Part 1: The Descriptive Set Theory of Finite Structures

Part 1: The Descriptive Set Theory of Finite Structures(Joint work with Sam Buss, Yijia Chen, Jörg Flum, Moritz Müller)Standard way to compare NP problems: Polytime reducibility.However, what if we want to compare isomorphism problems?ISO(C) = {(A, B) | A, B ∈ C and A ≃ B}Polytime reducibility gives a polytime f : C × C → D × D such that(A, B) ∈ ISO(C) ↔ f (A, B) ∈ ISO(D)But better would be a polytime f : C → D such that(A, B) ∈ ISO(C) ↔ (f (A), f (B)) ∈ ISO(D)This is called a strong isomorphism reduction

Strong Isomorphism Reductions in Complexity TheoryIn general: For binary relations E, F on classes C, D we dene astrong equivalence reduction to be a polytime f : C → D such thatA E B ↔ f (A) F f (B)This is the nitary analog of the similar theory for countablestructures, the Descriptive Set Theory of Equivalence Relations (bigliterature)Summary:1. Even if P = NP there is a rich structure to strong isomorphismreducibility.2. The ner questions about strong isomorphism reducibility areclosely connected with deep open problems in complexity theory.

Invariant Polytime ClassesFix a nite vocabulary (relation, function and constant symbols)Consider structures for this vocabulary with universe[n] = {1, ..., n} for some nite n. (We allow the empty structure.)Structures can be identied with 0, 1-strings of length a xedpolynomial in nThe number of structures of size n is exponential in nA class C of structures is invariant i it is closed underisomorphism: A ∈ C, A ≃ B → B ∈ C(∗) We consider polytime, invariant classes with structures ofarbitrarily large nite size

Invariant Polytime ClassesExamples:SET (Empty vocabulary)FIELD (Fields)ABELIAN (Abelian groups)CYCLIC (Cyclic groups)ORD (Linear orders)LOP (Linear orders with a distinguished point)BOOLE (Boolean algebras)GROUP (Groups)LOU (Linear orders with a unary relation)GRAPH (Irreexive binary relations)

Strong Isomorphism ReductionsIf C, D are polytime classes then we write: C ≤ iso D ithere is a polytime f : C → D such that A ≃ B ↔ f (A) ≃ f (B)We say that f is a strong isomorphism reduction from C to DA strong isomorphism degree is an equivalence class under theequivalence relation (C ≤ iso D and D ≤ iso C)C ≤ iso D is stronger than "ISO(C) is polytime reducible toISO(D)".Indeed, there are many distinct strong isomorphism degrees ofclasses C with ISO(C) in P

Strong Isomorphism ReductionsRecall that we assume that all of our classes are in P (and thereforeall isomorphism relations are in NP)PropositionC ≤ iso GRAPH for all C.Proof Idea: Arbitrary structures can be coded as graphs.Also not dicult:PropositionSET, FIELD, ABELIAN, CYCLIC, ORD and LOP all have the samestrong isomorphism degree. BOOLE is strong isomorphismreducible to them.But how do we show that a class C is NOT strong isomorphismreducible to another class D?

Potential ReducibilityThe strong isomorphism complexity of a class is related to thenumber of isomorphism types it has in dierent cardinalitiesLet C be a class.C(n) = those structures in C of size at most n#C(n) = number of isomorphism types of structures in C(n)Examples:#BOOLE(n) = [logn]#CYCLIC(n) = n, #SET(n) = #ORD(n) = n + 1#LOP(n) = ∑ ni=1i = n · (n + 1)/2#LOU(n) = ∑ ni=0 2i = 2 n+1 − 1#GROUP(n) is superpolynomial, subexponential(#GROUP(n) ≤ n O(log 2 n) )

Potential ReducibilityFor any vocabulary there is a polynomial p such that for any class Cin that vocabulary, #C(n) ≤ 2 p(n)DenitionC is potentially reducible to D (C ≤ pot D) i there is a somepolynomial p such that #C(n) ≤ #D(p(n)) for all n.LemmaIf C ≤ iso D then C ≤ pot D.Proof. Suppose that f is a strong isomorphism reduction. Thenthere is a polynomial p such that for A in C, f (A) has size at mostp(|A|). As f induces a 1-1 map on isomorphism types, it followsthat #C(n) ≤ #D(p(n)). □

Potential ReducibilityNow one can show that there is a rich structure of strongisomorphism degrees below LOP:TheoremThe partial order of the countable atomless Boolean algebra isembeddable into the potential reducibility degrees below LOP. Thisembedding is also into the strong isomorphism degrees below LOP.In particular, there are innite chains and antichains in the strongisomorphism degrees below LOP.SET, FIELD, ABELIAN, CYCLIC, ORD and LOP all have the samepotential reducibility degree, as they have the same strongisomorphism degree.

