σ ⋠ Y for each σ enumeratedinto G after stage s 0 .ToshowB ≤ T Y ,givenaninputx ≥ s 0 ,usingY as anoracle compute t>xsuch that ∀i [n 0 ≤ it.Letn ≤ x be thelargest number such that f r (i) =f t (i) foralli, n 0 ≤ i

8.6 Summary and discussion 3618.5.25. ⋄ Prove that for each nonempty Π 0 1 class P there is a superlow set Y ∈ P andac.e.K-trivial set B such that B ≰ T Y .Concludethatthereisac.e.K trivial setB ∉ H ✸ ,andinfactB ∉ ˜H ✸ . (Actually, for each benign cost function c there is sucha Y and a B obeying c.)8.5.26. ⋄ Problem. Is the class Shigh of superhigh sets Σ 0 3?8.6 Summary and discussionWe summarize the results on the absolute computational complexity of sets. Wealso discuss the interplay between computational complexity and the degree ofrandomness and interpret some results from this point of view.AdiagramofdownwardclosedpropertiesFigure 8.1 displays most of our properties closed downward under ≤ T .Itcontainsthe lowness properties and the complements of highness properties. Theproperties occurring frequently are framed in boldface.The lines indicate implications from properties lower down in the diagram toproperties higher up in the diagram. Up to a few exceptions, we know that nonot ≥ T 0'bounds only GL 2not of PAdegreenot ≥ LR 0'not superhigharray computablebounds only GL 1not of d.n.c.degreenot highLow(ΩΩ)lowc.e. traceablecomputablydominatedLow(MLR)superlowstrongly superlowjump traceablestrongly jumptraceablecomputablytraceablecomputableFig. 8.1. Downward closed properties.

362 8 Classes of computational complexityother implications hold. Open questions include whether strong jump traceabilityis equivalent to strong superlowness, or at least whether it implies superlowness.For further questions regarding possible implications see Problem 8.6.4.There are interesting intersection relations:Low(Ω) ∩ computably dominated ⇔ computable (8.1.18), andc.e. traceable ∩ computably dominated ⇔ computably traceable (8.2.16).The original, much larger diagram is due to Kjos-Hanssen (2004). It includes forinstance the properties of computing no ML-random set, or of being computedby a 2-random set; see Exercise 8.6.1 below.For the c.e. sets, several classes coincide, as Fig. 8.2 shows. No further implicationshold. The coincidences, starting from the bottom, are proved in 8.4.35,5.1.27, 8.4.23, and 8.4.26.< T 0'< LR 0'not superhighnot highlow 2lowarray computable⇔ c.e. traceablesuperlow ⇔ jump traceableLow(Ω) ⇔ Low(MLR)strongly jump traceable⇔ strongly superlowcomputableFig. 8.2. Downward closed properties for c.e. sets.Weak reducibilities, defined in Section 5.6, provide a general framework forlowness properties and highness properties. A weak reducibility ≤ W determinesthe lowness property C ≤ W ∅ and the dual highness property C ≥ W ∅ ′ .Table8.3gives an overview. For related results see Simpson and Cole (2007).Exercises.8.6.1. Insert the classes (i) BN1R = {A: ∀Z ≤ T A [Z ∉ MLR]} and(ii) BB2R = {A: ∃Z ≥ T A [Z is 2-random]} into Fig. 8.1.(iii) Identify the conull classes in Fig. 8.1.8.6.2. ⋄ (Kjos-Hanssen and Nies, 20xx) Note that a jump traceable set is not JT-hardand hence not LR-hard. However, show that there is a jump traceable superhigh set.Hint. Use the proof of the jump inversion theorem for truth-table reducibility due toMohrherr (1984): for every set A ≥ tt ∅ ′ there is a set B such that B ′ ≡ tt A.

8.6 Summary and discussion 363Table 8.3. Some weak reducibilities.Weak reducibility Lowness property Highness prop. Defined in≤ T computable ≥ T ∅ ′ 1.2.6≤ LR Low(MLR) u.a.e.d 5.6.1≤ JT jump traceable ≥ JT ∅ ′ 8.4.13A ′ ≤ tt B ′ superlow superhigh 1.2.20A ′ ≤ T B ′ low high pg. 25≤ CT comp. traceable ≥ T ∅ ′ 8.4.21≤ cdom comp. dominated ≥ T ∅ ′ 5.6.88.6.3. ⋄ For as many classes C in Fig. 8.1 as possible, show that C is only contained inthe indicated classes. (Most counterexamples are somewhere in this book.)8.6.4. ⋄ Problem. We ask whether no further implications hold in Fig. 8.1.(i) Can a c.e. traceable set be LR-hard? (ii) Can a set that bounds only GL 1 sets beLR-hard? (iii) Can a set that is low for Ω be superhigh?8.6.5. ⋄ Problem. Decide whether the sets that are computably dominated and in GL 1are closed downward under ≤ T .Computational complexity versus randomnessMany results relate the computational complexity of a set with its degree ofrandomness. As much as we would like a paradigm such as “to be more randommeans to be less complex”, in fact the relationship has no overall direction. Aparticular computational complexity class only exposes one aspect of the informationthat can be extracted from the set, and the direction of the relationshipdepends on this aspect. It also depends on further properties of the sets underdiscussion, such as being in a class of descriptive complexity. For instance, withinthe left-c.e. sets, more random means more complex: Schnorr randomness impliesbeing high, and ML-randomness even implies being weak truth-table complete(3.2.31). If we widen the scope to the ω-c.e. sets, the direction of the relationshipbetween complexity and randomness is already less clear because there are superlowML-random sets. There still is a promptly simple set Turing below all theω-c.e. ML random sets (8.5.23). On the other hand, a set that is Turing belowall the ∆ 0 2 ML-random sets is computable (1.8.39).We now look at ML-random sets in general, without imposing extra restrictionson their descriptive complexity. Numerous results suggest now that more randommeans computationally less complex. A ML-random set either computes ∅ ′ ,or is not even of PA-degree (4.3.8). Randomness notions stronger than MLrandomnessare incompatible with computing ∅ ′ . In fact, if Z is ML-random, thenZ is weakly 2-random if and only if Z and ∅ ′ form a minimal pair (page 135). Inparticular, a weakly 2-random set is not of PA degree. We review further resultsin this direction.1. For Z ∈ MLR we have Z is 2-random ⇔ Z is low for Ω (3.6.19).

364 8 Classes of computational complexity2. For any set C, MLR C is closed downward under ≤ T within MLR (5.1.16).3. If A and B are ML-random then A ≤ vL B ⇔ A ≥ LR B (5.6.2).For some particular aspects of the information extracted from the set, theopposite can happen.1. Each 2-random set is of hyperimmune degree (3.6.15), whileaML-randomsetcanbecomputablydominated.2. For n>1, each n-random set computes an n-fixed point free function(4.3.17), while an (n − 1)-random set may fail to do so.Some updatesIn recent work two classes have been shown to coincide with the strongly jumptraceable c.e. sets: (1) the class (ω-c.e.) ✸ of Exercise 8.5.23 and (2) the classShigh ✸ of Theorem 8.5.15. Each of these results implies that one can obtain apromptly simple strongly jump traceable set A via an injury-free construction.Namely, one builds a promptly simple set A that obeys the non-adaptive costfunction for being in H ✸ for the appropriate Σ 0 3 class H (Theorem 5.3.15). Theresults also yield new proofs of the theorem of Cholak, Downey and Greenberg(2008) that the strongly jump traceable c.e. sets are closed under ⊕.(1) For the first result, after Corollary 8.5.5 due to Greenberg and Nies (20xx)it remains to show the implication that each c.e. set in (ω-c.e.) ✸ is strongly jumptraceable. This has been proved by Greenberg, Hirschfeldt and Nies (20xx). Givenan order function h, theybuildaML-randomω-c.e. set Z such that if A is asuperlow c.e. set, then A ≤ T Z implies that A is jump traceable with bound h.Roughly, they make Z a “bad set” in that it tries to not compute A by exploitingthe changes of A. This forces A to change little.(2) The second result is due to Nies (20xx). Using appropriate benign costfunctions he shows that each strongly jump traceable c.e. set A is Turing beloweach superhigh ML-random set. For the converse implication, he combines thetechniques used for (1) with a version of Kucera coding introduced in Kjos-Hanssen and Nies (20xx) to make the bad set Z superhigh.The bound in Theorem 8.5.1 can be improved to λz.max(1, ⌊ √ log z⌋). Onemodifies the construction by letting P e j (b) callP e j−1 (b2e )in(1).InClaim1onenow proves that P e j (b) forj

9Higher computability and randomnessRecall that a randomness notion is introduced by specifying a test concept. Sucha test concept describes a particular kind of null class, and a set is random inthat sense if it avoids each null class of that kind. (For instance, in 3.6.1 weintroduced weak 2-randomness by defining generalized ML-tests, which describenull Π 0 2 classes. So a set is weakly 2-random iff it is in no null Π 0 2 class.)Up to now,allthe null classes given by tests werearithmetical. In this chapterwe introduce tests beyondthe arithmetical level. We give mathematical definitionsof randomness notions using tools from higher computability theory. Themain tools are the following. A relation B ⊆ N k ×(2 N ) r is Π 1 1 if it is obtained froman arithmetical relation by a universal quantification over sets. The relation Bis ∆ 1 1 if both B and its complement are Π 1 1.Thereisanequivalent representationof Π 1 1 relations where the members are enumerated at stages that are countableordinals. For Π 1 1 sets (of natural numbers) these stages are in fact computableordinals, i.e., the order types of computable well-orders.Effective descriptive set theory (see Ch. 25 of Jech 2003) studies effective versionsof notion from descriptive set theory. For instance, the ∆ 1 1 classes are effectiveversions of Borel classes. Higher computability theory (see Sacks 1990)studies analogs of concepts from computability theory at a higher level. For instance,the Π 1 1 sets form a higher analog of the c.e. sets, and, to someextent,the ∆ 1 1 sets canbeseenasananalogofthe computable sets.Amaindifference between effective descriptive set theory and higher computabilityisthat the latter also studies sets of countable ordinals. However,we still focus on sets of natural numbers: only the stages of enumerations areordinals. Thus, for us there is no need to distinguish between the two.While analogs of many notions from the computability setting exist in thesetting of higher computability, the results aboutthem often turn out different.The reason is that there are two new closure properties.(C1) The Π 1 1 and ∆ 1 1 relations are closed under number quantification.(C2) If a function f maps each number n in a certain effective way to acomputableordinal, then the range of f is bounded by a computable ordinal. Thisis the Bounding Principle 9.1.22.The theory of Martin-Löf randomness and K-complexityof strings is ultimatelybased on c.e. sets, so it can be viewed within the new setting. In Section 9.2,we transfer several results to the setting of Π 1 1 sets, for instance the MachineExistence Theorem 2.2.17. However, using the closure properties above we alsoprove that the analog of Low(MLR) coincideswith the ∆ 1 1 sets, while the analog

366 9 Higher computability and randomnessof the K-trivial sets doesnot. Theseresults aredueto Hjorth and Nies (2007).In Section 9.3 we study ∆ 1 1-randomness and Π 1 1-randomness. The tests aresimply the null ∆ 1 1 classes and the null Π 1 1 classes, respectively. The implicationsareΠ 1 1-randomness ⇒ Π 1 1-ML-randomness ⇒ ∆ 1 1-randomness.The converse implications fail.Martin-Löf (1970) was the first tostudy randomness in the setting of highercomputability theory. He suggested ∆ 1 1-randomness as the appropriate mathematicalconcept of randomness. His main result was that the union of all ∆ 1 1 nullclasses is a Π 1 1 class that is not ∆ 1 1 (see 9.3.11). Later it turned out that ∆ 1 1-randomness is the higher analog of both Schnorrandcomputable randomness.The strongest notion we consider is Π 1 1-randomness, which has no analog inthe setting of computability theory. This notion was first mentioned in Exercise2.5.IV of Sacks (1990), but called Σ 1 1-randomness there. Interestingly, there is auniversal test, namely,alargest Π 1 1 null class (9.3.6).In Section 9.4 we study lowness properties in the setting of higher computability.For instance, we consider sets that are ∆ 1 1 dominated, the higher analog ofbeing computably dominated. Using the closure properties (C1) and (C2), weshow that each Π 1 1-random set is ∆ 1 1 dominated. We also prove that lownessfor ∆ 1 1-randomness is equivalent toahigheranalogof computable traceability.Lowness for Π 1 1-randomness implies lowness for ∆ 1 1-randomness. It is unknownwhether some such set is not ∆ 1 1.Acitation such as “Sacks 5.2.I” refers to Sacks(1990),ourstandard referencefor background on higher computability. References to the original results canbe found there. Some of the results weonlyquote relyonelaborate technicaldevices such as ranked formulas.9.1 Preliminaries on higher computability theoryΠ 1 1 and other relationsFor sets X 1 ,...,X m ,welet X 1 ⊕ ... ⊕ X m = {mz + i: i

9.1 Preliminaries on higher computability theory 367For a set Z let p n (Z) ={x: 〈x, n〉 ∈Z}. Let ̂R = {〈e, n, Z〉: R(e, n, p n (Z))}.Then ∀n ∃Y R(e, n, Y ) ⇔ ∃Z ∀n ̂R(e, n, Z): to showthe implication from leftto right, supposethat for each n there is Y n such that R(e, n, Y n ). Using thecountable axiom of choice one can combine these witnesses into asinglewitnessZ = ⋃ n Y n × {n}, andp n (Z) =Y n for each n. Then ̂R(e, n, Z).9.1.2 Exercise. If R ⊆ 2 N is open then in a uniform way we have the followingequivalence: R is a Π 1 1 class ⇔ A R = {x: [x] ⊆ R} is a Π 1 1 set.Well-orders and computable ordinalsIn the following we will consider binary relations W ⊆ N × N with domain aninitial segment of N. Theycanbeencodedbysets R ⊆ N via the usual pairingfunction. We identify the relation with its code. Linear orders will be irreflexive.Note that a c.e. linear order is computable. A linear order R is a well-order ifeach nonempty subset of its domainhasaleast element. The class of well-ordersis Π 1 1.Furthermore, the index set {e: W e is a well-order} is Π 1 1.Given a well-order R and an ordinal α, we let |R| denote the order type of R,namely, the ordinal α such that (α, ∈) isisomorphicto R. Wesaythat α iscomputable if α = |R| for a computable well-order R. Each initial segmentof acomputable well-order is also a computable well-order, so the computableordinals are closed downwards. We let ω1Y denote the least ordinal that is notcomputable in Y .Theleast incomputable ordinal is ω1 ck (which equals ω1 ∅).An important example of a Π 1 1 class isC = {Y : ω Y 1> ω ck1 }. (9.1)To see that this class is Π 1 1,note thatY ∈ C ↔∃e [ Φ Y e is well-order & ∀i [W i is computable relation → Φ Y e ≁ = Wi ] ] .This can be put into Π 1 1 form because the Π 1 1 relations are closed under numberquantification. If ω1Y = ω1 ck we say that Y is low for ω1 ck .Representing Π 1 1 relations by well-ordersAc.e.set can be either described by a Σ 1 formula in arithmetic or by a computableenumeration. A Σ 0 1 class, ofthe form {X : ∃yR(X ↾ y )} for computable R,can be thought of as being enumerated at stages y ∈ N.In Definition 9.1.1 we introduced Π 1 1 classes by formulas in second order arithmetic.Equivalently, they can be described by a generalized type of enumerationwhere the stages are countable ordinals. It becomes possible to look at awholeset X in the course of ageneralizedcomputation (this is the global view of Section1.3). While one could introduce a formal model of infinite computations, wewill be content with viewingthe order type of a well-order computed from X viaaTuringfunctional as the stage when X enters the class.9.1.3 Theorem. Let k, r ≥ 0. GivenaΠ 1 1 relation B ⊆ N k × (2 N ) r , one caneffectively obtain a computable function p: N k → N such that

368 9 Higher computability and randomness〈e 1 ,...,e k ,X 1 ⊕ ...⊕ X r 〉∈B ↔ Φ X1⊕...⊕Xrp(e 1,...,e k )is a well-order. (9.2)Moreover, the expression on the right hand side is always a linear order. If r =0the oracle is the empty set by convention. If k =0then p is a constant.The order type of Φ X1⊕...⊕Xrp(e 1,...,e k )acountable ordinal α, weletis the stage at which the element enters B, soforB α = {〈e 1 ,...,e k ,X 1 ⊕ ...⊕ X r 〉: |Φ X1⊕...⊕Xrp(e 1,...,e k )| < α}. (9.3)Thus, B α contains the elements that enter B before stage α.The right hand side in (9.2) can be expressed by a universal set quantificationof an arithmetical relation. So, conversely, each relation given by (9.2) is Π 1 1.The idea to prove Theorem 9.1.3 is as follows, say in the case of a Π 1 1 class B.(1) One expresses B by universal quantification over functions, rather than sets:X ∈ B ↔ ∀f ∃nR(f(n),X), where f(n) denotes the tuple (f(0),...,f(n − 1)) as inSacks 5.2.I. The advantage of taking functions is that one can choose R computable,rather than merely Σ 0 2.(2) This yields a Turing functional Ψ B such that, for each X, Ψ B(X) isasetofcodesfor tuples in N ∗ (the sequence numbers as defined before 1.8.69), and X ∈ B iff Ψ B(X)is well-founded under the reverse prefix relation ≻ on sequence numbers (Sacks 5.4.II).(3) Using the length-lexicographical (also called Kleene–Brouwer) ordering, one caneffectively “linearize” Ψ B(X) (Sacks 3.5.III), and thus obtain a Turing functional Φ psuch that X ∈ B ↔ Φ X p is a well-order. Also see Jech (2003, Thm. 25.3, 25.12).IndicesAn index for p as a computable function (or the constant p itself in the caseof Π 1 1 classes) serves as an index for B in (9.2). Via (9.3) such an index yieldsan approximation of B at stages which are countable ordinals. We are mostlyinterested in the cases of Π 1 1 sets (k =1andr =0)andΠ 1 1 classes (k =0andr = 1). In the former case, by the convention that the oracle is ∅, wehavee ∈ B ↔ R e is a well-order,where R e = Φ ∅ p(e) . We may replace R e by a linear order of type ωR e + e +1,sowe will assume that at each stage at most one element is enumerated, and noneare enumerated at a limit stage. By the above,O = {e: W e is a well-order}is a Π 1 1-complete set. That is, O is Π 1 1 and S ≤ m O for each Π 1 1 set S. Theset of ordinal notations is Π 1 1-complete aswell(Sacks2.1.I).Inmost cases it isinessential which Π 1 1-complete set one chooses; we will stay with the one givenabove.For p ∈ N, welet Q p denote the Π 1 1 class with indexp. Thus,Q p = {X : Φ X p is a well-order}. (9.4)Note that Q p,α = {X : |Φ X p | < α}, sothat X ∈ Q p implies X ∈ Q p,|Φ Xp |+1.

