NiesBookWithErrata

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.