Potential ReducibilityAny class is potentially reducible to LOU, as LOU has themaximum number of isomorphism classes.Corollary(a)BOOLE < iso LOP < iso LOU ≤ iso GRAPH.(b) LOP < iso GROUP < iso GRAPH.(c) LOU ≰ iso GROUP.GROUP ≤ iso LOU? GRAPH ≤ iso LOU?

Potential Reducibility versus Strong IsomorphismReducibilityDoes potential reducibility ≤ pot imply strong isomorphismreducibility ≤ iso ? We situate this question between two openquestions in complexity theory:TheoremIf U2EXP ∩ co-U2EXP ≠ 2EXP then ≤ pot and ≤ iso are distinct.2EXP = DTIME(2 2nO(1) ).N2EXP = NTIME(2 2nO(1) ).U2EXP consists of those problems in N2EXP for which on everyinput there is exactly one accepting run.co-U2EXP is the complement of U2EXP.

Potential Reducibility versus Strong IsomorphismReducibilityTheoremIf ≤ pot and ≤ iso are distinct then P ≠ #P.For a nondeterministic polytime machine M, dene:f M (x) = the number of accepting runs of M on input x.Then the class #P consists of all such functions f M .

NP Equivalence RelationsHow do isomorphism relations compare to NP equivalence relationsas a whole?For equivalence relations E, F on classes C, D, a strong equivalencereduction is a polytime f : C → D such thatA E B ↔ f (A) F f (B)PropositionIf the Polytime Hierarchy does not collapse, then GRAPH is notmaximal among NP equivalence relations under strong equivalencereducibility.The reason is that the maximality of GRAPH would imply thatGRAPH is NP-complete in the usual sense, which is known to implythat the Polytime Hierarchy collapses.

Future WorkMore facts from Innite Descriptive Set Theory:1. Smooth → Borel selector for iso relations, not in general.2. Each closed subgroup of S ∞ is an automorphism group.3. Iso relations with classes of bounded Borel rank are Borel.4. Borel ER's with ctble classes are equivalent to iso relations.5. There is a maximal analytic ER, no maximal Borel ERDo these have analogues for nite structures?

Part 2: Computational Complexity in Set TheoryJoint work with Arnold Beckmann and Sam BussA central notion in Finite computation:Polytime functions on nite stringsHow can we generalise this notion to arbitrary sets?When is a function F : V → V computable in polynomial time?Consider some standard models for polynomial-time computation:

Polynomial-Time Set Recursion1. Turing machinesDicult to write an arbitrary set on a tape.2. Fixed point logicEven for nite structures, this works well only if there is an ordering.Moreover, on innite ordered structures, it is too powerful (it goesbeyond the hyperarithmetic).3. SchemesThis works, by generalising an idea of Bellantoni-Cook!

Bellantoni-Cook RecursionIdea behind Bellantoni-Cook recursion:Dene functionsf (⃗x/⃗y)where ⃗x,⃗y are nite sequences of nite (binary) strings and thevalues of f are nite stringsThe components of ⃗x are the Normal Inputs and those of ⃗y theSafe InputsWhen performing primitive recursions, the previous value is placedon the Safe side:f (0,⃗x/⃗y) = g(⃗x/⃗y)f (z ∗ i,⃗x/⃗y) = h i (z,⃗x/⃗y, f (z,⃗x/⃗y)), i = 0 or 1

Bellantoni-Cook RecursionWhen composing, one is careful not to allow safe inputs to becopied onto the normal side:f (⃗x/⃗y) = h(k(⃗x/−)/l(⃗x/⃗y))Net eect: the depth of recursions performed are bounded not bythe values obtained but by the sizes of the inputsOn normal inputs one gets exactly the polynomial-time computablefunctionsOn safe inputs string-length is increased by only a constant amount

Bellantoni-Cook RecursionA few examples (illustrated with numbers, not strings):The successor function S(x/y, z) = z + 1 is in the classAddition A(x/y) = x + y is in the class, by a primitive recursion:A(0/y) = yA(x + 1/y) = S(x/y, A(x/y)) = A(x/y) + 1A ∗ (x, y/z) = A(y/z) = y + z is also in the classThen multiplication M(x, y/−) = x × y is in the class, by a secondprimitive recursion on x:M(0, y/−) = 0M(x + 1, y/−) = A ∗ (x, y/M(x, y/−)) = y + M(x, y/−)

Bellantoni-Cook RecursionBUT exponentiation is not in the class!To run the previous argument for exponentiation one would needM ∗ (x, y/z) = M(y/z) = y × zin the class; but we only haveM(y, z/−) (multiplication of Normal Inputs)and no functionM(y/z) = y × z,which has z as a Safe Input.