9.1 Preliminaries on higher computability theory 369An index for a ∆ 1 1 set S ⊆ N is a pair of Π 1 1 indices for S and N − S. An indexfor a ∆ 1 1 class S ⊆ 2 N is a pair of Π 1 1 indices for the classes S and 2 N − S.RelativizationThe notions introduced above can be relativized toaset A.It suffices toincludeAas a further set variable in (9.2). For instance, S ⊆ N is a Π 1 1(A) set if S ={e: 〈e, A〉 ∈B} for a Π 1 1 relation B ⊆ N × 2 N ,andC ⊆ 2 N is a Π 1 1(A) classifC = {X : 〈X, A〉 ∈B} for a Π 1 1 relation B ⊆ 2 N × 2 N .The following set is Π 1 1(A)-complete:O A = {e: W A eis a well-order}.We frequently apply that a Π 1 1 object can be approximated by ∆ 1 1 objects.9.1.4 Approximation Lemma. (i) For each Π 1 1 set S and each α < ω ck1 ,theset S α is ∆ 1 1.(ii)ForeachΠ 1 1 class B and each countable ordinal α, the class B αis ∆ 1 1(R), for every well-order R such that |R| = α.Proof. We prove (ii) and leave (i) as an exercise. Let p ∈ N be an index for B.Then X ∈ B α ↔ ∃g [g is an isomorphism of Φ X p and a proper initial segmentof R], which is Σ 1 1(R). Moreover, X ∉ B α ↔ Φ X p is not a well-order ∨∃g [g is anisomorphism of R and an initial segment of Φ X p ], which is also Σ 1 1(R). ✷9.1.5 Remark. Borel classes were introduced on page 66. One can show that C ⊆ 2 Nis Borel ⇔ C is ∆ 1 1(A) forsomeA. FortheimplicationfromlefttorightoneletsA bea Borel code for C, namely,asetthatcodeshowC is built up from open sets using theoperations of complementation and countable intersection. The implication from rightto left follows from the Lusin separation theorem (Kechris, 1995, 14.7).Exercises.9.1.6. (Reduction Principle for Π 1 1 classes) Given Π 1 1 classes P and Q, onecaneffectivelydetermine disjoint Π 1 1 classes ˜P ⊆ P and ˜Q ⊆ Q such that P ∪ Q = ˜P ∪ ˜Q.9.1.7. Improve the indexing of the ∆ 1 1 classes introduced after (9.4): there are computablefunctions g and h such that (Q g(n) ) n∈N ranges over all the ∆ 1 1 classes and, foreach n, wehaveQ g(n) ∩ Q h(n) = ∅, andQ g(n) ∪ Q h(n) =2 N unless Q g(n) = ∅.9.1.8. Show that ω ck1 < ω O 1 .Π 1 1 classes and the uniform measure9.1.9 Theorem. (Lusin; see Sacks 6.2.II) Each Π 1 1 class is measurable.Proof. ℵ 1 denotes the least uncountable ordinal. Recall from (9.4) that Q p isthe Π 1 1 class with indexp. Foreachα < ℵ 1 , Q p,α is Borel by 9.1.4 and hencemeasurable. Since R has a countable dense subset, for each j there is a countableordinal β j such that λQ j,α = λQ j,βj for each α ≥ β j .Let β = sup j β j .It sufficesto showthat Q p −Q p,β is null for each p. If Y ∈ Q p −Q p,β then the well-order Φ Y phas an initial segment of order type β, whichcanbecomputed by Y .Therefore|Φ Y j | = β for some j. Thisshowsthat Q p − Q p,β ⊆ ⋃ j∈N {Y : |ΦY j | = β}, whichis null by the choice of β.✷

370 9 Higher computability and randomnessFor the following frequently used result see Sacks 1.11.IV. It states that themeasure of aclasshasthe same descriptive complexity asthe class itself. Notethat (ii) follows from (i).9.1.10 Measure Lemma. (i) For each Π 1 1 class B, λB is left-Π 1 1.AΠ 1 1 indexfor {q ∈ Q 2 : q

9.1 Preliminaries on higher computability theory 371To prove it, firstly, by Sacks 1.4.III and its proofthere is a set A ≤ T O,A< h O such that A ∈ S. Thenω1 A = ω1 ck .Secondly,bySacks7.6.II,ω1 A < ω1Band A ≤ h B imply O A ≤ h B.Thisisappliedfor B = O to obtain O A ≤ h O.Spector (1955) proved that if A is a Π 1 1 set then either A ∈ ∆ 1 1 or O ≤ h A(Sacks 7.2.II). Recall that we view the Π 1 1 sets as analogs of the c.e. sets. Thereducibility ≤ h on the Π 1 1 sets certainly behaves differently from ≤ T on the c.e.sets. Finite hyperarithmetical reducibility ≤ fin-h is more like ≤ T .9.1.17 Definition. A fin-h reduction procedure is a partial function Φ: {0, 1} ∗ →{0, 1} ∗ with Π 1 1 graph such that dom(Φ) is closed under prefixes and, if Φ(x) ↓and y ≼ x, then Φ(y) ≼ Φ(x). We write A = Φ Z if ∀n∃m Φ(Z ↾ m ) ≽ A ↾ n ,andA ≤ fin-h Z if A = Φ Z for some fin-h reduction procedure Φ.If A is hyperarithmetical then Φ = {〈x, A↾ |x| 〉: x ∈ {0, 1} ∗ } is Π 1 1,soA ≤ fin-h Zvia Φ for any Z.Many constructions of c.e. sets can be adapted to the setting of Π 1 1 sets and ≤ fin-h .For instance, one can transfer the proof of Theorem 1.6.8 to the new setting to showthat there are Π 1 1 sets A and B such that A | fin-h B. Note that, even though the use ofa fin-h reduction procedure is finite, the stages are now computable ordinals.Exercises.9.1.18. Prove the statements after Definition 9.1.12.9.1.19. Define a binary function g ≤ T (O) ′ such that for each ∆ 1 1 function h there is esuch that h(x) =g(e, x) foreachx. Conclude that O is not ∆ 1 1.A set theoretical viewReasoning about Π 1 1 sets andΠ 1 1 classes can often be simplified by emphasizingasettheoretical view. From now on we will use the symbols N and ω interchangeably.We assume familiarity with the constructible hierarchy, defined byrecursion on ordinals: for a set S we let L(0,S)bethe transitive closure of {S}∪S.L(α +1,S)contains the sets that are first-order definable with parameters in(L(α,S), ∈), and L(η,S)= ⋃ α

372 9 Higher computability and randomnessThe principle of transfinite recursion in L A (Sacks 1.6.VII) states that for eachΣ 1 over L A function I : L A → ω A 1 there is a Σ 1 over L A function f : ω A 1 → ω A 1such that f(α) =I(f ↾ α ) for each α < ω A 1 . (Here f ↾ α denotes the restrictionof f to α = {β : β < α}.)By Theorem 9.1.3 we can view Π 1 1 sets as being enumerated at stages that arecomputable ordinals. The following important theorem provides a further viewof this existential aspect of Π 1 1 sets.9.1.21 Theorem. S ⊆ N is Π 1 1 iff there is a Σ 1 -formula ϕ(y) such thatS = {y ∈ ω : (L(ω ck1 ), ∈) |= ϕ(y)}.To see that each Π 1 1 set S is of this form, suppose y ∈ S ↔ R y is well-orderedwhere R y is a computable relation effectively determined from y. Theformulaϕ(y) expressesthat R y is isomorphic to anordinal:∃α ∃g [g : R y∼ = (α, ∈)]. Suchan isomorphism g is in L(ω1 ck )bytransfinite induction. For the converse see forinstance Sacks 1.3.VII. Note that the proof of the converse works even when ϕcontains parameters from L(ω1 ck ).We say that D ⊆ (L A ) k is Σ 1 over L A ifthere is a Σ 1 formula ϕ such thatD = {〈x 1 ,...,x k 〉∈(L A ) k : (L A , ∈) |= ϕ(x 1 ,...,x k )}. Thus, by 9.1.21, S ⊆ Nis Π 1 1 iff S is Σ 1 over L(ω1 ck ).We often consider partial functions from L A to L A with agraphdefinedbyaΣ 1 formula with parameters. We say the function is Σ 1 over L A .Suchfunctionsare an analog of functions partial computable in A. Weprovideauseful tool.9.1.22 Bounding Principle. Suppose f : ω → ω A 1 is Σ 1 over L A .Thenthereis an ordinal α < ω A 1 such that f(n) < α for each n.Proof. Since L A satisfies Σ 1 replacement by 9.1.20, there is a set y ∈ L A suchthat ran(f) ⊆ y. SinceL A satisfies ∆ 0 separation, the set s of ordinals in y isin L A .Henceβ = ⋃ ran(f) ⊆ ⋃ s is in L A .Nowlet α = β +1.✷9.1.23 Remark. One can use Theorem 9.1.21 to build Π 1 1 sets S. Anenumerationof S over L(ω1 ck )isaΣ 1 function f : ω1 ck → ω ∪ {nil} such that S =ranf ∩ ω (here nilis a further element, say ω). A construction C of S is given by a Σ 1 function tellingus what to enumerate at stage α given the enumeration up to α. Formally,C is a Σ 1over L(ω1 ck ) function mapping f ↾ α to the number to be enumerated at α, ortonilifno number is enumerated. By transfinite recursion in L(ω1 ck ), for each construction Cauniqueenumerationf exists.9.1.24 Exercise. Use the Bounding Principle to show that each ∆ 1 1 well-order R isisomorphic to a computable well-order.9.2 Analogs of Martin-Löf randomness and K-trivialityWe develop an analog ofthe theory of ML-randomness and K-triviality basedon Π 1 1 sets. To do so we view some important questions of Chapters 2, 3, and 5in the higher setting. The new results aredueto Hjorth and Nies (2007).

9.2 Analogs of Martin-Löf randomness and K-triviality 373Π 1 1 Machines and prefix-free complexity9.2.1 Definition. A Π 1 1-machine is a possibly partial function M : {0, 1} ∗ →{0, 1} ∗ with aΠ 1 1 graph. For α ≤ ω1 ck we let M α (σ) =y if 〈σ,y〉∈M α .Wesaythat M is prefix-free if dom(M) isprefix-free.9.2.2 Proposition. There is an effective listing (M d ) d∈N of all the prefix-freeΠ 1 1-machines.Proof. Let (S d ) d∈N be an effective listing of the Π 1 1 subsets of {0, 1} ∗ × {0, 1} ∗ .Thus 〈σ,y〉∈S d ↔ Rσ,y d is a well-order, where (Rσ,y)isauniformly d c.e. sequenceof linear orders as after Theorem 9.1.3. Now let 〈σ,y〉∈M d ↔Rσ,y d is a well-order &∀〈ρ,z〉∀g [(ρ ≺ σ ∨ (ρ = σ & z ≠ y)) →g is not an isomorphism of Rρ,z d with aninitial segment of Rσ,y].d(Informally, the machine has not halted on a proper prefix of σ, andhasnotproduced an output other than y on input σ, before the stage given by theorder type of Rσ,y.) e Clearly this is a Π 1 1 condition uniformly in d. If S d is aprefix-free Π 1 1-machine then M d = S d .✷As a consequence, there is an optimal prefix-free Π 1 1-machine.9.2.3 Definition. The prefix-free Π 1 1-machine U is given by U(0 d 1σ) ≃ M d (σ).Let K(y) = min{|σ|: U(σ) =y}.Forα ≤ ω ck1 let K α (y) = min{|σ|: U α (σ) =y}.Since U has Π 1 1 graph, the relation “K(y) ≤ u” isΠ 1 1 and, by 9.1.4, for α < ω ck1the relation “K α (y) ≤ u” is∆ 1 1.MoreoverK ≤ T O.(Here(ρ) 0 is the firstcomponent r of the pair ρ = 〈r, y〉.)Similar to 2.2.15,aΠ 1 1 set W ⊆ N×{0, 1} ∗ is called a Π 1 1 bounded request set if∑ρ 2−(ρ)0 [[ρ ∈ W ]] ≤ 1. (Here (ρ) 0 is the first component r of the pair ρ = 〈r, y〉.)9.2.4 Theorem. From a Π 1 1 bounded request set W one can effectively obtain aprefix-free Π 1 1-machine M such that ∀〈r, y〉 ∈W ∃w [|w| = r & M(w) =y].Proof. By Theorem 9.1.3 let g be a computable function such that x ∈ W ↔ S x isawell-order,whereS x = Φ ∅ g(x)is a computable linear order. Recall that for x ∈ W ,we view the order type α = |S x| as the stage when x is enumerated into W .Asnotedafter 9.1.3 we may assume that at each stage at most one element is enumerated, andnone at a limit stage. We turn this enumeration into a construction (in the sense ofRemark 9.1.23) of a prefix-free Π 1 1-machine M. WeletR 0 = {∅}. Ateachstageγ ≥ 0we define an antichain R γ ∈ L(ω1 ck )ofstrings.ThesetofextensionsofstringsinR γ isour reservoir of future w-values. Strings in this set are called unused. WearegivenR αfor α < γ. AsinTheorem2.2.17,witheachstringx we associate the half-open intervalI(x) ⊆ [0, 1) of real numbers with binary representation extending x.Construction of M and of finite sets R γ ⊆ {0, 1} ∗ (γ < ω ck1 ).At a successor stage α = β +1,ifarequest 〈r, y〉 is enumerated into W ,wewillfindastring w of length r and put 〈w, y〉 into M. Letz be the longest string in R β of length≤ r. Choosew = w α so that I(w) istheleftmostsubintervalofI(z) oflength2 −r ,i.e., let w = z0 r−|z| . To obtain R α,firstremovez from R β .Ifw ≠ z then also add thestrings z0 i 1, 0 ≤ i

374 9 Higher computability and randomnessFor a limit ordinal η, weletR η = {x: ∃γ < η ∀α [γ < α < η → x ∈ R α]}.Verification. We will see that a string x can appear in R α at most once, so that actuallyR η(x) =lim γ→ηR γ(x) foralimitordinalη. Intheclaimbelowweverifyanumberofproperties in order to show that for each request 〈r, y〉, z as above exists, and thereforewe can assign a string w of length r to the request. LetE α = ⋃ {I(x): x ∈ R α}be the set of real numbers corresponding to R α.Atalimitstageη, letG η = ⋂ β

9.2 Analogs of Martin-Löf randomness and K-triviality 375(iv) Again, this is clear for successor stages α = β +1, in which case we may defineP α = P β .Ifα = η is a limit stage, let P η be the intersection of the sets P γ and thecomplements of the null classes G γ − E γ from (i) for γ ≤ η. Thenforeachβ < η, theset P η is partitioned by E β and I(w γ), γ ≤ β, w γ defined. So P η is partitioned intoG η and the sets I(w γ)forγ < η. Since G η is partitioned on P η into the intervals I(z),z ∈ R η,wehaveshown(iv)forη.By (iv) the Π 1 1-machine M is prefix-free.✷One can use the foregoing Theorem 9.2.4 tocharacterize K by its minimality,similar to Theorem 2.2.19. We skip this, but we do adapt the Coding Theorem2.2.25 by applying 9.2.4. Theprobability that a prefix-free Π 1 1-machine Doutputs astring x is P D (x) =λ[{σ : D(σ) =x}] ≺ .Forα ≤ ω1 ck ,let P D,α (x) =λ[{σ : D α (σ) =x}] ≺ .Clearly2 −K(x) ≤ P U (x). We show ∀x 2 c 2 −K(x) >P D (x)for some constant c. Thisalsoholdsat certain ordinal stages. For g : ω1 ck → ω1 ck ,we say that a limit ordinal η ≤ ω1 ck is g-closed if ∀α < η [g(α) < η].9.2.5 Theorem. For each prefix-free Π 1 1-machine D there is a Σ 1 over L(ω1 ck )function g D : ω1 ck → ω1 ck and a constant c such that for each g D -closed η ≤ ω1ckwe have ∀x 2 c 2 −K η (x) >P D,η (x).Proof. We build a Π 1 1 bounded request set W , accounting the enumeration of requests〈r, x〉 against the open sets generated by the D-descriptions of x. Atstageα, ifx is astring, r ∈ N is least such that P D,α(x) ≥ 2 −r+1 ,andtherequest〈r, x〉 is not in Wyet, then we put 〈r, x〉 into W .For a string x, letα x be the greatest stage at which a request 〈r, x〉 is put into W .Then P D,αx (x) ≥ 2 −r+1 . Hence all such requests together contribute at most 1/2. Thetotal weight of all requests 〈r ′ ,x〉 enumerated at previous stages is at most 2 −r ,sincer ′ >rfor such a request, and there is at most one for each length r ′ .ThusW is abounded request set.Let c W be the coding constant for W given by Theorem 9.2.4. The function g D isthe delay it takes the optimal Π 1 1-machine to react to an enumeration of a requestinto W .Thusg D(α) =µβ. ∀〈r, x〉 ∈W α [K β (x) ≤ r + c W ]forα < ω1 ck ,whichisΣ 1over L(ω1 ck ). If r is least such that P D,η(x) > 2 −r+1 ,thenattheleaststageα < η whereP D,α(x) ≥ 2 −r+1 ,weenumerate〈r, x〉 and cause K η(x) ≤ K gD (α)(x) ≤ r + cW ,bythe hypothesis that η is g D-closed. Since r was chosen least, we have 2 −r+2 ≥ P D,η(x),hence 2 c W +3 2 −K η (x) > 2 −r+2 ≥ P D,η(x). Thus c = c W +3isasrequired. ✷As in Theorem 2.2.26, one can apply Theorem 9.2.5 to boundthe number ofstrings of a given length with alowK-complexity.9.2.6 Theorem. There is a constant c ∈ N and a Σ 1 over L(ω1 ck ) functiong : ω1 ck → ω1 ck such that, for each g-closed η ≤ ω1 ck ,∀b ∀n #{x: |x| = n & K η (x) ≤ K η (n)+b} ≤ 2 c 2 b .Proof. Let D be the prefix-free machine given by D(σ) =|U(σ)|. Let g be thefunction g D obtained in the foregoing theorem for D, andlet c be the constantsuch that 2 c 2 −K η (n) >P D,η (n) for each n and each g-closed η. Nowweconcludethe argument as in the proof of Theorem 2.2.26.✷9.2.7. ⋄ Carry out the Π 1 1 version of 2.2.24(ii) for a simple proof of a variant of 9.2.4.

376 9 Higher computability and randomnessA version of Martin-Löf randomness based on Π 1 1 setsA Π 1 1-ML-test is a sequence (G m ) m∈N of open setssuchthat ∀m ∈ N λG m ≤ 2 −mand the relation {〈m, σ〉: [σ] ⊆ G m } is Π 1 1 (by Exercise 9.1.2 it is equivalent torequire that the classes G m be uniformly Π 1 1). A set Z is Π 1 1-ML-random ifZ ∉ ⋂ m G m for each Π 1 1-ML-test (G m ) m∈N .Let MLR denote the class of Π 1 1-ML-random sets. For b ∈ N let R b =[{x ∈ {0, 1} ∗ : K(x) ≤ |x| − b}] ≺ .9.2.8 Proposition. (R b ) b∈N is a Π 1 1-ML-test.Proof. By the remark after 9.2.3 the relation {〈b, σ〉: [σ] ⊆ R b } is Π 1 1. Oneshows λR b ≤ 2 −b as in the proof of Proposition 3.2.7.✷Hjorth and Nies (2007) proved the higher analog of Schnorr’s Theorem 3.2.9.Thus Z is Π 1 1-ML-random iff Z ∈ 2 N − R b for some b. (Theproof is harder inthe new setting because of the limit stages.) Since ⋂ b R b is Π 1 1, this implies thatthe class of Π 1 1-ML-random sets isΣ 1 1.We provide two examples of Π 1 1-ML-random sets.1. By the Gandy Basis Theorem 9.1.16 there is a Π 1 1-ML-random set Z ≤ T Osuch that O Z ≤ h O.2. Let Ω = λ[dom U] ≺ = ∑ σ 2−|σ| [[U(σ)↓]]. Note that Ω is left-Π 1 1.Theproof ofTheorem 3.2.11 shows that Ω is Π 1 1-ML-random (the stage variable t now denotesacomputable ordinal).The reducibility ≤ fin-h was introduced in 9.1.17. We say that A ≤ wtt-h Z ifA = Φ Z for some fin-h reduction such that the use function is computablybounded. The following is analogous to Theorem 3.3.2.9.2.9 Theorem. Let Q be a closed Σ 1 1 class of Π 1 1-ML-random sets such thatλQ ≥ 1/2 (say Q =2 N −R 1 ). For each set A there is Z ∈ Q such that A ≤ wtt-h Z.Proof. Let f be the function from the proof of Theorem 3.3.2. Let ̂Q = Paths(T )where T is the tree defined by (3.4). Then ̂Q is a closed Σ 1 1 class. Using the purelymeasure-theoretic Lemma 3.3.1, we may define Z exactly as in that proof, andZ ∈ ̂Q because ̂Q is closed. For a Σ 1 1 class S given by an index, emptiness is aΠ 1 1 property. Hence it becomes apparent at an ordinal stage α < ω1 ck .We describe a reduction for A ≤ wtt-h Z,wheref(n+1) is the computable boundon the use for input n. Todetermine A(r), let x = Z ↾ f(r) ,andlet y = Z ↾ f(r+1) .Find a stage α < ω1 ck such that ̂Q α ∩ [{v : x ≼ v & |v| = |y| & v< L y}] ≺ = ∅ or̂Q α ∩ [{v : x ≼ v & |v| = |y| & v> L y}] ≺ = ∅. Inthe first case output 0, in thesecond case output 1.✷9.2.10 Exercise. Show Ω ≡ wtt O. Hint. Modify the proof of Proposition 3.2.30.An analog of K-trivialityRecall from 9.2.3 that K(x) denotes the prefix-free complexity of x in the settingof Π 1 1 sets. We study sets that are K-trivial. For technical reasons we also considerK-triviality at certain limit stages.

9.2 Analogs of Martin-Löf randomness and K-triviality 3779.2.11 Definition. A is K-trivial if ∃b ∀nK(A ↾ n ) ≤ K(n) +b. Given a limitordinal η ≤ ω1 ck ,wesaythat A is K-trivial at η if ∃b ∀nK η (A ↾ n ) ≤ K η (n)+b.(Thus, being K-trivial is the same as being K-trivial at ω1 ck .)One can modify the proof of Proposition 5.3.11 in order to show:9.2.12 Proposition. There is a K-trivial Π 1 1 set A that is not ∆ 1 1. ✷The proof is by a construction in the sense of Remark 9.1.23. It uses the MachineExistence Theorem in the version 9.2.4.Fix b and η ≤ ω1 ck .Thesets that are K-trivial via b at η are the paths throughthe tree T η,b = {z : ∀u ≤ |z| K η (z ↾ u ) ≤ K η (u) +b}. If η < ω1 ck then T η,bis ∆ 1 1 because U η is a ∆ 1 1 set by 9.1.4(i) and ∆ 1 1 sets areclosedundernumberquantification. Let g D be the function obtained in 9.2.5 where D(x) ≃ |U(x)|.9.2.13 Proposition. Let η ≤ ω1 ck be g D -closed.(i) There is c ∈ N such that the following holds: for each b, atmost2 c+b setsare K-trivial at η with constant b.(ii) If η < ω1 ck and A is K-trivial at η then A is hyperarithmetical.(iii) Each K-trivial set is computable in O.Proof. Let c be as in Theorem 9.2.6. The size of each level of T η,b is at most2 c+b ,whichshows(i).Note that each path A of T η,b is isolated, and hence computable in T η,b .For(ii), if η < ω1 ck this shows that A is hyperarithmetical. For (iii), note that sinceK ≤ T O, the tree T ω ck1 ,b is computable in O. Nowweargueasin(ii). ✷9.2.14 Lemma. If A is K-trivial and ω A 1 = ω ck1 then A is hyperarithmetical.Proof. Suppose A is K-trivial via b. Weshowthat A is K-trivial at η via b forsome g D -closed η < ω1 ck ,whereg D is the function from the proof of 9.2.13. Wedefine by transfinite recursion(seebefore 9.1.23) a function h: ω → ω1 ck whichis Σ 1 over L A :let h(0) = 0 andh(n +1)=µβ >g D (h(n)). ∀m ≤ nK β (A↾ m ) ≤ K β (m)+b.Since A is K-trivial, h(n) is defined for each n ∈ ω. Let η =sup(ranh), thenη < ω1A = ω1 ck by the Bounding Principle 9.1.22, so η is as required. ✷Adapting the relevant proofs from Section 5.2, one can show that the class K of K-trivial sets is closed under ⊕ and closed downward under ≤ wtt-h .Inparticular,O isnot K-trivial, as Ω ≡ wtt O by 9.2.10. The golden run method of Section 5.4 can beadapted to show that K is even closed downward under ≤ fin-h .However,K is not closeddownward under ≤ h ,sinceO ≤ h A for the Π 1 1 set A ∈ K from Proposition 9.2.12.Lowness for Π 1 1-ML-randomnessMLR A denotes the class of sets whichareΠ 1 1-ML-random relative to A, andAis low for Π 1 1-ML-randomness if MLR A = MLR. A is called a base for Π 1 1-MLrandomnessif A ≤ fin-h Z for some Z ∈ MLR A (see Definition 9.1.17 for ≤ fin-h ).

378 9 Higher computability and randomnessBy Theorem 9.2.9, a set that is low for Π 1 1-ML-randomness is a base for Π 1 1-MLrandomness.The following contrasts with Theorem 5.1.19:9.2.15 Theorem. A is a base for Π 1 1-ML-randomness ⇔ A is ∆ 1 1.Proof. ⇐: A ≤ fin-h Z for each Z, soA is a base for Π 1 1-ML-randomness.⇒: Suppose A = Φ Z for some fin-h reduction procedure Φ (see 9.1.17) andZ ∈ MLR A .Assumefor a contradiction that A is not ∆ 1 1.1. We show that ω1 A = ω1 ck .Note that the class {Y : A = Φ Y } is null by Theorem9.1.13 (or by directly adapting Theorem 5.1.12 to the setting of ≤ fin-h ).Similar to Remark 5.1.13, for each n letV n = S Φ A,n =[{ρ: A↾ n≼ Φ ρ }] ≺ =[{ρ: ∃α < ω ck1 A↾ n ≼ Φ ρ α}] ≺ .If ω1 ck < ω1 A then by the Approximation Lemma 9.1.4(i) relative to A, the set{〈n, ρ〉: ∃α < ω1 ck A ↾ n ≼ Φ ρ α} is ∆ 1 1(A). So by the Measure Lemma 9.1.10(ii)the function h(n) =µn. λV n ≤ 2 −n is ∆ 1 1(A). Then (V h(n) ) n∈N is a Π 1 1-ML-testrelative to A which succeeds on Z, contrary to the hypothesis that Z ∈ MLR A .2. We show that A is K-trivial, which together with ω1 A = ω1 ck implies that Ais ∆ 1 1 by Lemma 9.2.14. Wewanttoadaptthe proof of Theorem 5.1.22, which (inthe setting of computability theory) states that each base for ML-randomness Ais low for K. First we observe that (still in that setting) the proof can be modifiedto showdirectly that A is K-trivial: we replace U by the oracle prefix-freemachine M given by M X (σ) ≃ X ↾ U(σ) .Themodifiedproof now shows that∀yK(y) ≤ + KM A (y). For y = A↾ n,weobtain K(A↾ n ) ≤ + KM A (A↾ n) ≤ + K(n).Now we adapt this (already modified) proof to the present setting, where Φis a fin-h reduction procedure. Fix a parameter d ∈ N. If U τ (σ) convergesatastage α < ω1 ck ,westart feeding open sets Cd,σ τ ,andcontinue to dosoaslong as λCd,σ τ < 2−|σ|−d .Werestrict the enumeration into Cd,σ τ tosuccessorstages. For limit stages η we define Cd,σ τ [η] =⋃ α

9.3 ∆ 1 1-randomness and Π 1 1-randomness 379Schnorr tests andhyperarithmetical martingales are all equivalent in strength tonull ∆ 1 1 classes (9.3.3). Moreover, Chong, Nies and Yu (2008) proved that eachnull Σ 1 1 class is contained in a null ∆ 1 1 class, so Σ 1 1-randomness also coincideswith ∆ 1 1-randomness. However, adapting the methods of Section 7.4, there isa ∆ 1 1-random set of slowly growing initial segment complexity, which thereforeis not Π 1 1-ML-random (Theorem 9.3.4).9.3.2 Definition. Z is Π 1 1-random if Z avoids each null Π 1 1 class.Each Π 1 1-random set Z satisfies ω1Z = ω1 ck because the Π 1 1 class {A: ω1 A > ω1 ck }is null by 9.1.15. Thus, the Π 1 1-ML-random set Ω is not Π 1 1-random, as Ω ≡ wtt Oby 9.2.10. In fact we show that Z is Π 1 1-random iff Z is ∆ 1 1-random and ω1Z = ω1 ck .Therefore, within the ∆ 1 1-random sets, the Π 1 1-random sets arecharacterized byalownessproperty (for the new setting, one that is closed downward under ≤ h ).This is similar to the characterizations of the weakly 2-random and the 2-randomsets within the ML-random sets inSection 3.6.Notions that coincide with ∆ 1 1-randomnessA Π 1 1-Schnorr test is a Π 1 1-ML-test (G m ) m∈N such that λG m is left-∆ 1 1 uniformlyin m. Asupermartingale M : {0, 1} ∗ → R + ∪{0} is hyperarithmetical if its undergraph{〈x, q〉 : q ∈ Q 2 & M(x) >q} is ∆ 1 1.Recallthat the success class Succ(M),defined in 7.1.1, is null by 7.1.15. By the following, ∆ 1 1-randomness coincides withthe higher analogs of both Schnorrandcomputable randomness. This differenceto the computability theoretic case stems from the closure properties (C1) and(C2) in the chapter introduction.9.3.3 Theorem. (i) Let A be a null ∆ 1 1 class. Then A ⊆ ⋂ G m for some Π 1 1-Schnorr test {G m } m∈N such that λG m =2 −m for each m. Infact,therelation {〈m, σ〉: [σ] ⊆ G m } is ∆ 1 1.(ii) If (G m ) m∈N is a Π 1 1-Schnorr test then ⋂ m G m ⊆ Succ(M) for a hyperarithmeticalmartingale M.(iii) Succ(M) is a null ∆ 1 1 class for each hyperarithmetical supermartingale M.Proof. (i) By Sacks 1.8.IV, for each m we may effectively obtain a ∆ 1 1 openset G m such that A ⊆ G m and λ(G m − A) ≤ 2 −m .It now suffices to showthat from a ∆ 1 1 open set S and a rational q ≥ λS we may in an effective wayobtain a ∆ 1 1 open set ˜S such that S ⊆ ˜S and λ ˜S = q. Onecaneasilyadaptthe proof of Lemma 1.9.19 to the new setting. As before, we define a functionf : [0, 1) R → [0, 1) R by f(r) =λ(S ∪ [0,r)). For x ∈ Q 2 , f(x) isauniformlyleft-∆ 1 1 real by 9.1.10. So, by the closure under number quantification, the least tsuch that f(t) =q is ∆ 1 1 effectively in q. Nowlet ˜S = S ∪ [0,t).(ii) Let f(m, k) be the least α such that λG m − λG m,α ≤ 2 −k . By 9.1.15f(m, k) < ω1 ck for each m, k. By9.1.10and9.1.21,f is Σ 1 over L(ω1 ck ), soby the Bounding Principle 9.1.22 there is β < ω1 ck such that ∀m, k f(m, k) < β.By 7.1.15, ⋂ m G m ⊆ Succ(M) for the martingale M = ∑ m M G m. Since λG m =λG m,β we have M = ∑ m M G m,β,soM is hyperarithmetical by 9.1.10 and 9.1.4.

380 9 Higher computability and randomness(iii) We have Z ∈ Succ(M) ↔ ∀d ∃nM(Z ↾ n ) ≥ d, soSucc(M) is∆ 1 1 by theclosure of ∆ 1 1 relations under number quantification.✷The foregoing characterization of ∆ 1 1-randomness via hyperarithmetical martingalescan be used to separate itfrom Π 1 1-ML-randomness.9.3.4 Theorem. For every unbounded nondecreasing hyperarithmetical functionh there is a ∆ 1 1-random set Z such that ∀ ∞ nK(Z ↾ n | n) ≤ h(n).Proof. We adapt the proof of 7.4.8. Most of the basics developed for computablemartingales carry over to the case of hyperarithmetical martingales, forinstance Proposition 7.3.8 that one can assume the values are in Q 2 .Thepartialcomputable martingales of Definition 7.4.1 now become partial martingales witha Π 1 1 graph. As before, they can be listed effectively. The partial computablefunction F in Lemma 7.4.10 is now a function with aΠ 1 1 graph.✷The higher analog of Schnorr’s Theorem 3.2.9 in Hjorth and Nies (2007) implies:9.3.5 Corollary. There is a ∆ 1 1-random set that is not Π 1 1-ML-random. ✷By Theorem 9.1.11 the class of ∆ 1 1-random sets isnot Π 1 1.Inparticular, there isno largest null ∆ 1 1 class. However, the class of ∆ 1 1-random sets isΣ 1 1 by 9.3.11.More on Π 1 1-randomnessThere are only countably many Π 1 1 classes, so the union of all null Π 1 1 classes isnull. We show that it is also a Π 1 1 class. In particular, there is a universal testfor Π 1 1-randomness. Theorem 1A-2 in Kechris (1975) is more general: it not onlyimplies this result, but also shows that there is a largest countable Π 1 1 class, anda largest thin Π 1 1 class (a class is thin if it has no perfect subclass). Also seeMoschovakis (1980, Thm 4F.4).9.3.6 Theorem. There is a null Π 1 1 class Q such that S ⊆ Q for each nullΠ 1 1 class S.Proof. We claim that one may effectively determine from a Π 1 1 class S a nullΠ 1 1 class Ŝ ⊆ S such that Ŝ = S in case S is already null. Assuming this, let Q pbe the Π 1 1 class with indexp as defined in (9.4). Then Q = ⋃ ̂Q p p is Π 1 1,soQ isas required.To prove the claim, if S = Q p let Ŝ = {X ∈ S : S |Φ X p |+1 is null}. BytheApproximation Lemma 9.1.4(ii) (and its proof) wecancompute from p (independentlyof X) anumberthat is an index for S |Φ X p |+1 as a ∆ 1 1(X) classifR = Φ X p is well-ordered.The proof of (ii) of the Measure Lemma 9.1.10 can be extended to showthatthere is a computable function taking us (still independently of X) from an indexfor a ∆ 1 1(X) classto anindexfor its measure as a left-∆ 1 1(X) real number. Toexpress that this real number equals 0 merely involves quantification over therationals. Therefore Ŝ is a Π1 1 class, and its index can be computed from an indexfor S. Since{X : ω1X > ω1 ck } is null by 9.1.15, Ŝ is contained in the union of a

9.4 Lowness properties in higher computability theory 381null class and all the S α , α < ω1 ck that are null, hence Ŝ is null. When S is nullevery S α for α < ω 1 is null, and therefore Ŝ = S.✷Applying the Gandy Basis Theorem 9.1.16 to the Σ 1 1 class S =2 N − Q yields:9.3.7 Corollary. There is a Π 1 1-random set Z ≤ T O such that O Z ≤ h O. ✷This contrasts with the fact that in the computability setting already a weakly2-random set forms a minimal pair with ∅ ′ (page 135) that {Y : ω1Y > ω1 ck } is anull Π 1 1 class:For each Π 1 1 class S we have S ⊆ {Y : ω1Y > ω1 ck } ∪ ⋃ α ω1 ck } is a null Π 1 1 class:9.3.8 Fact. Q = {Y : ω1 Y > ω1 ck } ∪ ⋃ α

382 9 Higher computability and randomnessProof. Let g ≤ T (O) ′ be as in Exercise 9.1.19. If ω1 A > ω1 ck then O ≤ h A bySpector’s Theorem 9.1.14, andhenceg ≤ h A.Let f = λn. g(n, n) +1, thenf ≤ h A,and∃nf(n) >h(n) for each hyperarithmetical function h. Therefore Ais not hyp-dominated.✷Chong, Nies and Yu (2008) proved that the converse implication fails: if G ⊆ Nis a member of each dense ∆ 1 1 open set (this property is a higher analog of weak1-genericity) then, adapting 1.8.48, p G is not dominated by any hyperarithmeticalfunction. A stronger genericity condition (corresponding to 1-genericity) implies thatω1G = ω1 ck .9.4.3 Theorem. Each Π 1 1-random set A is hyp-dominated.Proof. Suppose the Π 1 1 relation S is a reduction procedure for f ≤ h A asin 9.1.12 such that {〈n, y〉: 〈n, y, X〉 ∈S} is the graph of apartial functionfor each X. Thefunction n ↦→ α n =min{α: ∃y〈n, y, A〉 ∈S α } is Σ 1 over L A ,so by the Bounding Principle 9.1.22 relative to A and since ω1A = ω1 ck , there isβ < ω1 ck such that α n < β for each n.Let r(n) bethe least k such that 2 −n ≥ λ{X : ∃i ≥ k 〈n, i, X〉 ∈S β }.Notethat r(n) exists byourfunctionality assumption on S X ,andthat the function ris hyperarithmetical because S β is ∆ 1 1 and by the Measure Lemma 9.1.10(ii).Then {X : ∃ ∞ n ∃i ≥ r(n)[〈n, i, X〉 ∈S β ]} is a null ∆ 1 1 class. Since A avoids thisclass, f is dominated by r.✷As a consequence, by Corollary 9.3.7 there is a hyp-dominated set Z ∉ ∆ 1 1 suchthat Z ≤ T O and O Z ≤ h O.Exercises.9.4.4. An analog of highness is being hyp-high, namely,somef ≤ h A dominates eachhyperarithmetical function. Show that if A is not hyp-high, then ω1A = ω1 ck .(Thisstrengthens 9.4.2).9.4.5. Show that there is a hyp-high set A such that A ≤ T O and O A ≤ h O (andhence ω1A = ω1 ck ).9.4.6. (Kjos-Hanssen, Nies, Stephan and Yu 20xx) We say that Z is weakly ∆ 1 1-randomif Z is in no closed null ∆ 1 1 class. Show that Z is Π 1 1-random ⇔ Z is hyp-dominatedand weakly ∆ 1 1-random.TraceabilityTraceability in the computability setting was studied in Section 8.2. In contrastto the computability setting, the analogs of c.e. and of computable traceabilitycoincide, again because of the Bounding Principle. Moreover, this class characterizeslowness for ∆ 1 1-randomness.9.4.7 Definition. (i) Let h be a nondecreasing ∆ 1 1 function. A Π 1 1 trace(∆ 1 1 trace) with bound h is a uniformly Π 1 1 (uniformly ∆ 1 1)sequenceof sets(T n ) n∈ω such that ∀n #T n ≤ h(n). (T n ) n∈ω is a trace for the function f iff(n) ∈ T n for each n.(ii) A is Π 1 1 traceable (∆ 1 1 traceable) if there is an unbounded nondecreasinghyperarithmetical function h such that each function f ≤ h A has a Π 1 1 trace(∆ 1 1 trace) with boundh.

9.4 Lowness properties in higher computability theory 383The argument in Theorem 8.2.3 shows thatthe particular choice ofthe bound hdoes not matter in (ii). In the higher setting the two notions coincide:9.4.8 Proposition. If A is Π 1 1 traceable then A is ∆ 1 1 traceable.Proof. First we show that ω1A = ω1 ck .Otherwise A ≥ h O,soit is sufficient toshow that O is not Π 1 1-traceable. Since each Π 1 1 set is many-one reducible to O,there is a uniformly O-computable list (Tn) e e,n∈N of all the Π 1 1 traces for thebound h(e) =e +1. Define a function g ≤ T O by g(e) =µn. n ∉ Te e , then g doesnot have a Π 1 1 trace with boundh.Now let f ≤ h A and choose a reduction procedure S for f as in 9.1.12. Choosea Π 1 1-trace (T n ) n∈ω for f with boundh. Letα n = min{α < ω A 1 : ∃m ∈ T n,α [〈n, m, A〉 ∈S α ]}Then by the Bounding Principle 9.1.22 relative to A there is β < ω1A = ω1 ck suchthat α n < β for each n. Forafixedβ < ω1 ck , the Approximation Lemma 9.1.4(i)provides an index for the ∆ 1 1 set R β uniformly in an index for a Π 1 1 set R.Therefore (T n,β ) n∈N is a ∆ 1 1 trace as required.✷Chong, Nies and Yu (2008) showed that there are 2 ℵ0 ∆ 1 1 traceable sets (ananalog of 8.2.17). In fact each generic set for forcing with perfect ∆ 1 1 trees (Sacks4.5.IV) is ∆ 1 1 traceable. There is a perfect class of generic sets inthat sense. Alsoby Sacks 4.10.IV there a generic Z ≤ h O.ThenZ is ∆ 1 1 traceable and Z ∉ ∆ 1 1.∆ 1 1 traceability characterizes lowness for ∆ 1 1-randomness. The result is analogousto Theorem 8.3.3, and also to someofthe equivalences in Theorem 8.3.9.9.4.9 Theorem. The following are equivalent for a set A.(i) A is ∆ 1 1-traceable (or equivalently, Π 1 1 traceable).(ii) Each null ∆ 1 1(A) class is contained in a null ∆ 1 1 class.(iii) A is low for ∆ 1 1-randomness.(iv) Each Π 1 1-ML-random set is ∆ 1 1(A)-random.Proof. (i) ⇒ (ii): This implication corresponds to Proposition 8.3.2. While theproof there relied on computable measure machines, here we give a direct proof.Let S be a null ∆ 1 1(A) class.ByTheorem9.3.3(i)relativized to A, S ⊆ ⋂ G mfor a Π 1 1-Schnorr test {G m } m∈N relative to A such that {〈m, σ〉: [σ] ⊆ G m }is ∆ 1 1(A) andλG m =2 −m for each m. We view the k-th finite set D k as a subsetof {0, 1} ∗ .Defineafunction f ≤ h A as follows: let f(〈m, s〉) bethe least ksuch that [D k ] ≺ ⊆ G m , D f(〈m,s−1〉) ⊆ D k if s > 0, [σ] ⊆ G m for |σ| ≤ simplies σ ∈ D k ,andλ[D k ] ≺ > 2 −m (1 − 2 −s ). Let G m,s =[D f(〈m,s〉) ] ≺ , thenG m,s ⊆ G m,s+1 and G m = ⋃ s∈N G m,s.Let T =(T e ) e∈N be a ∆ 1 1 trace for f such that #T e ≤ e+1 for each e. Wedefinea ∆ 1 1 trace ( ̂T e ) e∈N .Byrecursionons, define ̂T 〈m,s〉 tobethe set of k ∈ T 〈m,s〉such that 2 −m (1 − 2 −s ) ≤ λ[D k ] ≺ ≤ 2 −m and D k ⊇ D l for some l ∈ ̂T 〈m,s−1〉(where ̂T 〈m,−1〉 = N). LetV m = ⋃ { [D k ] ≺ : k ∈ ̂T 〈m,s〉 , s ∈ N } .