The Safe-Recursive Set FunctionsWe adapt the Bellantoni-Cook idea to set theory by combining theGandy-Jensen rudimentary set functions with the set-theoreticanalogue of Bellantoni-Cook safe recursionBasic FunctionsΠ m,ni(x 1 , . . . , x m /x m+1 , . . . , x m+n ) = x i (1 ≤ i ≤ m + n)Pair(−/a, b) = {a, b}Di(−/a, b) = a \ bRudimentary Union Schemef (⃗x/⃗a, y) = ∪{g(⃗x/⃗a, z) | z ∈ y}Safe Recursionf (y,⃗x/⃗a) = h(y,⃗x/⃗a, {f (z,⃗x/⃗a) | z ∈ y})Safe Compositionf (⃗x/⃗a) = h(⃗r(⃗x/−)/⃗s(⃗x,⃗a))

The Safe-Recursive Set FunctionsSome examples of SR set functions:S(−/a) = a ∪ {a} is SRS ∗ (a/b, c) = b ∪ c is SR⊕(a/b) = {⊕(c/b) | c ∈ a} ∪ b is SR:⊕(a/b) = S ∗ (a/b, {⊕(c/b) | c ∈ a})For ordinals α, β, ⊕(α/β) = β + α

The Safe-Recursive Set FunctionsIn analogy to getting multiplication from addition throughsafe-recursion, we have:⊗(a, b/−) = ⊕(b/{⊗(c, b) | c ∈ a}) is SRFor ordinals α, β, ⊗(α, β/−) = β × αBut ordinal exponentiation is not SR:PropositionIf f (⃗x/⃗y) is a safe-recursive set function then there is a polynomialfunction p f on ordinals such that:rank(f (⃗x/⃗y)) ≤ max(rank(⃗y)) + p f (rank(⃗x))

The Safe-Recursive Set FunctionsHow powerful are the SR (safe-recursive) functions?Following Jensen, dene:SR-closure(A) = least SR-closed B ⊇ AFor transitive T , SR(T ) = SR-closure(T ∪ {T })We have:TheoremFor transitive T , SR(T ) = L T rank(T ) ω , where L T is the L-hierarchyrelativised to T .

Characterisation of SRSF FunctionsSimilarly we have a characterisation of SR functions in terms ofdenability. For any ⃗x let TC(⃗x) be the transitive closure of ⃗x.The function ⃗x ↦→ TC(⃗x) is SR. Also dene:SR(⃗x) = SR(TC(⃗x)) = L TC(⃗x)rank(⃗x) ωSR n (⃗x) = L TC(⃗x)rank(⃗x)for nite nnTheoremSuppose that f (⃗x/−) is SR. Then for some Σ 1 formula ϕ and somenite n we have:f (⃗x, −) = y i SR n (⃗x) ϕ(⃗x, y).Conversely, any function so dened is SR (on innite inputs).

The SR HierarchyAnalog of Jensen's J-hierarchy:SR 1 = HF, the collection of hereditarily nite setsSR α+1 = SR(SR α ) for α > 0SR λ = ⋃ α

Safe-Recursion on Restricted InputsBinary strings of length ωIf ⃗x is a nite sequence of binary ω-strings then of course rank(⃗x) isless than ω + ω so we simply have SR(⃗x) = L ω ω[⃗x].Thus the SR functions on ω-strings look like:where ϕ is Σ 1 and n is nite.f (⃗x, −) = y i L ω n[⃗x] ϕ(⃗x)These functions coincide with those computable by an innite-timeTuring machine in time ω n for some nite n, and were consideredby Hamkins, Schindler, Welch and others.

Safe-Recursion on Restricted InputsFinite stringsThere are many ways to code nite strings as setsA natural coding:#(i ∗ s) = the ordered pair (i, #(s)) = {{i}, {i, #(s)}}Theorem(Beckmann-Buss) With the above coding, the SR functions onnite strings are exactly those in the class ATIME(EXP, POLY) offunctions which are computable by an alternating Turing machinerunning in exponential time with polynomially-many alternations.Interestingly, the same class arises as the complexity of therst-order theory of the reals with order and addition

Some Recent Work of Tosi AraiArai weakened our schemes for SR functions, obtaining his PC(predicatively computable) functions. Recall that we used:Rudimentary Union Scheme:f (⃗x/⃗a, y) = ∪{g(⃗x/⃗a, z) | z ∈ y}Arai weakens this to:f (⃗x, y/⃗a, ) = ∪{g(⃗x, z/⃗a) | z ∈ y}Theorem(Arai) The PC functions on nite strings are exactly the PTIMEfunctions.On innite inputs, SR = PC

Future Work1. Another strategy to dene complexity classes in Set Theory:Denability in Weak Set Theories.There is work of Arai, Buss and others on this for PTIME.2. Further machine models for complexity classes in set theory(wellordered inputs).3. Descriptive complexity in set theory: logics that capturecomplexity classes?