384 9 Higher computability and randomnessThen λV m ≤ 2 −m # ̂T 〈m,0〉 + ∑ s∈N 2−s−m # ̂T 〈m,s〉 .Since#̂T 〈m,s〉 ≤ #T 〈m,s〉 ≤〈m, s〉 +1≤ (m + s +1) 2 +1, it is clear that lim n∑s∈N 2−s−m # ̂T 〈m,s〉 = 0, and⋂hence lim m λV m =0.Since∀m ∀sf(〈m, s〉) ∈ ̂T 〈m,s〉 ,wehave∀mG m ⊆ V m . Som G m is contained in the null ∆ 1 1 class ⋂ m V m.(ii) ⇒ (iii) and (iii) ⇒ (iv) are immediate.(iv) ⇒ (i) is a straightforward adaptation of the implication “⇒” in Theorem8.3.3. We are now given a function g ≤ h A,andwanttodefineaΠ 1 1 trace(T n ) n∈N for g with boundλn.2 n .Nowlet R = R 2 as defined before 9.2.8. Themeasure-theoretic Lemma 8.3.4 can be used as before. We define a Π 1 1-Schnorrtest (U g d ) d∈N relative to A. Thedefinitions of ( ˜T n ) n∈N and (T n ) n∈N are as before.The relation {〈n, k〉: λ(B n,k − ˜R) < 2 −(k+4) } is now Π 1 1.✷For each set A there is a largest null Π 1 1(A) classQ(A) byrelativizing 9.3.6.Clearly Q ⊆ Q(A); we say that A is low for Π 1 1-randomness if they are equal.9.4.10 Lemma. If A is low for Π 1 1-randomness then ω A 1 = ω ck1 .Proof. Otherwise, A ≥ h O by Theorem 9.1.14. ByCorollary9.3.7there is a Π 1 1-random set Z ≤ h O,andZ is not even ∆ 1 1(A) random.✷9.4.11 Open question. Is each set that is low for Π 1 1-randomness in ∆ 1 1?By the following result each such set is low for ∆ 1 1-randomness. We say that Ais Π 1 1-random cuppable if A ⊕ Y ≥ h O for some Π 1 1-random set Y .9.4.12 Theorem. A is low for Π 1 1-randomness ⇔(a) A is not Π 1 1-random cuppable & (b) A is low for ∆ 1 1-randomness.Proof. ⇒: (a) By 9.4.10 A ≱ h O.Therefore the Π 1 1(A) class{Y : Y ⊕ A ≥ h O}is null, by relativizing Corollary 9.1.15 to A. ThusA is not Π 1 1-random cuppable.(b) Suppose for a contradiction that Y is ∆ 1 1-random but Y ∈ C for a null ∆ 1 1(A)class C. TheunionD of all null ∆ 1 1 classes is Π 1 1 by 9.3.11. Thus Y is in theΣ 1 1(A) classC − D. Bythe Gandy Basis Theorem 9.1.16 relative to A there isZ ∈ C − D such that ω1 Z⊕A = ω1 A = ω1 ck .ThenZ is ∆ 1 1-random but not ∆ 1 1(A)-random, so by Theorem 9.3.9 and its relativization to A, Z is Π 1 1-random butnot Π 1 1(A)-random, a contradiction.⇐: By Fact 9.3.8 relative to A we haveQ(A) ={Y : ω1 Y ⊕A > ω1 A } ∪ ⋃ α ω1 A is equivalent toO ≤ h Y ⊕ A. If Y is Π 1 1-random then firstly O ≰ h Y ⊕ A by (a), and secondlyY ∉ Q(A) α for any α < ω1A by hypothesis (b). Therefore Y ∉ Q(A) andY isΠ 1 1(A)-random.✷

Solutions to the exercisesSolutions to Chapter 1Answers to many of the exercises in this chapter can be found in Soare (1987) orOdifreddi (1989). Most of the solutions we provide are for exercises that are exceptionsto this.Section 1.11.1.12 Use an effective forth-and-back argument.Section 1.21.2.25 ⇒: By 1.2.22.⇐: Let Z be the graph of f. The Turing functional Φ on input n with oracle set Ylooks for the least w ≤ h(n) such that 〈n, w〉 ∈Y . If found it outputs w, otherwise0.Section 1.41.4.7 (i) is like the proof of the Limit Lemma, substituting g for Z and g s(x) forZ s(x).(ii) ⇐: Similar to the proof of the corresponding implication in 1.4.4.⇒: If g is ω-c.e. we obtain a weak truth-table reduction of g to the change set byadapting the proof of the Limit Lemma. However, we cannot define a Boolean expressionas in 1.4.4, because the outputs are now natural numbers.1.4.8 Suppose Z = Φ E for a Turing functional Φ with use function bounded by g.Given i ∈ N and a string σ of length g(i), “Φ σ (i) =1”isaΣ 0 1 property uniformly,so there is a computable binary function h such that Φ σ (i) =1 ↔ E satisfies theBoolean expression h(i, σ). Then i ∈ Z ↔ E satisfies the Boolean expression which isadisjunctionoverallσ of length g(i) ofexpressionsstatingthatσ is an initial segmentof the oracle and the oracle satisfies h(i, σ).1.4.19 By the Recursion Theorem there is an e such that W e = {e}. BythePaddingLemma 1.1.3 there is i ≠ e such that W i = W e;theni ∉∅ ′ .1.4.20 For (i), (iii), and (iv) see Soare (1987). (ii) Choose a computable sequence(Z s) s∈N as in Proposition 1.4.17. If n ∉ Z s then put [0,s)intoX n.1.4.22 By 1.4.20(ii) there is a uniformly c.e. double sequence (X e,n) e,n∈N such that eachX e,n is an initial segment of N and e ∈ S ↔ ∃n X e,n = N. NowletA 〈e,n〉 = W e ∩ X e,n[let A 〈e,n〉 = V e ∩ X e,n], then (A k ) k∈N is the required uniform listing of S.1.4.23 Let U, V be disjoint Σ 0 3 sets that are effectively inseparable relative to ∅ ′′ (seeSoare 1987, pg. 44). Note that if Ũ is Σ0 3 and U ⊆ Ũ, Ũ ∩ V = ∅ then Ũ is Σ0 3 complete.Fix a c.e. set G ∉ S. Uniformlyine we will enumerate a c.e. set A e ⊆ G such thate ∈ U → A e is computable, and e ∈ V → A e = ∗ G.Thereisacomputabletriplesequence (x e,i,s) e,i,s∈N ,nondecreasingins, such that e ∈ U ↔ ∃n lim sx e,2n,s = ∞, ande ∈ V ↔ ∃n lim sx e,2n+1,s = ∞. Atstages>0, if i is least such that x e.i,s >x e,i,s−1do the following. If i is even restrain A e up to s with priority i. Ifi is odd put into A eall x ∈ G s not restrained with priority

386 Solutions to the exercisescomputable function r. IfY = Φ C e then Y ≤ tt ∅ ′ because x ∈ Y ↔ Θ(∅ ′ ; r(〈e, x〉)) = 1.Auseboundish(x) =max k≤x q(r(〈k, x〉)) for all x ≥ e, whichsuffices.1.5.6 ⇒: Let f(x) =max{Φ A e (x): e ≤ x & Φ A e (x) ↓}. Sincef ≤ T A ′ ≡ T A ⊕∅ ′ ,thefunction f is as required. ⇐: Since ψ(x) ≃ µs Js A (x) ↓ is partial computable in A, wehave J A (x) ↓↔ Jf(x)(x) A ↓ for almost all x. ThusA ′ ≤ T A ⊕∅ ′ .1.5.13 (a) By the Limit Lemma 1.4.2 relative to C, thereisaC-computable approximation(A s) s∈N of A. Ifg is as in 1.5.12 then g ≤ T A since C ≤ T A. Nowargueasbefore in order to show that A ≤ T C.(b) If A ′ ≡ T C ′ then C< T A< T C ′ ,soA is not computably dominated.1.5.14 By 1.4.17 there is a computable sequence of strong indices (A s) s∈N such thatx ∈ A ↔∀ ∞ x [x ∈ A s]. Define g ≤ T A by{µs > t. x ∉ A s if x ∉ A,g(〈x, t〉) =t if x ∈ A. .There is a computable function f such that f(〈x, t〉) ≥ g(〈x, t〉) for all x, t. Thenx ∈ A ↔ ∃t ∀u ∈ [t, f(〈x, t〉) [ x ∈ A u]so A is c.e. and hence computable.1.5.15 By 1.4.20 relative to A, thesetTot A = {e: dom(Φ A e )=N} is Π 0 2(A)-complete.Hence A ′′ ≡ T Tot A and Tot A is Π 0 1(A ′ ). Further,e ∈ Tot A ↔ ∃i [ Φ i total & ∀x Φ A e,Φ i (x)(x)↓ ] .Therefore Tot A is Σ 0 1(A ′ ⊕∅ ′′ ). This implies A ′′ ≡ T Tot A ≤ T A ′ ⊕∅ ′′ .1.5.16 Suppose the function g ≤ T A is not dominated by any computable function.We build a set Y ≤ T A such that Y ≰ tt A. SupposeY ↾ m has been defined. To defineY (m), look for the least e ≤ m such that Φ A e,g(m) ↾ m= Y ↾ m and r = Φ A e,g(m)(m)↓, anddefine Y (m) tobenotequaltor. Ifthereisnosuche define Y (m) =0.Clearly each e is considered at most once. If Y ≤ tt A, thenbyProposition1.2.22Y = Φ A e for some e such that the number of steps to compute Φ A e (m) is bounded byt(m), for a strictly increasing computable function t. Sincethereareinfinitelymanymsuch that g(m) ≥ t(m) wecauseY ≠ Φ A e , contradiction.1.5.17 (J. Miller) Clearly a computable set is of that kind. Now assume A is uniformlycomputably dominated via r.By1.5.12itsuffices to show that A ≤ T ∅ ′ .Infact,A ≤ T ∅ ′follows already from the weaker hypothesis that 1 + Φ r(e) is not dominated by Φ A e foreach e. Letp be a computable binary function such that for each e, n we have{Φ A 1+Φ r(e) (x) if n ∈ Ap(e,n)(x) ≃0 else.Let q be the function obtained when we require “n ∉ A” inthefirstlineinstead.Bythe Recursion Theorem with Parameters there are computable functions h, l such thatΦ A p(h(n),n) = Φ A h(n) and Φ A q(l(n),n) = Φ A l(n) for each n. Ifn ∈ A then α n = Φ r(h(n)) ispartial, otherwise Φ A h(n) would be total and dominate 1 + α n.Ifn ∉ A then α n is total.Similarly, β n = Φ r(l(n)) is partial iff n ∉ A. Using∅ ′ as an oracle we can decide whichof α n, β n is partial.1.5.21 Relativize Theorem 1.5.19 to A.Section 1.61.6.7 The strategy for the lowness requirements L e is as before. To ensure that A ′ isnot ω-c.e. we have to rule out each potential approximation Φ 2 i (x, t), with the computablebound Φ 1 j(x) onthenumberofchanges,ofA ′ .LetU 〈i,j〉 be the corresponding

Solutions to Chapter 1 387requirement. We build a Turing functional Γ Y (x) (see Section 6.1), and by Fact 1.2.15and the Recursion Theorem for functionals we are given a reduction function p for Γ,namely ∀Y ∀x Γ Y (x) ≃ J Y (p(x)).The strategy for U 〈i,j〉 is as follows. Choose x greater than the last stage when U 〈i,j〉was initialized. Let z = p(x). Wait for n = Φ 1 j(z)↓. Lett ∗ =0.Foratmostn +1 times,do the following:(a) whenever t is greater than t ∗ and Φ 2 i (z, t − 1) = 1, Φ 2 i (z, t) = 0 then declareΓ A (x)↓ with large use u (in particular u>〈n +1, 〈i, j〉〉). Wait for a stage v such thatJ A (p(x))↓ [v] andlett ∗ = v;(b) whenever t is greater than t ∗ and Φ 2 i (z, t − 1) = 0, Φ 2 i (z, t) =1thenmakeΓ A (x)undefined by enumerating the greatest number m ∈ N [〈i,j〉] such that m0, if there is x and esuch that x ∈ W e,at s, lett be the least stage such that x ∈ Ãt or x ∈ W q(e),t. This stageexists since W e − A = W q(e) − A. IfthefirstalternativeappliesletA s = A s−1 ∪ Ãt,otherwise let A s = A s−1 ∪ Ãs. IfWe is infinite then for some x, s we have x ∈ We,at sand x ∉ W q(e) .Thenx ∈ A s.1.7.12 See Soare (1987, pg. 283).Section 1.81.8.9 “⇐ ”istheuseprinciple.For“⇒” letA = {〈σ,x〉: ∀Z ≽ σ L(Z) ≽ x}.1.8.12 For each pair of sets X, Y we have abs(F (X) − F (Y )) ≤ d(X, Y )(see1.8.7for the definition of the distance function d), so ˜F is continuous. To show that ˜F −1 iscontinuous, note that for any cylinder [σ] wehave˜F ([σ] ∩ X )=(0.σ, 0.σ +2 −|σ| ) − Q 2,which is open in [0, 1) R − Q 2.1.8.13 The homeomorphism is given by 1 n 001 n 101 n 20 ...→ (n 0,n 1,n 2,...).1.8.18 Given k ∈ N, foreachi ≤ k we may compute q i ∈ Q 2 such that 0 ≤ r i − q i

388 Solutions to the exercisesthere is a Turing functional Ψ such that Ψ X is total for each oracle X and Ψ(B ′ ; e) =τ e,where τ e is the leftmost string τ of length e +1 such that Q τ ≠ ∅. LetY ∈ ⋂ e Qτe .Then e ∈ (Y ⊕ B) ′ ↔ τ e(e) =1,so(Y ⊕ B) ′ ≤ tt B ′ .1.8.46 Construction relative to ∅ ′ of Π 0 1 classes (P i ) i∈N . Let P 0 = P .Stage 2i +1. If P 2i ∩ {X : J X (i) ↑} ≠ ∅, thenletP 2i+1 be this class. Otherwise, letP 2i+1 = P 2i .Stage 2i +2.Seewhetherthereise ≤ i which has not been active so far such that forsome m ≤ c B(i) wehaveQ i e,m := P 2i+1 ∩ {X : Φ X e (m)↑} ≠ ∅. Ifsolete be least, let mbe least for e, andletP 2i+2 = Q i e,m. Saythate is active. Otherwise,letP 2i+2 = P 2i+1 .Verification. Let Y ∈ ⋂ r P r .SinceB can determine an index for each P r ,wehaveY ′ ≤ B by the usual argument of the Low Basis Theorem. Each e is active at mostonce, and if so then Φ Y e is partial. Suppose now that Φ Y e is total. We claim that thereis r such that Φ Z e is total for each Y ∈ P r , and therefore Φ Z e is computably dominatedby the argument in the proof of 1.8.42. If the claim fails then B ≤ T ∅ ′ ,asfollows.Lets 0be a stage such that no jsW e,t(n) =Y (n). For the computable sets apply thisand 1.1.9.1.8.64 Suppose the class of c.e. sets is contained in a Π 0 2 class S = ⋂ nRn where theR n are uniformly Σ 0 1. Then each R n is dense, so S contains every weakly 1-generic set.The same argument works for the computable sets. Alternatively, if the class of computablesets is Π 0 2,thentheindexset{e: W e is computable} is in Σ 0 3∩Π 0 2(∅ ′ ), and hencein ∆ 0 3. This contradicts the fact that this index set is Σ 0 3 complete, Exercise 1.4.20(iv).1.8.65 Let A = ⋂ y By where the class By is Σ0 1 uniformly in y. ThenBy ∗ = {Z : ∃m[Ψ Z↾m ] ⊆ B y} is Σ 0 1 uniformly in y, soC = {Z : Ψ Z is total} ∩ ⋂ y B∗ y is Π 0 2.1.8.66 Let B = ⋃ n An for an effective sequence of Π0 2 classes. For each n, e let C n,ebe the class uniformly obtained via 1.8.65 from A n and Ψ = Φ e.ThenG = ⋃ n,e Cn,eis Σ 0 3.1.8.67 (i) Suppose C = {Z : ∀nZ↾ n∈ R} where R ≤ T ∅ ′ .BytheLimitLemmachooseacomputableapproximation(R t) t∈N for R. ThenC = {Z : ∀n ∀s ∃t >sZ↾ n∈ R t}, soC is Π 0 2.(ii)EveryΠ 0 1(∅ ′ )singletonis∆ 0 2 by Fact 1.8.33.1.8.71 ⇒: If there is σ such that σu is on T P for infinitely many u, then{f : σ ⊀ f} ∪ ⋃ u[σu] isanopencoveringofP without a finite subcovering.⇐: If T P ⊆ {σ ∈ N ∗ : ∀i

Solutions to Chapter 2 389(C i) i∈N and ɛ > 0, chose open sets A i ⊇ C i such that µ r(A i) ≤ µ r(C i)+ɛ2 −i .Thenµ ⋃ r(i Ci) ≤ ⋃i µr (Ai) ≤ ∑ i µr(Ai) ≤ 2ɛ + ∑ i µr(Ci).1.9.13 If ɛ > 0 and 2 −n #(S ∩ {0, 1} n ) ≥ ɛ then the strings of length n contribute atleast ɛ to λ[S] ≺ .Sothereareatmost1/ɛ such n.1.9.14 Assume for a contradiction that U is measurable. Since X ∈ U ↔ N − X ∉ Uand the complementation operation on subsets of N preserves λ, wehaveλU =1/2.But U is closed under finite variants whence λU ∈ {0, 1} by the zero-one law.1.9.15 (J. Miller) For a class C let I C denote the function given by I C(Z) = 1 ifZ ∈ C and I C(Z) =0otherwise.Considerthefunctiong : 2 N → {0,...,N} defined byg(Z) =#{i: Z ∈ C i} = ∑ Ni=1 IC (Z). Since ∫ ig(Z)λ(dZ) ≥ ∑ N2 N i=1 ɛ >k,theclass{Z : g(Z) >k} is not a null class.1.9.20 Let (q n) n∈N be a nondescending computable sequence of rationals such thatlim nq n = r, Fact1.8.15.Introducingsomerepetitionsifnecessary,wemayassumethateach q n is of the form k2 −n for some k ∈ N. We define an effective sequence (C n) n∈Nof clopen sets such that λC n = q n and C n ⊆ C n+1 for each n, andR = ⋃ nCn is therequired c.e. open set. Each set C n is generated by strings of length k n,wherek n 0. Let b ∈ N + be least such that q n − q n−1 ≤ (1 − 2 −b )(1 − q n−1), and letk n = k n−1 + b. PickaclopensetD such that λD = q n − q n−1, D ∩ C n−1 = ∅ and Dis given by strings of length k n none of which are of the form σ1 b for |σ| = k n−1. LetC n = D ∪ C n−1.1.9.21 Define a computable Z ∈ P , Z = ⋃ nσn, as follows. Let σ0 = ∅. If σn is definedsuch that λ(P | σ n) > 0, then let σ n+1 be the leftmost extension σ of σ n of length n +1such that λ(P | σ) > 0. Use 1.9.18 to show that we can compute the sequence (σ n) n∈N .1.9.22 For n = 1, this is Fact 1.9.16. Now let n>1, and inductively assume the resultholds for Π 0 n−1 classes. A Σ 0 n class C is an effective union of Π 0 n−1 classes D i whereD i ⊆ D i+1 for each i. EachλD i is left-Π 0 n−1 and hence left-Σ 0 n,uniformlyini. ThusλC =lim iλD i is left-Σ 0 n. The case of Π 0 n classes is similar.Solutions to Chapter 2Section 2.12.1.3 Consider the machine M given by M(τ) ≃ V(V(τ)). Each V-description of σ isan M-description of x, henceC M (x) ≤ C(σ). If M = Φ b , b>0, then C(x) ≤ C M (x)+b,so that |σ| = C(x) ≤ C(σ)+b.2.1.4 When the computation V(σ) =x converges at a stage, declare R(σ) =x unless xalready has an R-description of the same length at a previous stage, or there is a V-description ρ < L σ, |ρ| = |σ|, such that V(ρ) =x converges at the same stage.2.1.5 If V(σ)↓ and |σ| ≡ i mod d for 0 ≤ i

390 Solutions to the exercises2.1.17 Since 2 n ≥ r n ≥ 2 n−d+1 ,wehaven ≥ log r n ≥ n − d + 1, so we can obtain nfrom r n and a constant number of bits. Let z n be the leftmost d-incompressible C stringof length n. Sincethesetofd-compressible C strings is c.e., if we know r n (and hence n)we can produce all the d-compressible C strings of length n, andthereforedeterminez n.Thus n − d

Solutions to Chapter 3 391Now use the Low Basis Theorem.(ii) For n

392 Solutions to the exercises3.2.33 See Downey, Hirschfeldt and Nies (2002).Section 3.33.3.3 Given a number d>1, we build a bounded request set L (we think of d as a codingconstant for L provided by the Machine Existence Theorem 2.2.17). Let c = d +1.By(1.17) on page 55, each Π 0 1 class P (given by an index for Π 0 1 classes) has an effectiveapproximation (P s) s∈N by clopen sets P s,whicharegivenbyastrongindexforasetof strings of length s. Thehypothesis“λ(P e ∩ Q) ≤ 2 −K(e)−c ”isaΣ 0 1-property of e,because λ(Pte ∩ Q t)isnonincreasingint and converges to λ(P e ∩ Q), while 2 −K t(e)−cis nondecreasing and converges to 2 −K(e)−c .Construction of L. If t is least such that λ(Pte ∩ Q t) ≤ 2 −Kt(e)−c ,thenforeach y ∈ Pt e ∩ Q t enumerate a request 〈|y| − c, y〉 into L at stage t.The enumeration for P e adds a weight of at most 2 −K(e) to L. Then, for any given d,L is a bounded request set since ∑ e 2−K(e) ≤ 1. Now suppose that, by Remark 2.2.21,d>1 is a coding constant for L (note that this fixed point, and hence c, can be foundeffectively). If λ(P e ∩Q) ≤ 2 −K(e)−c then at some stage t the enumeration into L causesK(Z ↾ t) ≤ t − 1foreachZ ∈ P e ∩ Q. This contradicts Q =2 N − R 1,henceP e ∩ Q = ∅.3.3.4 Apply 3.3.3 to the particular effective listing of Π 0 1 classes given by P x =[x].Section 3.43.4.3 (i) Use the binary machine M(σ, α) ≃ V α (σ).(ii) Let α =0 n ,thenC α (n) = + C(n).3.4.4 Suppose that A = Φ(B) for a Turing functional Φ. Define a prefix-free oraclemachine M by M X (σ) ≃ U Φ(X) (σ). Namely, on input σ, M looks for α ≺ Φ(X) suchthat U α (σ) =y and, if it is found, outputs y. Then∀yK B (y) ≤ + K M B (y) = + K A (y).3.4.9 For “⇐” itsuffices to adapt the proof of this implication in 3.4.6. (V d ) d∈N is nowaML-testrelativeto∅ (k−1) , and (S d ) d∈N is a Solovay test relative to ∅ (k−1) .Theopensets H d are uniformly c.e. in B ⊕∅ (k−1) .IfA ∈ H d for each d ≥ d 0,thenA is notML-random relative to B (k−1) .The proof of “⇒” needssomeadditionalidea;seevanLambalgen(1987).3.4.14 If Z is ω-c.e., then f is ω-c.e. in the sense of Exercise 1.4.7, and hence f ≤ wtt ∅ ′ .Let{[Z s ↾ e+1] if s is the stage such that JsAs (e) convergesA-correctlyĜ e =∅ if J A (e)↑ .Then for almost all e, J A (e)↓↔ J A (e)[f(e)]↓, whichshowsthatA ′ ≤ wtt ∅ ′ .3.4.21. If0 n 1 ≺ X, letM X (σ) ↓ for all strings σ of length n such that σ ≠1 n .ThusΩ X M =1− 2 −n .Wehaver 1 =1,andthissupremumisnotassumedbyΩ M .3.4.22 By Fact 3.4.19, for every δ > 0, {X : abs(Ω A M −Ω X M ) < δ} = {X : Ω X M > Ω A M −δ}is open.3.4.23 If the real r is right-c.e. and r is left-c.e. relative to A, thenr ≤ T A, sor is notML-random relative to A. Therefore,ifr =1− Ω = Ω A then r is not ML-random in A,contradiction.Section 3.53.5.6 Clear when B is finite. Otherwise, consider the Π 0 1 class P = {Z : B ⊆ Z}. Foreach finite set F ,wehaveλ{Z : F ⊆ Z} ≤ 2 −#F .ThusP is a null class containing B.

Solutions to Chapter 4 3933.5.7 By Theorem 3.3.7, if Z ∈ P is not autoreducible then the Π 0 1 class P is uncountable.3.5.16 Let L 0 be the bounded request set {〈|σ|,x〉: M(σ) = x}, andletL 1 be abounded request set with total weight 1 − Ω M . Then the total weight of L = L 0 ∪ L 1is 1, so the prefix-free machine for L is as required.3.5.17 Compute a stage s such that Ω M − Ω M,s < 2 −r .Then{y : K M (y) ≤ r} ={y : K M,s(y) ≤ r} as in the proof of Proposition 3.5.15.Section 3.63.6.6 Relativize Proposition 3.5.5 to ∅ ′ .3.6.7 Proceed similar to the solution for 3.5.6, but relative to ∅ ′ .3.6.16 The class of compression functions for K forms a bounded Π 0 1 class describedby the condition ∀n [F ↾ n∈ S], where S is the computable set { α ∈ N ∗ :∀i, j < |α| [ i ≠ j → α(i) ≠ α(j) ] & ∑ i

394 Solutions to the exercises4.1.7 By the remark after Fact 1.2.15, let π be a computable permutation of N suchthat Ĵ(x) ≃ J(π(x)) for each x. Let ̂f = f ◦ π. Then ̂f ≡ T f, andforeachx,Ĵ(x) = ̂f(x) ↔ J(π(x)) = f(π(x)).4.1.8 By Ex. 1.1.12, let π be a computable permutation of N such that ∀x Ŵx = W π(x).Let ĝ = π −1 ◦ g ◦ π. Thenĝ ≡ T g, andforeachx, Ŵĝ(x) = Ŵx ↔ W g(π(x)) = W π(x) .4.1.14 ∅ ′ is of d.n.c. degree relative to A. Hence∅ ′ is Turing complete relative to A,that is, A is in GL 1.4.1.15 See Odifreddi (1989, III.8.16).4.1.16 Let h be a computable function such that ∀np A(n) ≤ h(n). To compute f(e)in the proof of Proposition 4.1.13, we merely need to query the oracle A on numbersk less than h(g(e)) + 1. Thus f ≤ wtt A, andA is wtt-complete by Theorem 4.1.11 andthe proof of 4.1.4.Section 4.24.2.7 In the following let i ∈ {0, 1}. Suppose that Y i is of d.n.c. degree via f i ≤ T Y i.We define partial computable functions α i.BytheDoubleRecursionTheoremwearegiven reduction functions p i such that ∀e [α i (e) ≃ J(α i(e))]. We replace (4.1) byx ∈ W e,s − W e,s−1 & ∀i ∈ {0, 1}∀t x

Solutions to Chapter 5 395range over sequences of pairs of this kind, and write dom(F )fortheset{r 0,...,r i−1}.(Note that F is a finite assignment in the sense of Section 1.3.)Step i for even i. We code D(n) intoB. By the Gödel incompleteness theorem, fromany F we may effectively determine a sentence that cannot be decided on the baseof PA ∪ F : we find a number c(F ) ∈ N, c(F ) > max dom(F ) such that, if PA ∪ F isconsistent, then so are PA ∪ F ∪ {α c(F ) } and PA ∪ F ∪ {¬α c(F ) }.Letr i = c(F )andB(r i)=D(n).Step i for odd i. We ensure that the next sentence left open is decided by B. Thisis done in a way similar to (ii)⇒(iii) in the proof of Theorem 4.3.2. Consider thepartial computable binary function ψ given by ψ(F, k) = 1 if we first find a proof ofa contradiction from PA ∪ F ∪ {¬α k },andψ(F, k) = 0 if we first find a proof of acontradiction from PA ∪ F ∪ {α k }.Letg ≤ T D be a total {0, 1}-valued extension of ψ.Let r i be the least number not in dom(F i), and let B(r i)=g(F i,r i).Clearly this construction determines a set B ≤ T D that is a completion of PA.AlsoD ≤ T B since each number r i is determined effectively from F i,andD(n) =B(r 2n).4.3.10 (Kučera) One can extend 1.8.40 to infinite sequences (B i) i∈N of incomputablesets. Letting (B i) i∈N list the incomputable ∆ 0 2 sets, for each nonempty Π 0 1 class P weobtain a set A ∈ P such that A, ∅ ′ form a minimal pair. Now let P be the class of{0, 1}-valued d.n.c. functions. If Z ≥ T A is ML-random then Z ≥ T ∅ ′ by 4.3.8, so Z isnot weakly 2-random.4.3.18 Let B> T ∅ ′ be a Σ 0 2 set such that B ′ ≡ T ∅ ′′ .By1.8.46thereisacomputablydominated ML-random set Y such that Y ≤ T B.ThusY is weakly 2-random. If g ≤ T Yis 2-f.p.f then there is 2-d.n.c. function f ≤ T Y , whence ∅ ′′ ≤ T B⊕∅ ′ by Theorem 4.1.11relative to ∅ ′ , contradiction.Solutions to Chapter 5Section 5.15.1.4 K(Z ↾ n)= + K A (Z ↾ n)= + K A ((Z△A)↾ n)= + K((Z△A)↾ n).5.1.5 Y is 2-random by Exercise 3.6.21, so A forms a minimal pair with both Y andZ (page 135). Since A ≤ T Y ⊕ Z, thisimpliesY | T Z.5.1.6 Suppose A is low for K, andconsiderΦ A e .Foreachn, ifm = Φ A e (n), thenK(m) = + K A (Φ A e (n)) ≤ + K A (n) ≤ + 2logn.Let F (n) =2max{m: K(m) ≤ 3logn}. ThenF dominates Φ A e for each A, e as above.Also, F is ω-c.e., and hence F ≤ wtt ∅ ′ by Exercise 1.4.7(ii).5.1.11 W e ∈ Low(MLR) ↔ ∃R c.e. open∃k ∀s λ[R s] ≺ ≤ 1 − 2 −k & ∀z ∀s ∃t ≥ s (K We (z)[t] ≥ |z| ∨ [z] ⊆ [R t] ≺ ).5.1.15 Otherwise, the class is conull by the 0-1 law, and therefore its intersection withthe class of ML-random sets is conull. However, by Theorem 4.3.8, each ML-randomset of PA degree is Turing above ∅ ′ ,sobyTheorem5.1.12thisintersectionisanullclass.It is possible to only apply a technique used in the proof of Theorem 4.3.8 for asimple direct solution. If the class is conull then by the Lebesgue Density Theorem1.9.4 there is a Turing functional Φ such that Φ X (w) ∈ {0, 1} if defined, and{Z : Φ Z is total and d.n.c. } has measure at least 3/4. We define a partial computable αand are given a reduction function p for α:

396 Solutions to the exercisesIf s is least such that 1/4 < λ{Z : Φ Z s (p(0)) = b}, defineα(0) = b. Then Φ Z (p(0)) =J(p(0)) for more than 1/4 of all sets, contradiction.5.1.16 Suppose that A = Φ Y for a Turing functional Φ, and let c be the constant ofProposition 5.1.14.(i) and (ii): Given a c.e. open set R, wewilleffectively obtain a c.e. open set ̂R such thatλ ̂R ≤ 2 c λR. IfA fails a test (G n) n∈N for the randomness notion in question, then Yfails the test (Ĝn+c) n∈N.For x ∈ {0, 1} ∗ ,letS x be the effectively given c.e. set which follows the canonicalcomputable enumeration of {σ : x ≼ Φ σ } as long as the measure of the open setgenerated does not exceed 2 c−|x| .Fromac.e.opensetR we can effectively obtain a(finite or infinite) c.e. antichain {x 0,x 1,...} such that R = ⋃ i [xi]. Let ̂R = ⋃ i [Sx i ]≺ .Since [S xi ] ≺ ∩ [S xj ] ≺ = ∅ for i ≠ j, wehaveλ ̂R = ∑ i λSx ≤ i 2c λR. Moreover,A ∈ Rimplies x i ≺ A for some i and hence Y ∈ ̂R by the hypothesis on c. Clearly(Ĝn+c) n∈Nis a test for the same randomness notion that succeeds on Y .(iii): argue in a similar way but with R and ̂R being Σ 0 1(C) classes.5.1.27 Since Ω ≡ T ∅ ′ , such an A is a base for ML-randomness and hence low for K.5.1.28 Let A be a superlow ML-random set and let Z = A.5.1.29 Relativizing the Low Basis Theorem to A, oneobtainsa{0, 1}-valued function fsuch that ∀e ¬f(e) =J A (e) and (f ⊕ A) ′ ≡ T A ′ .LetD = f ⊕ A.5.1.30 Z is ML-random relative to Y by Theorem 3.4.6, and hence ML-random relativeto A. ThusA is a base for ML-randomness.5.1.31 Suppose that A is c.e. and not low for K, Φ Z = A, and∅ ′ ≰ T Z. Wewillshowthat Z is not ML-random by building a Solovay test (E d ) d∈N for Z.Define C η d,σ as in the proof of Theorem 5.1.22. For each d, theboundedrequestsetL dfails to show that A is low for K, sowemusthaveλC η d,σ < 2−|σ|−d for some η ≺ Aand σ such that U η (σ) converges.C η d,σkeeps asking for new strings, thus, for each d,there are n and σ such that Z ↾ n∈ C η d,σfor some η ≺ A. Nowletg(d) betheleaststage s such that Z ↾ n is in C η d,σ,sfor some n, σ g(d). Now define E d = ∅ if d ∉∅ ′ ,andotherwiseE d = ⋃ {C η d,σ,m(d) : η ≺ A m(d) & σ ∈ {0, 1} ∗ }.Then the sets E d are uniformly c.e. and λE d ≤ 2 −d ,so([E d ] ≺ ) d∈N is a Solovay test.Furthermore, if m(d) >g(d), there are n, σ such that Z ↾ n is in C η d,σ for some η ≺ A m(d),so Z ↾ n is in E d .ThusZ is not ML-random.Section 5.25.2.7⇒: If A is low for K then for each n we have K(A↾ n)= + K A (A↾ n)= + K A (n).⇐: For each n we have K(n) ≤ + K(A↾ n) ≤ + K A (n), so A is low for K.5.2.8 (Stephan) We deal with the case of K; for the case of C one uses the plain optimalmachine V instead of U.The prefix-free machine M operates as follows on input σ.(1) Wait for U(σ)↓= n. (2)Ift is least such that n ∈∅ ′ t,outputA t ↾ n.Since A is wtt-incomplete, there are infinitely many n ∈∅ ′ such that A t ↾ n= A ↾ n,where n enters ∅ ′ at stage t. ThusK M (A ↾ n) ≤ |σ| for each U-description σ of suchan n, and hence K(A↾ n) ≤ + K(n).

Solutions to Chapter 5 3975.2.9 (i) For each partial computable one-one f, ∀nK(f(n)) = + K(n); see Fact 2.3.1.Also, from A↾ rn one can compute A↾ n.(ii) We have K(A ↾ n) ≤ + K(A ↾ rn ) ≤ + K(r n) ≤ + K(n). The increase of the constantthat is hidden in the inequality K(A ↾ n) ≤ + K(n) does not depend on b because it isdue to the first and third inequalities.5.2.10 To make A not K-trivial, meet the requirementsR b : ∃nK(A↾ n) >K(n)+b.The strategy is as follows. Start by picking a sufficiently large candidate n, andensurethat K(n) issmallbyenumeratingarequestintoaboundedrequestsetW .Fromnowon, change A ↾ n whenever K(A ↾ n) ≤ K(n) +b, butonlyforupto2 c+b times foreach approximation to K(n). If the approximation to K(n) isfinal,thenthisnumberof A-changes will suffice by (i) of Theorem 5.2.4. One needs to know in advance howmany A-changes are needed necessary so that n can be chosen large enough.To ensure that A is low, meet the usual lowness requirements L e from Theorem 1.6.4:initialize the requirements R b , b>e,wheneverJ A (e) newlyconverges.Here are the details for the R b -strategy. Let d be the coding constant for the boundedrequest set W given in advance via the Recursion Theorem.Phase 1. If R b has been initialized for k times so far, let r = k + b + d +2. Choosen = n 0 +2 c+b r,wheren 0 is the largest number mentioned so far in the construction.Put the request 〈k + b +1,n〉 into W and wait for K(n)

398 Solutions to the exercises5.2.24 For the implication (i)⇒(ii), let the binary machine M be given by M(∅,n)=A↾ n.Ifd is the coding constant for M then K(A↾ n| n) ≤ d for each n. Theimplication(ii)⇒(iii) is trivial. For the implication (iii)⇒(i), note that C(A ↾ n| n) ≤ b impliesC(A↾ n) ≤ C(n)+2b + O(1), and apply Theorem 5.2.20(ii).Section 5.35.3.7 Since K(2 j ) ≤ + 2logj, thereisanincreasingcomputablefunctionf and anumber j 0 such that ∀j ≥ j 0 K f(j) (2 j ) ≤ j − 1. Enumerate the set A = N in order,but so slowly that for each j ≥ j 0 the elements of (2 j−1 , 2 j ]areenumeratedonlyafterstage f(j), one by one. Each such enumeration costs at least 2 −(j−1) ,sothecostforeach interval (2 j−1 , 2 j ]is1.5.3.8 Suppose the limit condition fails for e. Chooses 0 such that∑s≥s 0∑xs 0,n such that c(n, s) > 2 −e .ThenA s(n) =A(n).5.3.9 The condition that Φ i is total is Π 0 2.GiventhatΦ i is total, the condition thatthe sum S in (5.6) is finite is Σ 0 2.Inthatcaselim sD Φi (s)(x) ↓, so the condition thatΦ i is an approximation of V e is Π 0 2.5.3.17 {Ω ∅′ } is a Π 0 2(∅ ′ )class,andhenceΠ 0 3.Being2-random,Ω ∅′ forms a minimalpair with ∅ ′ .Soeachc.e.setA ≤ T Ω ∅′ is computable.5.3.18 Let x

Solutions to Chapter 6 399adaptive cost function c A(x, s) =λ[{y : y ∈ B A [s − 1] with use >x}] ≺ and proceed inawaysimilartotheproofof5.3.35.Section 5.45.4.5 Awebsearch(2007)revealedthattheyarestillavailable.(Onebottlewillsetyou back around US$1500.) Recall that your bottle of wine corresponds to a weightof 1. In each of the constructions we show that L is a bounded request set. This meansthat we never need to pour more than one bottle into the topmost decanter(s).Section 5.55.5.9 Otherwise, for each superlow set A there is a c.e. superlow set C ≥ T A byProposition 5.3.6. This is clearly not the case: for instance A could be ML-random, andhence of d.n.c. degree, so C ≡ T ∅ ′ .5.5.10 For (i) use Exercise 1.8.53. For (ii), if a Schnorr random set is low then it isalready ML-random by 3.5.13.5.5.13 For some c, thesetL = {〈log h(m)+c, m〉: m ∈ N} is a bounded request set.It shows that each m has a U-description σ m such that |σ m| ≤ + log h(m). Now argueas before.5.5.16 If A is K-trivial then Ω A U is left-c.e.; since Ω A U is ML-random, this implies thatA ≤ T Ω A U .IfA ≤ T Ω A U then A is a base for ML-randomness and hence K-trivial.Section 5.65.6.6 ⇒: K B (A↾ n) ≤ + K A (A↾ n)= + K A (n).⇐: For each n, K B (n) ≤ + K B (A↾ n) ≤ + K A (n). Thus A ≤ LK B.5.6.7 Argue as in the proof of the implication “⇐” ofTheorem5.6.5.5.6.8 Suppose ∅ ′ ≤ cdom C.Thenthereisg ≤ T C such that g(n) ≥ µs. ∅ ′ (n) =∅ ′ s(n)for each n. Thus∅ ′ ≤ T C.5.6.10 Modify the proof of the implication (iii)⇒(i) of Theorem 5.6.9.The class U 1 = {Z : Z ≤ L A ′ } is Π 0 1(A ′ ), and hence a Π 0 2(A) classofmeasure0.A ′ .The class U 2 = {Z : Z ≤ L N − A ′ } is a Π 0 2(A) classofmeasure1− 0.A ′ .Nowargueasbefore that both 0.A ′ and 1 − 0.A ′ are right-c.e. relative to B ′ , whence A ′ ≤ T B ′ .5.6.22 Note that ∅ ′ ≤ T A ⊕ Z, forotherwiseZ ∈ MLR A by Proposition 3.4.13, whichimplies Z ≰ LR A. ByLemma5.6.20,A ∈ K(Z), and hence ∅ ′ ≤ T A ⊕ Z ∈ K(Z). Thus∅ ′ ∈ K(Z) byTheorem5.6.17(iii).5.6.23 (i) Let S> T ∅ ′ be a Σ 0 2 set in K(∅ ′ ). By the Sacks jump theorem (page 160)there is a c.e. set A such that A ′ ≡ T S. ThenA ′ ≡ LR ∅ ′ .(ii)Thisfollowsfrom5.6.21.5.6.25 We show that A ∈ Low(MLR X ). If R ∈ MLR X then R ⊕ X is ML-random, andhence ML-random relative to A. ByvanLambalgen’sTheorem3.4.6relativetoA, thisimplies that R is ML-random relative to A ⊕ X.5.6.32 Let G = ⋂ U n where the U n are uniformly Σ 0 2.Bytheuniformityoftheimplication“⇒” inLemma5.6.29,eachU n is covered by a class V n of the same measure,where the V n are uniformly Π 0 2(C). Then ⋂ n Vn is a Π0 2(C) classasrequired.5.6.33 Let H = ⋂ nSn as in Lemma 5.6.29. Similar to 5.6.28, given e, n, theoracle∅′⋂can determine clopen classes C n,e ⊆ S n such that λ(S n − C n,e) ≤ 2 −n−e .LetL e =n Cn,e. ThenLe is uniformly Π0 1(∅ ′ ), L e ⊆ H, andλ(H − L e) ≤ λ ⋃ n(Sn − Ce,n) ≤2 −e+1 .NowL = ⋃ eLe is as required.Solutions to Chapter 66.1.2 Consider a prefix-free machine M such that M(0 n 1) converges when n enters A.

400 Solutions to the exercises6.3.7 Let us first show how to make C incomputable. Run the construction almost asbefore, with the only difference that now the marker γ(2i) isusedforcoding∅ ′ (i), whileγ(2i +1) is used to satisfy the requirement C ≠ Φ i (i.e., C is not computed by Φ i). Atstage s, instep(1)findtheleasti such that i ∈∅ ′ s −∅ ′ s−1 or Φ i(γ s−1(2i + 1)) = 0. Inthe first case put γ s−1(2i) − 1intoC, in the second put γ s−1(2i + 1) into C.To ensure C is of promptly simple degree, meet the requirement PSD i from (6.4) viaγ(2i + 1). When a number x>γ s−1(2i + 1) enters W i at stage s then attempt to putγ s−1(2i + 1) into C.6.3.16 Use 6.3.15(iv) with A = ∅ ′ and B = S.Solutions to Chapter 77.1.10 (i) is clear. (ii) Suppose C = αE x + βE y =0forα, β ∈ R. Say|x| ≤ |y|, andchoose r ∈ {0, 1} such that xr ⋠ y. ThenC(xr) =α2 |x| =0,soα =0andhenceβ =0.(iii) B equals the convex combination ∑ |x|=n B(x)2−n E x.7.1.11 Let S(0 k )=2 k and S(x) =0forx ∉ {0} ∗ .Foreachb of the form 2 k we haveλ { Z : ∃nS(Z ↾ n) ≥ b } =2 −k =1/b.7.1.14 We may assume that S(x) > 0foreachx ∈ {0, 1} ∗ ,andS(∅) < 1. We letR = G+E, whereG(x) ∈ N is the current savings and E is a supermartingale boundedby 2. Whenever a ∈ {0, 1} and E(xa) > 1thendefineG(xa) =G(x) +1, otherwiseG(xa) =G(x). In the former case, for strings y ≽ xa, E begins with capital E(xa) − 1,using the same betting factors S(yb)/S(y) asS for b ∈ {0, 1}. Ify ≽ x then R(y) −R(x) ≥ E(y)−E(x) ≥−2. If Z ∈ Succ(S) then lim nG(Z ↾ n)=∞, whencelim nR(Z ↾ n)= ∞. WehaveR ≤ T S because the S-computable real numbers form a field.7.2.5 View W e as a subset of {0, 1} ∗ ×Q 2 and let U e = {〈x, q〉 ∈W e : x = ∅ → q ≤ 1}.The construction in Fact 7.2.4 applied to U = U e yields a supermartingale approximationof a c.e. supermartingale S e.IfU e is the undergraph of a c.e. supermartingale Ssuch that S(∅) ≤ 1thenS e = S.7.2.7 (i) is immediate by 7.1.7(ii), and (ii) is (almost) trivial. For (iii), let S be themartingale such that S(0 k )=2 k for each k (see the solution to 7.1.11). We show thatlim k FS(0 k )2 −k = 0, whence FS does not multiplicatively dominate S. NotethatFS(0 k )2 −k = ∑ inwe have ∑ n≤i

Solutions to Chapter 7 401let q = e −2/d2 < 1, then 2 −m #A m,d ≤ 2q m by (2.17), hence the tail sum taken from ron is bounded by 2 ∑ m≥r qm =2q r /(1−q) whicheffectively tends to 0 with r. Hence Zis not Schnorr random by 7.3.4.7.3.11 We may assume that S is Q 2 valued. By 7.3.10 the leftmost non-ascending pathfor S is computable.7.4.5 Suppose Y = ∗ Z and B k succeeds on Y .Choosen such that Y (m) =Z(m) forall m ≥ n, thenB k,n succeeds on Z.7.4.6 If x = E ↾ n has been defined already wait for B(x0) to converge. If B(x0) ≤ B(x1)let E(n) =0,elseE(n) =1.IfB(x0) diverges for some x define E = x0 ∞ .7.5.11 Some set Z ≡ T C is weakly 1-generic by 1.8.50, and hence weakly random. SuchasetisnotSchnorrrandombecauseitfailsthelawoflargenumbers.7.5.12 We meet the usual minimal pair requirements N e,andrequirementsT (k,r) : ∃nB k (Z r ↾ n)↑ ∨ B k (Z r) < ∞ (k ∈ N,r ∈ {0, 1}),where (k, r) denotes 2k + r. Weassociatestrategiesα of length |2e| with N e, andstrategies β of length 2i +1withT i. For such a β, ifi =(k, r), the outcome 0 meansthat B k (Z r ↾ n)isdefinedforarbitrarilylargen. TheT (k,r) strategy β is as follows:at stage s let s β B k (Z r ↾ n 1) and Z r(n) =0(β requires attention) then leave Z r ↾ n unchanged, let Z r(n) =1and Z r(m) =0forn

402 Solutions to the exercises(In z the positions left open by α have been filled in, up to the largest position indom(α).) The prefix-free machine M on input τ waits for U(σ) =α where τ = σy.Then it checks whether |α| + |y| =max(dom(α)) + 1; if so it prints the string z wherethe bits of y in ascending order are used to fill in those positions. For instance, ifα =(〈4, 0〉, 〈0, 1〉) andy = 110 then z =11100.7.6.28 Suppose this probability is 1. In the proof of 7.6.24, replace (7.20) byn i+1 = µn > n i. 2 −n #{y : |y| = n & ∃r [0,n i) ⊆ dom Θ y S (r)} ≥ 1 − 2−i .Let C i be the clopen set generated by the y that are left out, then λC i ≤ 2 −i .IfS doesnot scan all places of the set Y then Y ∈ C i for almost every i. Since( ⋃ i>m Ci) m∈N isa Schnorr test, Y is not Schnorr random.If Y passes this Schnorr test then a suitable variant of D succeeds on Y .SupposeY ∉ C i for i ≥ n 2r. D only bets from strings of length n 2r on. For k ≥ r, oneonlyconsiders strings u of length m such that [xu] ∩ C 2k+1 = ∅ in (7.21) and the subsequentdiscussion, because r u and w u are only defined for such strings.Solutions to Chapter 8Section 8.18.1.7 If P ⊆ MLR, eachsetZ in P ∩ Low(Ω) is2-random,andhencenotΣ 0 2.8.1.8 Let r =lim k r k where (r k ) k∈N is an effective sequence of rationals. Sincelim k Ω X↾ kk= Ω X ,wehaveX ∈ G r ↔∀n ∃k ≥ n [abs(Ω X↾ kk− r k ) ≤ 1/n].8.1.11 A is low for Ω, henceabaseforML-randomness,andthereforelowforK.8.1.12 Suppose C ≤ LR X, Y . We show that C is a base for ML-randomness.Note that X and Y are 2-random relative to each other by 3.4.9. They are low for Ω,and hence in GL 1.SinceY is 2-random in X, Y is ML-random in X ⊕ Ω, sobythevanLambalgen Theorem relative to X, Y ⊕ Ω is ML-random in X, and hence ML-randomin C. FinallyC ≤ T Y ′ ≡ T Y ⊕ Ω by Corollary 8.1.10, since Y is low for Ω.8.1.17 (i) “≤ + ”: Modify the request operator L in 8.1.16 as follows. For each m, i,if K s(X ↾ 〈m,i〉 )

Solutions to Chapter 8 403Section 8.28.2.9 It suffices to show that some function g ≤ wtt ∅ ′ dominates each computablefunction. To this end, let g(n) =max{Φ e(n): Φ e(n)↓ & e ≤ n}.8.2.10 As in Exercise 1.5.5, let h be a computable function such that Y ≤ T C impliesY ≤ tt ∅ ′ with use bound h for each Y .Weclaimthattheω-c.e. function g(m) =µs. ∅ ′ s ↾ h(m) = ∅ ′ ↾ h(m) dominates each function computed by C.Assume for a contradiction that the function f ≤ T C is not dominated by g. Webuildaset Y ≤ T C such that Y is not truth-table reducible to ∅ ′ with use bound h. Let(Ψ e) e∈Nbe an effective listing of all (possibly partial) truth-table reduction procedures with usebound h defined on initial segments of N, similartothelistingintroducedbefore1.4.5.We define Y (m) inductively.Lookfortheleaste ≤ m such that Ψ e,f(m) (∅ ′ f(m)) ↾ m=Y ↾ m and r = Ψ e,f(m) (∅ ′ f(m); m)↓, and define Y (m) tobenotequaltor. Ifthereisnosuch e define Y (m) =0.Each e eventually has strongest priority at a stage m such that f(m) ≥ g(m). Since∅ ′ f(m) ↾ h(m) is stable, we cause Y ≠ Ψ e(∅ ′ ).8.2.11 A is low for K. If f ≤ T A then K(f(x)) ≤ + K A (f(x)) ≤ + K A (x) ≤ + 2|x|. Toobtain a c.e. trace that works for all f on almost all inputs, let T x = {y : K(y) ≤ 3|x|}if this set is nonempty and {0} otherwise. Note that #T x = O(x 3 + 1).8.2.12 We meet the lowness requirements L e from the proof of Theorem 1.6.4. We builda functional Γ such that f = Γ A is total, and the requirements R k : ∃n >kf(n) ∉ V nare met, where (V n) n∈N is a universal c.e. trace as in Corollary 8.2.4. The priorityordering is R 0 >L 0 >R 1 >...;ifL e (e

404 Solutions to the exercises⇐: Let ˜f(x) =f ↾ I n(x) .Picka[total]functionψ for ˜f. Wemayassumethatψ(y) isatuple of length h(n(y)) for each y. LetS x be the union of the sets of entries occurringin ψ(y) forsomey ∈ I n(x) .Then(S x) x∈N is a c.e. [computable] trace for f. Notethat#S x ≤ h(n(x)) 2 .8.2.24 Modify the proof of Theorem 8.2.23.⇒: There is a c.e. trace (W g(n) ) n∈N with bound λn.2 n for f. DefineL as before butusing W g(n) instead of D g(n) .ThenL is a bounded request set, and the correspondingprefix-free machine N satisfies K A M (y) ≥ K N (y) − 1foreveryy.⇐: Given f ≤ T A, defineM as before. Then ∀yK(y) ≤ K A M (y)+b for some b. Let g bea computable function such that W g(x) = {y : K(y) ≤ 3logx}, then#W g(x) ≤ 2x 3 +1for each x, andf(x) ∈ W g(x) for almost all x.8.2.25 Adapt the proof of 5.2.3. Let ˜M be the computable measure machine relativeto A such that ˜M(σ) ≃ A↾ M(σ) for each σ. Chooseac.m.m.N for ˜M as in 8.2.22, thenfor each n we have K N (A↾ n) ≤ + K ˜M(A↾ n) ≤ + K M (n).8.2.31 ∀nK(Z ↾ n| n) ≤ + log n, soK(Z ↾ n) ≤ + K(Z ↾ n| n)+K(n) =O(log n).8.2.32 Let ˜h(n) =⌊h(n)/3⌋. Thereareinfinitelymanyn such that K(n) < ˜h(n) andK(Z ↾ n| n) ≤ ˜h(n). For every such n, wehaveK(Z ↾ n) ≤ + K(Z ↾ n| n)+K(n) ≤ + ˜h(n)+K(n) ≤ + 2˜h(n).Section 8.38.3.5 We show that the function g(n) =µs. Ω s ↾ 2n= Ω ↾ 2n dominates each functionf ≤ T A. Sinceg ≤ wtt Ω ≤ wtt ∅ ′ ,thisimpliesthatA is array computable.The set L = {〈n +1, Ω f(n) ↾ 2n〉: n ∈ N} is a bounded request set relative to A oftotal weight 1. Let M be the corresponding computable measure machine relative A.If g(n) ≤ f(n) thenKM A (Ω ↾ 2n) ≤ n +1.Ifthereareinfinitelymanysuch n, thenΩ isnot Schnorr random relative to A.8.3.6 (i.a) Let C n be the n-th clopen set in some effective listing of the clopen sets.Suppose (G m) m∈N is a Kurtz test relative to A (Definition 3.5.1). Then there is f ≤ T Asuch that ∀mC f(m) = G m.LetS m = C J(m) if J(m) ↓ and λC J(m) ≤ 2 −m ;otherwiselet S m = ∅. SinceA does not have d.n.c. degree there are infinitely many m such thatG m = S m.So,wheneverZ fails (G m) m∈N then Z fails the Solovay test (S m) m∈N .(i.b) Clearly Z ∉ WR D .ThusD ∉ Low(MLR, WR), hence D has d.n.c. degree by (i).(ii) Let f ≤ T A be a d.n.c. function. Let Q =2 N − R 1 be a nonempty Π 0 1 class ofML-random sets. Using f compute a sequence d 0

Solutions to Chapter 8 405Now let y 0 = ∅ and y i+1 = y îf(α(y i)) for i

406 Solutions to the exercises(here we used that A is c.e., and thus cannot turn back to a previous configuration).Therefore #T e ≤ g(e). A finite variant of (T e) e∈N is a trace for Φ A .8.4.36 Given an order function g, letn and f be as in the proof of Claim 2. Letq(k, t) =A ′ (k) fork < n.Fork ≥ n, ift

Solutions to Chapter 8 407Hence C Φ is Π 0 3.Also,C Φ is null by Sacks’ Theorem 5.1.12 relativized to ∅ ′ .Eachweakly3-random set Z is 2-random, and hence in GL 1.IfZ is also high then Z is in someclass C Φ, contradiction.8.5.22 The ∆ 0 2 sets form a Σ 0 4 class since “Y ≤ T ∅ ′ ”amountstotheΣ 0 4 condition∃e ∀n ∃s ∀t ≥ s [Y (n) =Φ ∅′e (n)[t]]. If this class would be Σ 0 3 then by Theorem 5.3.15there would be a low incomputable c.e. set B Turing below each ML-random ∆ 0 2 set,contrary to Theorem 1.8.39.8.5.23 By 1.4.4 Y ∈ H iff there is a truth-table reduction procedure Φ such that∀n, s ∃t ≥ s [Φ(∅ ′ ; n)[t] =Y (n)].For ˜H, useinsteadtheexpression∀n, s ∃t ≥ s [Φ(∅ ′ ; n)[t] =Y t ′ (n)].8.5.24 By 1.8.41 with P =2 N −R 1,thereisaML-randomsetY such that (Y ⊕B) ′ ≤ ttB ′ ≤ tt ∅ ′ .ThenY ∈ ˜H, soA ≤ T Y . Therefore (A ⊕ B) ′ ≤ m (Y ⊕ B) ′ ≤ tt ∅ ′ .8.5.25 The second assertion follows from the first by letting P =2 N − R 1. For the firstassertion, we combine the framework given by the proof of Theorem 1.8.39 with a costfunction construction of a c.e. K-trivial set B. Insteadofworkingrelativeto∅ ′ ,weprovide at each stage s approximations of the objects to be constructed. For each iL 0 >S 1 >L 1 >..., and when a strategy changes its Π 0 1 class it initializes the strategies ofweaker priority, because the environment they work in has changed.Construction of B and Y . For a Π 0 1 class G we let G[s] betheusualapproximationatstage s by a clopen class given by (1.17). Let P 0,s = P and f(e, s) =0foreache suchthat 2e +1≥ s.Carry out substages i for 1 ≤ i 2 −e−me ,then let x e = 〈s, e〉 and initialize the strategies of weaker priority. We say that S e acts.(b) If G[s] ≠ ∅ where G = P 2e+1,s ∩ {Y : ¬Φ Y e (x e)[s] =0}, letP 2e+2,s = G; otherwiselet P 2e+2,s = P 2e+1,s , put x e into B and declare S e satisfied. We say that S e acts.Verification. Let R i be the i-th requirement in the priority list. We show that thereis a computable function g such that R i acts no more than g(i) times.Supposewehave defined g(j) for1≤ j

408 Solutions to the exercisesthan g(i) := k +1 times. Now suppose that R i = S e.Aftereachinitialization,bythe benignity feature of the standard cost function c K, the candidate x e changes atmost 2 e+k times. Thus R e acts no more than g(i):= k(2 e+k +1)times.If f(e, s) ≠ f(e, s − 1) then some R i acts at stage s where i ≤ 2e +1, so the number ofchanges of f(e) iscomputablybounded.Clearlylim sf(e, s) =Y ′ (e), so Y is superlow.Let x be the final candidate of S e.If¬Φ Y e (x) =0thenx ∉ B. IfΦ Y e (x) =0theneventually the alternative (b) in S e applies at substage 2e +2, so x ∈ B. Ineithercase S e is met. Finally, B is K-trivial because its enumeration obeys c K.Section 8.68.6.1 (i) The implications are: “not of d.n.c. degree” ⇒ BN1R ⇒ “not of PA degree”. Toargue in terms of highness properties, each set of PA degree bounds a ML-random set,but a low ML-random set is not of PA degree. Each ML-random set is of d.n.c. degree,but there is a set of d.n.c. degree that does not compute a ML-random set by Ambos-Spies, Kjos-Hanssen, Lempp and Slaman (2004). (ii) The implications are: computable⇒ BB2R ⇒ Low(Ω). The second implication is strict because an incomputable set inLow(MLR) isnotcomputedbya2-randomset.(iii)Theminimalconullclassesinthediagram are Low(Ω), “not of PA degree” and “not high”.8.6.3 We discuss two classes. (1) The sets that are not of d.n.c. degree: by 6.3.14 letC be a Turing incomplete LR-hard c.e. set. Then C is not of d.n.c. degree. Also C ishigh and hence not in GL 2.(2)Thesetsthatarecomputablydominated:suchasetcan be of PA degree by 1.8.42. It can fail to be in GL 1 by the discussion after 8.4.7.Solutions to Chapter 99.1.2 ⇒: A R = {x: ∀Y ≻ x [Y ∈ R]}.⇐: The relation B = {〈n, Y 〉: Y ↾ n∈ A R} is Π 1 1,andR = {Y : ∃n 〈n, Y 〉∈B}.9.1.6 We are given indices e, i for the Π 1 1 classes P and Q. ReplacingΦ X e by a linearorder of type |Φ X e |+ω+1, and Φ X i by a linear order of type |Φ X i |+ω+2, we may assumethat sets enter P and Q at different stages. Let ˜P = {X ∈ P : Φ X i is not isomorphicto an initial segment of Φ X e } (X enters P at a stage when it has not entered Q).Symmetrically, let ˜Q = {X ∈ Q: Φ X e is not isomorphic to an initial segment of Φ X i }.9.1.7 For each pair i, j let n = 〈i, j〉, andletQ g(n) be the Π 1 1 class{X ∈ Q i : ∀Z [Z ∈ Q i ∨ Z ∈ Q j]}. Nowapply9.1.6totheclassesQ g(n) and Q j inorder to obtain Q g(n) and Q h(n) .9.1.8 Let ˜W e be the relation (with domain N) isomorphictoω + W e.LetR be thewell-order isomorphic to the ω-sum of all the ˜W e that are well-orders. Then |R| = ω1ckas each proper initial segment of R is a computable well-order. Since R ≤ T O we have|R| < ω1 O .9.1.18 The first statement is clear because 〈n, y〉 is not in the graph of f iff 〈n, z〉is in the graph for some z ≠ y. For the second, let g be a computable function suchthat 〈n, y, X〉 ∈S ↔ Φ X g(n,y)well-ordered. Replace S by the reduction procedure ˜S ={〈n, y, X〉 ∈S : ∀z ≠ y [ Φ X g(n,z) is not a proper initial segment of Φ X g(n,y) &[Φ X ∼ g(n,z) =Φ X g(n,y) → y

Solutions to Chapter 9 4099.1.24 We may assume the domain of R is N. NotethatR ∈ L(ω1 ck )by9.1.21.Letγ = |R|. Bytransfiniterecursiontheisomorphismg : γ → R is Σ 1 over L(ω1 ck )(sinceg(α) ≃ the least element with respect to R in ω − ran(g ↾ α).) Let f(n) be the ordinalisomorphic to the initial segment of R given by n. Thenf is Σ 1 over L(ω1 ck ). Hence bythe Bounding Principle the range of f is bounded, whence γ is a computable ordinal.9.2.7 By transfinite induction over over L(ω1 ck ), for each Π 1 1 set S the function α → S αis Σ 1 over L(ω1 ck ).We extend the solution to 2.2.24. We may assume that at most one request enters Wat each stage α. Define the prefix-free Π 1 1-machine M as follows. If 〈r, y〉 enters W atstage α, letk = r + 1. Let i α be the least i such that for X = W α∀F ⊆ X finite [ i2 −k ≥ ∑ l,z 2−l [〈l, z〉 ∈F ] ] .Note that the foregoing expression in variables X, i, k is ∆ 0 over L(ω1 ck ). Now as beforelet w α be the string w of length k such that 0.w = i2 −k . The function taking α to w α(and to {nil} if no element is enumerated at α) isΣ 1 over L(ω1 ck ). Define M(w α)=y.9.3.10 If S is null then S ⊆ Q. Otherwise,bytheSacks–TanakaTheorem9.1.11,Shas a hyperarithmetic member X, so{X} is a Π 1 1 null class, whence X ∈ Q.9.3.11 We claim that G = ⋃ ̂Q n g(n) ,whereg is as in Exercise 9.1.7 and Ŝ for a Π1 1class S is defined in the proof of 9.3.6. Then G is Π 1 1 because we uniformly in n havean index for ̂Q g(n) as a Π 1 1 class.If X ∈ G then X ∈ S for some null ∆ 1 1 class S. ThenS = Q g(n) for some n and Ŝ = S.Now suppose that X ∈ ̂Q g(n) .IfQ g(n) is null we are done. Otherwise let γ be the leastordinal such that Q g(n),γ is not null. Then γ < ω1 ck by 9.1.15, so X ∈ ̂Q g(n),α+1 whereα +1< γ. Therefore X is in a null ∆ 1 1 class by the Approximation Lemma 9.1.4(ii).9.3.12 ⇒: The class L = {X ⊕ Y : X ∈ R Y & Y ∈ R} is Σ 1 1.SinceR Y is co-null foreach Y ,byFubini’sTheoremL is conull. Hence R ⊆ L.⇐: Let R[Y ]={X : X ⊕ Y ∈ R}. Thentheclass{Y : R[Y ]is conull} is Σ 1 1 by theMeasure Lemma 9.1.10, and conull by Fubini’s Theorem: otherwise there are rationalsɛ > 0andq

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GeneralNotation indexThe absolute value of a number r ∈ R is denoted by abs(r).The cardinality of a set X is denoted by #X.See the preface, page viii, for conventions on variables.∀ ∞∃ ∞for almost every xfor infinitely many xα ≃ βboth expressions α, β are undefined, or both are defined with thesame value 4∅the empty string∅the empty set#X cardinality of the set XX△Y (X−Y )∪(Y −X), the elements on which sets X and Y disagree 70X = ∗ Y (X − Y ) ∪ (Y − X) is finite 70dom ψ or dom(ψ) domain of a function ψran ψ or ran(ψ) range of ψN set of natural numbers 0, 1, 2 ...Rset of real numbersR + 0 set of non-negative real numbers 68abs(r)absolute value of r ∈ RQ 2 {z2 −n : z ∈ Z,n∈ N}, setofdyadicrationals 50〈m, n〉m +(m + n)(m + n +1)/2, number coding the ordered pair of mand n(〈m, n〉) 0m, thefirstcomponentofsuchapair(〈m, n〉) 1n, thesecondcomponentofsuchapair〈n 0,...,n k 〉 〈...〈n 0,n 1〉,n 2〉,...,n k 〉X × Y{〈m, n〉: m ∈ X & n ∈ Y }, cartesian product of X, Y ⊆ NN [i] N × {i} = {〈m, i〉: m ∈ N}∑A [i]A ∩ N [i]n τ(n)[θ(n)] sum of terms τ(n) overalln such that condition θ(n) holds,asfor instance in∞ = ∑ n1/n [n is prime]. Note that the value of τ(n) canberepeated,which is the advantage of this notation over ∑ {τ(n): θ(n)}.Chapter 1P k e e-th Turing program with k inputs 3Φ k e partial computable function or functional given by P k e 3, 10Φ e short for Φ 1 e 3W e dom(Φ e), the e-th c.e. set 6∅ ′ {e: e ∈ W e}, thehaltingproblem 6Φ e,s(x) =y P e on input x yields y in at most s steps 7W e,s dom(Φ e,s) 7

Notation index 419D n the finite set with strong index n 7≤ m many-one reducibility 8≤ 1 X ≤ 1 Y if X ≤ m Y via a one-one function f 9We Y dom(Φ Y e ) 10f ≤ T Y f is Turing reducible to Y 10A ≤ T Y the characteristic function of A is Turing reducible to Y 10J Y (e) Φ Y e (e) 10Y ′ dom(J Y ), the Turing jump of Y 10Y (n) n-th jump of Y ,definedbyY (0) = Y and Y (n+1) =(Y (n) ) ′ 11{0, 1} ∗ set of strings of zeros and ones 12στ concatenation of σ and τ 12σa σ followed by the symbol a 12σ ≼ τ σ is a prefix of τ 12σ | τ neither of σ, τ is a prefix of the other 12σ < L τ σ is to the left of τ, thatis,∃ρ [ρ0 ≼ σ & ρ1 ≼ τ] 12|σ| the length of σ 12∅ the empty string, that is, the string of length 0. 12Z ↾ n Z(0)Z(1) ...Z(n − 1) 12log n max{k ∈ N: 2 k ≤ n} 13log 2 n usual real-valued logarithm of n in base two 13Φ Y e,s(x) =y (see page 13)Φ Y e,s(x)↓ Φ Y e,s(x) isdefined 13use Φ Y e (x)1+largest oracle question asked during the computationΦ Y e (x) 13Φ σ e (x) =y Φ F e (x) =y for F = {i 0 402 N Cantor space, the space of functions Z : N → {0, 1} with theproduct topology 45[y] {Z : y ≼ Z}, cylindergivenbythestringy 47Paths(B) paths of a binary tree B 47T P {x: [x] ∩ P ≠ ∅}, treecorrespondingtoaclosedsetP 47x is on P x ∈ T P 48A R {x: [x] ⊆ R}, setofstringscorrespondingtotheopensetR 48

420 Notation index[S] ≺ {X ∈ 2 N : ∃y ∈ Sy≺ X}, theopensetgeneratedbyS 48∑0.Zi∈Z 2−(i+1) , the real number corresponding to a co-infiniteset Z 50Q 2 {z2 −n : z ∈ Z,n∈ N}, thesetofdyadicrationals 50Σ 0 n, Π 0 n relation (see 1.8.55) 64N NBaire space, the space of functions f : N → N with the producttopology 67Γ f {〈n, f(n)〉: n ∈ N}, the graph of a function f 67D : N ∗ → N function D(n 0,...,n k−1 )= ∏ k−1i=0 pn i+1i ,wherep i is the i-thprime number 67Seq the range of D, thesequencenumbers 67µ(C |σ) µ(C ∩ [σ])/µ[σ], local outer measure 69λ uniform (outer) measure 70C | σ {X : σX ∈ C} for a class C and a string σ 71Chapter 2g ≤ + f ∃c ∈ N ∀n [g(n) ≤ f(n)+c] for functions f,g 75g = + f g ≤ + f ≤ + g 75g = O(f) ∃c ≥ 0 ∀ ∞ n [abs(g(n)) ≤ c abs(f(n))] 75f ∼ g lim nf(n)/g(n) =1 75C M (x) min{|σ|: M(σ) =x} for machine M 76C M,s(x) min{|σ|: M s(σ) =x} 77V plain optimal machine given by V(0 e−1 1ρ) ≃ Φ e(ρ) 76C(x) C V (x) plaindescriptivecomplexityofstringx 77x is d-compressible C C(x) ≤ |x| − d for x ∈ {0, 1} ∗ and d ∈ N 78Cpr d set of d-compressible C strings 78C g (x)min{|σ| : V(σ) =x in g(|x|) steps}, time-boundedversionof C 81K M (x) C M (x) for prefix-free machine M 83Ω M∑σ 2−|σ| [M(σ)↓] halting probability of the prefix-freemachine M 84U optimal prefix-free machine of Theorem 2.2.9 85K(x) K U (x) 85K s(x) min{|σ|: U s(σ) =x} 86wgt W (S)under W 87P M (x)∑ρ,x 2−(ρ) 0[ρ ∈ W & x ∈ S &(ρ) 1 = x], weight of S∑σ 2−|σ| [M(σ) =x] = λ[{σ : M(σ) =x}] ≺ ,probabilitythatMoutputs x 91V 2 optimal binary machine given by V 2 (0 e−1 1σ,y) ≃ Φ 2 e(σ,y) 92C(x | y) min{|σ|: V 2 (σ,y)=x} conditional C-complexity 92U 2optimal prefix-free machine when fixing the secondcomponent 92K(x | y) min{|σ|: U 2 (σ,y)=x} conditional K-complexity 92x ∗ a shortest U-description of x (see 2.3.2) 93x is d-incom- K(x) > |x| − d for x ∈ {0, 1} ∗ and d ∈ N 98pressible Kx is strongly K(x) > |x| + K(|x|) − d 98d-incompressible K

Notation index 421Chapter 3MLR class of Martin-Löf random sets 106non-MLR 2 ω − MLR 106RbM [{x ∈ {0, 1} ∗ : K M (x) ≤ |x| − b}] ≺ for machine Mand b ∈ N 107R b Rb U 107Ω Ω U = λ[dom U] ≺ 109Ω s λ[dom U s] ≺ 109β ≤ S α ∃d ∈ N ∃γ left-c.e. [ 2 −d β + γ = α ] ,Solovayreducibility 113K M A(x) |σ| for shortest string σ such that M A (σ) =x 121K A (x) K U A(x) 121Ω A M λ(dom M A ) 121Ω A Ω A U = λ(dom U A ) 122R A b [{x ∈ {0, 1} ∗ : K A (x) ≤ |x| − b}] ≺ 122W2R class of weakly 2-random sets 134Low(Ω) {A: Ω ∈ MLR A },thesetsthatarelowforΩ 140Chapter 4PA Peano arithmetic 156A ∼ 1 B A = B 160A ∼ 2 B A = ∗ B 160A ∼ n B A (n−3) ≡ T B (n−3) for n ≥ 3 160Chapter 5Low(C)class of sets A such that C A = C, foroperatorC : P(N) → P(P(N)) 163Low(MLR) class of sets A such that MLR A = MLR 163M class of sets that are low for K 165SA Φ {Z : A = Φ Z } for Turing functional Φ 169SA,n Φ [{σ : A↾ n≼ Φ σ }] ≺ 169Low(C, D) {A: C ⊆ D A } for randomness notions C ⊆ D 174K class of K-trivial sets 176T b{z : ∀u ≤ |z| [K(z ↾ u) ≤ K(u)+b]}, treeforK-trivials withconstant b 178G(b) #Paths(T b ), the number of sets that are K-trivial via b 179∑c K(x, s)x

422 Notation indexChapter 6γ Y (x) use Γ Y (x), for a Turing functional Γ 240use (e ∈ W C ) use Φ C (e), where the functional Φ defines the c.e.operator W 248Chapter 7S(Z) sup n S(Z ↾ n), for a supermartingale S 260, 262Succ(S) {Z : S(Z) =∞}, thesuccessclassofS. 260, 262B C(x)λ(C |x), the martingale associated with a measurableclass C 261∑FSy 2−K(y) E y,auniversalc.e.martingale 266CR class of computably random sets 268PCR class of partial computably random sets 273B k amartingalecopyingΦ k as long as possible 273TMG {k : B k total} 274B k,nasupermartingalecopyingascaledversionofB k from length non 274G n the martingale such that G n(x) =1for|x|

Index∆ 1 1class, 365measure of, 370dominated, 366relation, 365, 366set, 365, 366∆ 1 1 (A) function, 370∆ 0 2 set, 18∆ 0 separation, 371Π 0 1 class, 46, 53, 65examples of, 55in Baire space, 68, 138index for, 55measure of, 72, 116perfect, 311Π 1 1-random cuppable, 384Π 0 2class, 65presentation of, 66singleton, 2, 66, 104non-arithmetical, 66Π 0 nclass, 64complete, 22relation, 64relativized, 66set, 21singleton, 66Π 1 1 class, 366largest countable, 380largest null, 105, 380largest thin, 380measure of, 370Π 1 relation, 365, 366Π 1 1 set, 1, 365, 366analog of c.e. set, 365, 371complete, 67, 368, 369Σ 1 over L A , 372Σ 0 1 class, 53, 65Σ 0 nclass, 64complete, 22relation, 64relativized, 66relation on sets, 366set, 21Σ 1 1 relation, 366Σ 1 bounding, 371µ-measurable set, 68σ-algebra, 691-generic set, 63, 126, 146, 382existence, 64for higher computability, 382is generalized low, 64weakly, 62, 128, 382is hyperimmune, 62is weakly random, 128left-c.e., 62lowness for, 3182-random, see random, 2-accounting method, 171almost everywhere dominating, 236Ample Excess Lemma, 267arithmetical hierarchy, 23arithmetical set, 1, 21array computable, 313, 315, 343implied by Ω ∈ SR A , 326implied by Low(Ω), 326implies GL 2 , 315Arslanov, 148autoreducible set, 119and hat game, 119each ranked set is, 129axiom of choice, 367Baire category theorem, 62Baire space, 17, 50, 62, 67Barmpalias, 230, 233base for ML-randomness, 165, 170, 210,328Π 1 1-version, 377basis theoremand PA sets, 61and sets of PA degree, 156for Π 0 1 classes, 57for computably dominated sets, 19, 60,403Gandy, 370Kreisel, 57, 58, 135low, 57avoiding upper cone, 59low for Ω, 304Sacks–Tanaka, 370superlow, 58uniformity of, 57Bedregal, 323, 326

424 IndexBernoulli scheme, 73betting strategy, 259, 262, 268, 286KL, 297Achilles heel of, 298permutation, 294, 296Bickford, 241Bienvenue, 138, 305binary expansion, 50Binns, 235Biondi–Santi Brunello wine, 205, 210bit, 12, 17Boolean algebra, 46countable dense, 46of clopen sets, 49Borel class, 66, 68, 365, 369is measurable, 70Borel code, 1, 369Borel operator, 233bounded request set, 87, 90, 91, 114, 171,185, 199, 214, 308Π 1 1 , 373total weight of, 87Bounding Principle, 365, 372box promotion method, 348, 353, 356shrinking boxes, 355vs. decanter method, 356C (plain complexity), 74, 106characterization, 77conditional, 92, 106, 122continuity, 80, 95dips, 83, 105growth, 80incomputability, 82interaction with K, 94time-bounded version, 81, 82, 131, 139C-trivial set, 177, 182is computable, 182c.e., see computably enumerablec.e. open setindex for, 54measure of, 72with computable measure, 72c.e. operator, 10, 248inversion of, 249language for building, 248c.m.m., see machine, computable measureCalhoun, 150Calude, 81, 113, 115, 158Cantor space, 17, 45cappable, 37, 41, 243capricious destruction, 245Chaitin, 75, 109, 177halting experiment, 83change set, 19, 187charity, 262Chebycheff inequality, 73, 101Chernoff bounds, 101Cholak, 235, 258, 348, 352Chong, 379, 381Church, 288Church–Mises–Wald stochastic, 289Church–Turing thesis, 3with oracle, 10, 239class, 45class of similar complexity, 1clearing computations, 245clopen set, 45, 49strong index for, 49closed set, 45, 47in Baire space, 67coding constant, 76, 77, 84, 88, 99for prefix-free machine, 84, 308for request operator, 308given in advance, 90Coding Theorem, 91, 94, 375Π 1 1-version, 375Cohen genericity, 62Cole, 343collection schemeΣ 1 , 192combinatorial, 322compact space, 48completeTuring, 32, 37, 145, 148, 183, 238effectively simple is, 149Schnorr trivial can be, 321complete setr, 9many-one, 9, 22truth-table, 82weak truth-table, 40, 81, 82Completeness Criterion, 148, 149, 151complexitycomputational, 1, 18, 20, 64absolute, 2, 117relative, 2, 8, 117vs. degree of randomness, 364descriptive, 1, 18, 20, 64, 66of classes, 64relative, 2compressible C , 78, 98compressible K ,98compression function, 137computablerelative to a set, 10computable approximation, 18chunked, 195computable enumeration, 7, 193computable function, 3

Index 425computable ordinal, 365computable measure machine, seemachine, computable measurecomputable permutation, 34computable randomness, 130computable set, 6computably approximable from above, 77,86computably dominated, 1, 24, 27, 60, 135,141, 163, 164, 192, 260, 309,312, 316, 318, 321–323, 366and generalized low, 311, 318, 338, 364and low, 2and low for Ω, 309arithmetical complexity, 65as Low(C), 163basis theorem, 59, 60generalized low, 143is GL 2 ,28is array computable, 315ML-random, 364not of d.n.c. degree, 318of PA degree, 156perfect class of, 61, 311, 318same as tt-top, 28uniformly, 28weakly random is weakly 2-random,128, 135, 136computably enumerable, 6construction of a set, 40definition of a set, 40relative to a set, 10uniformly, 6computably inseparable sets, 159computably random, see random,computablycomputably traceable, see traceable,computablyconstructible hierarchy, 371continuous, 126function, 49lower semi-, 126for Ω M , 126upper semi-, 126controlled risk strategy, 202, 310, 330, 335conull class, 25, 70Cooper Jump Inversion Theorem, 133cost function, 186c Y for ∆ 0 2 set Y , 189adaptive, 164, 186, 198, 200, 344and injury-freeness, 200benign, 190, 348, 352, 359and s.j.t., 352, 357has limit condition, 352construction, 184criterion for K-triviality, 188method, 135, 176, 184, 348for ∆ 1 1 sets, 185for K-trivials, 195necessity of, 201, 215monotonic, 186obeying a, 186standard, 40, 186, 215with limit condition, 185–193, 198, 199countable additivity of measure, 70countable subadditivity of measure, 68counting condition, 77, 78, 86covering procedure, 167, 223, 226, 235,302, 323table, 229creative set, 37, 43, 113first-order definition, 44, 335Csima, 183cuppable, 242cupping partner, 359cylinder, 47and Baire space, 67and uniform measure, 68basic open, 47–50, 62, 66d.n.c., see diagonally noncomputabled.n.c. degree, set of, 144, 311characterization, 145, 147infinite subset of ML-random is, 146,326decanter method, 200, 294, 356decanter model, 205picture of, 206deficiency setof machine, 81deficiency stage, 39degreer-, 8, 16non-cappable, 37promptly simple, 37degree invariance, 35, 231, 233, 249, 253degree structure, 20delay function, 243, 246Demuth, 141, 158Demuth random, see random, Demuthdense along a set, 63dense set, 62dense simple, 81descriptionM-, 76close, 46of string, 74prefix-freesplitting off, 94, 95, 109, 138description system, 1, 18, 21for sets, 103

426 Indexfor strings, 74using language of arithmetic, 21via computable approximation, 21diagonalization, 30, 43witness for, 30diagonally noncomputable, 144n-, 160, 161two-valued, 55, 156, 304discrete c.e. semimeasure, 90Dobrinen, 235domination, 24, 26, 75, 225, 312, 313domination lemma for martingales, 279Downey, 131, 221, 253, 318, 319, 348, 352dyadic rational, 50effectively simple set, 32, 149is Turing complete, 32, 149exact pair, 219f-small, 226f.p.f., see fixed point freefacile, 276, 313, 320feasible computability, 81Figueira, 120, 149, 339, 344filter in Boolean algebra, 46finite assignment, 11, 17, 297, 395positions of, 297finite extension method, 30, 62finite variant of a set, 71first-order definablein E, 37, 44in R T ,37fixed point, 5fixed point free, 146n-, 146, 160–162, 364formula∆ 0 , 371Σ 1 , 371Franklin, 321Friedberg, 30, 35, 149jump inversion theorem, 336Friedberg–Muchnik theorem, 30, 35, 150and IΣ 1 , 192injury-free proof, 150, 152, 154Fubini’s Theorem, 25, 409function∆ 1 1 ,20ω-c.e., 20computable, 3growth rate of, 75partial computable, 3function tree, 317, 318, 337Gödel, 5, 118Gács, 91gambler, 260Gandy, 370garbage, 203, 207, 254, 329, 356quota, 204, 210, 211generalized low 1 ,GL 1 , 26, 140, 142, 143,166, 304, 311, 318and relative ML-randomness, 124can be high, 124computably dominated, 311generalized low 2 ,GL 2 ,29generic set for perfect trees, 383global view of sets, 16, 46, 66, 367goal of a procedure, 204golden run, 214golden run method, 176, 195, 200, 208,209, 214, 215, 294, 356, 377Main Lemma on, 201, 215, 216, 220,221, 223, 224, 228graph of a function, 10Greenberg, 189, 235, 258, 312, 318, 326,348, 352Griffiths, 131, 319Hájek, 192half-jump, 253halting probability, 84, 108, 109, 131, 318Π 1 1-version, 376halting problem, 6, 10, 11, 18, 21, 31, 57,59Harrington, 34, 44, 150, 242hat game, 119high set, 1, 24, 28, 130, 251, 256, 272, 361c.e. incomplete, 249c.e. minimal pair of, 252, 258, 287for higher computability, 382non-cuppable, 242that is not superhigh, 257high n set, 28relative to an oracle, 252higher computability theory, 104,105, 365highness property, 24, 190, 238for weak reducibility, 25, 225, 256, 363separating, 256Hirschfeldt, 170, 190, 192, 221, 328, 360Hjorth, 366, 376hungry set, 172, 210, 334hyp-dominated, 381hyp-high set, 382hyperarithmetical reducibility, 370finite, 370, 371hyperarithmetical set, see ∆ 1 1 sethyperimmune set, 27, 37, 139hyperimmune-free degree, set of, 27hypersimple set, 37, 39

Index 427i-set, 204ideal in Boolean algebra, 46ideal in uppersemilattice, 16, 219, 222principal, 16ideal of strings, 48, 54incompleteness theorem, 5, 118incompressible string, 74, 77, 78most are, 98, 103incompressible C , 78, 136incompressible K , 98, 100, 105strongly, 98, 137, 308incomputable, 6independent events, 227, 325index∆ 0 2 ,19ω-c.e., 20, 196for Π 0 class, 55for Π 1 relation, 368for Π 1 1 set, 368for c.e. open set, 54for c.e. set, 7for object, 7for Turing functional, 3for ∆ 1 class, 369for ∆ 1 1 set, 369index set, 23uniformly Σ 0 3 , 198indifferent set, 120induction scheme, 192∆ 0 , 192Σ 1 , 192initial segment complexity, 105, 129, 137,145, 147, 163, 259, 272, 275,289, 295, 308, 320, 321slowly growing, 275injury to a requirement, 33interval Solovay test, 112introreducible set, 40, 159inversionof c.e. operator, 238Ishmukametov, 343isolated point, 56Jech, 365, 368Jockusch, 26, 39, 57, 63, 157, 249JT-hardforms Σ 0 3 class, 342JT-reducibility, 340jump inversion, 336jump operator, 10, 26, 58, 147, 304definable in Turing degrees, 159not one-one on degrees, 26Jump Theorem of Sacks, 15, 160, 253, 399jump traceable, 302, 315, 336, 342by an oracle, 340characterization via C, 82, 339, 341closed downward, 337implies generalized low, 337strongly, 238, 336, 344can be promptly simple, 344closure under ⊕, 357obeys benign c.f., 352, 357versus superlow, 342with bound, 336K (prefix-free complexity), 74, 82, 85, 105conditional, 92incomputability, 82interaction with C, 94with oracle, 121K-trivial set, 34, 40, 92, 144, 163, 164,176, 238, 301, 343Π 1 1-version, 377can be not ∆ 1 1 , 377can be not s.j.t., 349can be promptly simple, 188closure properties, 181, 215constant for, 196cost function construction, 153is ∆ 0 2 , 177is below c.e. K-trivial, 215, 218is low for K, 200, 208, 335is superlow, 178, 207, 215listing with constants, 197≤ K (preordering), 225implies ≤ vL , 226, 309K-trivial set, 377at η, 377König’s Lemma, 47, 67Kechris, 380Khoussainov, 113Kjos-Hanssen, 133, 167, 223, 235, 323,326, 340, 363Kleene, 4, 10, 29Kleene–Brouwer ordering, 368Kleene–Post Theorem, 30, 35Kolmogorov, 75, 297Kolmogorov–Loveland (KL) bettingstrategy, 297Achilles heel of, 298Kraft–Chaitin Theorem, 86Kurtz, 63, 140, 234Kurtz test, 128, 129definition, 128Kučera, 34, 40, 61, 113, 167, 170, 219,253, 256Kučera’s Theorem, 151, 165for ML-random, 153for pairs of sets, 154for two valued d.n.c., 152for weak truth-table, 152uniform versions, 152, 153

428 Indexwith permitting, 153for Π 1 -ML-randomness, 377Kučera–Gács Theorem, 117for Π 1 1-randomness, 366, 384lowness index, 173, 218Lachlan, 148lowness property, 24, 140, 163, 184, 234,LaForte, 319, 321238, 301law of iterated logarithm, 133for weak reducibility, 25, 225, 363law of large numbers, 17, 109, 129, 270,in a Π 0 1 class, 57401in higher computability, 366fails for weakly 1-generic, 128on c.e. sets, 315Lebesgue Density Theorem, 69, 168orthogonal, 25, 309left-c.e. set, 51LR-base for ML-randomness, 231length-of-agreement function, 246LR-reducibility, 225Lerman, 235characterization, 226Levin, 75, 90, 93, 95, 105degrees countable, 233Lewis, 230, 231equivalence with ≤ LK , 225, 227lexicographical order, 12lower cones are null, 230length, 13size of lower cones, 233limit condition, 185, 186Lusin separation theorem, 369Limit Lemma, 19–21, 23, 29, 32, 160, 163Maass, 41LK-reducibility, 225machine, 75equivalence with ≤ LR , 227Π 1 local view of sets, 16, 66, 731 , 373optimal prefix-free, 373Loveland, 183, 297binary, 92Low Basis Theorem, 46optimal, 92low cuppable, 42, 233, 238, 242–244low for ω1 ck,370coding constant for, 76computable measure, 131, 132, 301,low for Ω, 364318, 323, 383same as weakly low for K, 305lowness for, 313, 318low for Ω, 140, 301, 304relative to an oracle, 318basis theorem, 301, 304within ∆ 0 2 , 301copying, 76, 84, 139low for ω1 ck,367optimal, 76, 103optimal prefix-freelow for C, 183, 345deficiency set of, 81low for K, 165, 322oracle, 121each K-trivial is, 207prefix-free, 82, 83, 87is generalized low, 166for a bounded request set, 88is superlow, 166prefix-free in second component, 92Muchnik construction, 165, 199uniformly optimal prefix-free, 127weakly, 305universal, 76low set, 1, 24, 26, 193Machine Existence Theorem, 74, 86–88,can be not superlow, 24294, 108, 131, 308relative to an oracle, 248, 249Π 1 1-version, 373low n set, 26relative to oracle, 308relative to an oracle, 252many-one reducible, 8lowly for C, 346Marchenkov, 34lownessMartin, 28, 59, 148for 2-randomness, 232, 233Martin-Löf, 102, 366for ML-randomness, 165, 167, 215coverable, 173, 358for operator C, 163cuppable, 359for pairs of randomness notions, 174,random, 2, 34, 40, 47, 174322Π 1 1 , 376for Schnorr randomness, 238base for, 170for weak 1-genericity, 318base for Π 1 1 , 377for weak 2-randomness, 165, 215closure under comp. permutations,for ∆ 1 1 -randomness, 366, 383 294

Index 429far from, 163, 177has d.n.c. degree, 145infinite subset of, 146, 326low, 124lowness for Π 1 1 , 377nonrandom features of, 117, 266not autoreducible, 119of PA degree, 157relative to an oracle, 121superlow, 108, 115, 124symmetry of relative, 121, 123Turing degree of, 159universal test for, 105, 107, 108, 266universal test for Π 1 1 , 105random difference c.e. real, 115random left-c.e. real, 108, 113is wtt-complete, 115random right-c.e. real, 115, 116test, 46, 104, 111, 134, 141Π 1 1 , 105, 376definition of, 106generalized, 104, 134, 169, 365martingaleQ 2 -valued, 270computable, 238, 268definition, 260elementary, 261equality, 260functional, 329partial computable, 272, 273simple, 289success of, 260vs. measure representation, 261maximal set, 28, 39measurable class, 70µ, 70measure, 68, 70countable additivity, 68Lebesgue, 68representation of, 68, 261, 263uniform (λ), 68, 261Merkle, 110, 166, 289metric space, 49Mihailovich, 318Miller, D., 242Miller, J., 2, 95, 135, 137, 190, 223, 226,253, 267, 268, 294, 305, 308,309, 311, 340, 386Miller, W., 59minimal pair, 25, 41, 154, 191, 287of high ML-random sets, 258of PA sets, 159, 258relative to ∅ ′ , 160satisfying highness property, 25, 258Mohrherr, 26, 256, 363Montalbán, 183movable marker, 240, 241, 250, 337Muchnik, Albert, 30Muchnik, Andrej A., 165, 199splitting technique, 295multiplicative domination, 266, 274n-c.e. set, 19n-random, see random, n-natural, 34natural property, 34Ng, 258, 316, 345, 348Nies, 61, 115, 170, 219, 223, 233, 294, 311,316, 323, 328, 336, 339, 341,344, 348, 352, 366, 376, 379,381non-cappable, 37, 41, 238non-cuppableΠ 1 1-random, 384wtt, 242ML, 359nonuniformity, 165, 215null class, 25, 70, 104, 134, 169, 365for randomness notion C, 322low for, 323O(log) boundedset, 289, 321string, 289Ω, 15, 52, 109, 149, 158Π 1 1-version, 376is wtt-complete, 115is Solovay complete, 114Ω A , 122, 125Ω M , 84, 318and output probability, 91as an operator, 121, 125, 144, 221, 248,304ω-c.e.function, 20, 315, 359, 360set, 19, 196O(f) error term, 75open set, 45, 47generated by set of strings, 48, 54in Baire space, 67operatorΣ 0 n , 233antimonotonic, 163c.e., 308closed under ⊕, 233monotonic, 163, 233optimal prefix-free machine, 84oracle incomparability, 239oracle machine, 121optimal, 121optimal prefix-free, 122

430 Indexprefix-free, 121uniformly optimal, 127oracle set, 1order function, 75, 80, 148convention on positive values, 313order type, 367ordinalg-closed, 375computable, 367notation, 4, 368Otto, the opponent, 36, 195, 201, 204,239, 309, 311, 353, 356outer measure, 68, 69countable subadditivity, 68definition, 68local, 69output probability, 91Π 1 1-version, 375output uniqueness, 239PA degree, 156computably dominated, 156low, 156minimal pair of, 159Parameter Theorem, 4partial computable, 3scan rule, 297total, 297universal function, 11, 147path of binary tree, 47Peano arithmetic, 5, 56, 157, 174completion of, 156fragments of, 192perfect class, 56, 61, 311–313, 316–318,337, 338Π 0 1 ,56and function trees, 317permitting, 38permitting methoddelayed, 42prompt, 243relative to ∅ ′ ,61Post, 10, 29, 31Post property, 34definable in E, 34structural, 34Post’s problem, 29, 34, 40degree-invariant solution, 35finite-injury solution, 32, 344injury-free solution, 34, 40, 145, 150,153, 189prefix-free code, 86, 87prefix-free set, 54prioritystronger (higher), 32weaker (lower), 32priority method, 35, 239with finite injury, 30, 32with infinite injury, 251with injury, 344priority ordering, 32dynamic, 344probability measure, 70promotionearly, 203, 205regular, 203, 205prompt permitting, 243prompt simplicity requirement, 41promptly simple degree, 243delay function for sets of, 243, 244, 246same as low cuppable, 244promptly simple set, 37, 40, 150, 151, 184,187, 191, 243as highness property, 238promptly simple wtt-degree, 243pseudojumpinversion, 249comparison with cupping, 250via c.e. set, 249via ML-random set, 253operator, 249r-complete, 9Raichev, 222random∆ 1 , 366, 378Π 1 1 , 366, 379universal test for, 366Π 1 1-weakly 2-, 378n-, 121, 160, 1622-, 82, 104, 106, 136, 304, 308, 311, 364and initial segment complexity, 137,308are of hyperimmune degree, 364lowness for, 232, 233not computably dominated, 137, 139,3093-, 136, 137computably, 28, 130, 259, 268, 289can be facile, 276closure under comp. permutations,294, 299higher analog, 366in high degree, 272, 283initial segment complexity, 272separation from ML-random, 271,283Demuth, 134, 141, 142are of hyperimmune degree, 143lowness for, 224Kolmogorov–Loveland (KL), 259, 294,298

Index 431lowness for, 329partial computably, 259, 289closure under comp. permutations,295definition, 273initial segment complexity, 272, 277lowness for, 328permutation, 295initial segment complexity, 296separation from PCR, 296polynomially, 104Schnorr, see Schnorr random, 259weakly, 128, 129, 135not ≥ tt ∅ ′ , 158relative to ∅ ′ , 136weakly 2-, 104, 106, 108, 117, 134–136,191, 311, 316, 365and lowness properties, 135can be not 2-f.p.f., 162lowness for, 175, 223not superhigh, 358weakly 3-, 237, 361weakly ∆ 1 1 , 382random variable, 73, 101independence, 73randomnessand being hard to describe, 103and exceptional properties, 102intuitive idea of, 102, 127mathematical notion of, 102, 103ranked formula, 366ranked set, 129real closed field, 222real number∆ 0 2 ,51computable, 51difference left-c.e., 51, 52, 222left-c.e., 51right-c.e., 51Recursion Theorem, 4, 90, 148–150, 153,157, 193Double, 12, 155, 246, 311for functionals, 11with parameters, 5reducibility, 2, 8, 224arithmetical, 224many-one, 8one-one, 9Solovay, 113, 114relativized, 303strong, 14, 224truth-table, 14Turing, 10van Lambalgen, 226, 364weak, 2, 224, 225, 227, 256, 340≤ JT , 340≤ LK , 225≤ LR , 225, 303weak truth-table, 14reduction function, 11, 148, 149, 151, 153reduction principlefor c.e. sets, 8for Π 1 1 classes, 369reduction procedure, 8≤ fin-h , 371≤ h , 370regular subset of model of I∆ 0 , 192relationarithmetical, 64, 366relativization, 2, 10request, 78, 86request operator, 308, 309requirement, 30initialization of, 33lowness, 32meeting a, 30simplicity, 31restraint of a strategy, 32Rettinger, 115reverse mathematics, 174right-c.e. set, 51Robinson guessing method, 193, 194, 246,345run of zeros, 99Russell’s paradox, 6Sacks, 35, 365, 370Sacks preservation strategy, 174Sacks Splitting Theorem, 36, 193, 241Schnorr, 105, 259, 270criticism by, 104, 127, 259null class, 322random, 127, 129, 174, 268and highness, 130characterization by c.m.m., 132higher analog, 366no universal test, 129not K-trivial, 219not computably random, 285relative to ∅ ′ , 143test, 104, 127, 129, 130, 132Π 1 1 , 379and martingales, 268definition, 129Theorem, 74, 105, 163, 267Π 1 1-version, 376for Schnorr randomness, 132relativized, 167trivial, 319, 321Scott, 157, 174Scott class, or set, 174

432 Indexsecond-order arithmetic, 234selection rule, 288advisor for, 290, 292firing, 290hiring, 290positions selected by, 288set selected by, 288success of, 288sequence number, 67, 368Shoenfield, 18Limit Lemma, 18Shore, 249, 253, 258simple set, 31, 37, 78, 81relative to an oracle, 248simplicity requirement, 32, 38, 41Simpson, 192, 226, 235, 256, 258, 340, 357Slaman, 113, 219Smullyan, 12Soare, 34, 44, 57Solomon, 223, 226, 235, 340Solomonoff, 75Solovay, 95, 177complete, 113equations, 95reducibility, 113test, 111, 130, 141, 185failing a, 111interval, 112, 153, 189, 190passing a, 111Soskova, 230Spector, 370, 371standard cost function, 186standard optimal plain machine, 76statistical properties of stringsbalance of zeros and ones, 100run of zeros, 99Stephan, 81, 149, 157, 166, 167, 170, 231,318, 321, 323, 328, 339stochasticcomputably (or Church), 288KL, 299partial computably, 289and simple martingales, 289not O(log) bounded, 289not weakly random, 289Stone duality, 46Stone space, 45string, 12string complexitymonotonic, 105, 108plain (C), see C (plain descriptivecomplexity)prefix-free (K), see K (prefix-freecomplexity)strong index, 7for clopen set, 49strongly d-incompressible, 98superhigh set, 235, 238, 256, 341c.e. minimal pair of, 258can be not LR-hard, 257superlow cuppable, 247, 361superlow set, 26, 58, 166, 176, 193, 207,218, 241, 336, 342, 343not base for ML-randomness, 170not low for K, 166strongly, 346with bound, 346supermartingale, 261approximation for, 265computable, 268computably enumerable, 169, 265multiplicatively optimal, 266universal, 266, 267hyperarithmetical, 379inequality, 261, 262generalized, 263leftmost non-ascending path of, 263,275, 276, 278leftmost non-ascending string of, 271Savings Lemma, 264, 299success of, 262undergraph of, 264tail of a set, 110Tanaka, 370Terwijn, 167, 316, 323test, 102, 103, 322Π 1 1-ML, 376Demuth, 141failing a, 127for randomness, 365generalized ML-, 134generalized Π 1 1-ML-, 378Kurtz, 128passing a, 127polynomial time, 104universal, 105, 129, 135, 322trace, 313bound for, 313c.e., 313bound for, 313computable, 316bound for, 316jump, 336traceable, 301, 312∆ 1 , 382Π 1 1 , 382c.e., 174, 302, 313, 322, 336and computably dominated, 316and double lowness notions, 323

Index 433characterization by machines, 319implies array computable, 315not of d.n.c. degree, 315order function irrelevant, 314weakly, 320computably, 164, 238, 301, 313, 316,322, 366can be generalized low, 338can fail to be generalized low, 338characterization by machines, 319perfect class of, 316same as facile on c.d. sets, 321same as Schnorr trivial on c.d. sets,321jump, see jump traceabletransfinite recursion in L A , 372treeΠ 0 1 , 53–56{0, 1} ∗ as, 12, 17binary, 47, 56computable, 53, 65dead branch of, 47extendable nodes on, 53, 56finitely branching, 68for Baire space, 67for closed class, 48, 57function, 317path of, 47without dead branches, 58without isolated paths, 56truth-table reduction, 14Turing, 3degree, 20, 159∆ 0 2 , 21, 159c.e., 21, 193minimal, 21, 338functional, 10, 11, 14, 47axiomatic approach, 239language for building, 238, 239oracle incomparability, 239output uniqueness, 239universal, 11, 12jump, 10, 16machine, 1, 3program, 3, 239universal, 4, 220reducibility, 10, 21u.a.e.d., see uniformly a.e. dominatingultrafilter, 71uniform measure, 17, 46of Π 0 n class, 71of Σ 0 n class, 71uniformity, 2, 4, 173, 198, 218uniformly a.e. dominating, 225, 234, 238,256uniformly c.e., 6universality, 2uppersemilattice, 16, 21use, 13clearing a computation of, 245lifting, 241, 244use principle, 3, 13, 22, 30, 47, 65, 126,387van Lambalgen Theorem, 9, 18, 122, 123,226, 305, 359, 393, 402fails for Schnorr randomness, 133for Π 1 1-randomness, 124, 381for n-randomness, 124Ville’s Theorem, 264weakly 1-generic, see 1-generic set, weaklyweakly 2-random, see random, weakly 2-weakly random, see random, weaklyweight condition, 86, 89weight of a set, 87well-order, 365, 367computable, 365form Π 1 1 class, 367no Borel on 2 N ,71worker’s method, 150Yu, 226, 268, 318, 379, 381, 382Zambella, 167, 316, 323zero-one law, 71, 357