TTHE COMPLEXITY OF SATISFACTION PROBLEMSIN REVERSE MATHEMATICS
LUDOVIC PATEYAbstract. Satisfiability problems play a central role in computer science and engineeringas a general framework for studying the complexity of various problems. Schaefer provedin 1978 that truth satisfaction of propositional formulas given a language of relations iseither NP-complete or tractable. We classify the corresponding satisfying assignment con-struction problems in the framework of reverse mathematics and show that the principlesare either provable over
RCA or equivalent to WKL . We formulate also a Ramseyanversion of the problems and state a different dichotomy theorem. However, the differentclasses arising from this classification are not known to be distinct.
1. IntroductionA common way to solve a constrained problem in industry consists in reducing it toa satisfaction problem over propositional logic and using a SAT solver. The generalityof the framework and its multiple applications make it a natural subject of interest forthe scientific community and constraint satisfaction problems remains an active field ofresearch.In 1978, Schaefer [10] gave a great insight in the understanding of the complexity of sat-isfiability problems by studying a parameterized class of problems and showing they admita dichotomy between NP-completeness and tractability. Many other dichotomy theoremshave been proven since, about refinements to AC reductions [1], variants about counting,optimization, 3-valued domains and many others [4, 7, 3]. The existence of dichotomiesfor n -valued domains with n > WKL ) which allows formalization of many compactness arguments and the solution tomany satisfiability problems. We believe that studying constraint satisfaction problems(CSP) within this framework can lead to insights in both fields: in reverse mathematics, wecan exploit the generality of constraint satisfaction problems to compare existing principlesby reducing them to satisfaction problems. In CSP, reverse mathematics can yield a betterunderstanding of the computational strength of satisfiability problems for particular classesof formulas. In particular we answer the question of Marek & Remmel [8] whether thereare dichotomy theorems for infinite recursive versions of constraint satisfaction problems. Definition 1.1.
Let (cid:66) = { F , T } be the set of Booleans. An (infinite) set of Boolean formu-las C is finitely satisfiable if every conjunction of a finite set of formulas in C is satisfiable. Date : November 5, 2018. This paper is an extended version of a conference paper of the same name published in CiE 2014. a r X i v : . [ m a t h . L O ] J a n LUDOVIC PATEY
SAT is the statement “For every finitely satisfiable set C of Boolean formulas over an in-finite set of variables V , there exists an infinite assignment ν : V → (cid:66) satisfying C .” Thepair ( V , C ) forms an instance of SAT .The base axiom system for reverse mathematics is called
RCA , standing for RecursiveComprehension Axiom. It consists of basic Peano axioms together with a comprehensionscheme restricted to ∆ formulas and an the induction restricted to Σ formulas. Theorem 1.2 (Simpson [11]) . RCA (cid:96) WKL ↔ SAT
Proof.
WKL → SAT : Let C be a finitely satisfiable set of formulas over a set of variables V . Let (cid:104) x i | i ∈ (cid:78) (cid:105) enumerate V . For each σ ∈ < (cid:78) , identify σ with the truth assignment ν σ on { x i | i < | σ |} given by ( ∀ i < | σ | )( ν σ ( x i ) = T ↔ σ ( i ) = ) . Let T ⊆ < (cid:78) be thetree T = { σ ∈ < (cid:78) | ¬ ( ∃ θ ∈ C (cid:150) | σ | )( ν σ ( θ ) = F ) } , where C (cid:150) | σ | is the set of formulasin C coded by numbers less than | σ | , and ν σ ( θ ) is the truth value assigned to θ by ν σ (note that ν σ ( θ ) is undefined if θ contains a variable x m for m ≥ | σ | ). T exists by ∆ comprehension and is downward closed. T is infinite because for any n ∈ (cid:78) , any satisfyingtruth assignment of C (cid:150) n restricted to { x i | i < n } yields a string in T of length n . By WKL let P ⊆ (cid:78) be a path through T . We show that every finite C ⊆ C can be satisfied by thetruth assignment ν : V → (cid:66) defined for all x i ∈ V by ν ( x i ) = T ↔ i ∈ P . Given C ⊆ C finite, let n be such that C ⊆ C (cid:150) n and such that Var ( C ) ⊆ { x i | i < n } . Now let σ ≺ P be such that | σ | = n . Then ( ∀ θ ∈ C )( ν σ ( θ ) = T ) because ν σ ( θ ) is defined for all θ ∈ C and ν σ ( θ ) (cid:54) = F for all θ ∈ C . Thus ν σ satisfies C . SAT → WKL : Let V = { x i | i ∈ (cid:78) } be a set of distinct variables, and to each string σ ∈ < (cid:78) , associate the formula θ σ ≡ (cid:86) i < | σ | (cid:96) i , where (cid:96) i ≡ x i if σ ( i ) = (cid:96) i ≡ ¬ x i if σ ( i ) =
0. Let T ⊆ < (cid:78) be an infinite tree, and, for each n ∈ (cid:78) , let T n = { σ ∈ T | | σ | = n } .Let C = { (cid:87) σ ∈ T n θ σ | n ∈ (cid:78) } . We show that every finite C ⊆ C is satisfiable. Given C ⊆ C finite, let n be maximum such that (cid:87) σ ∈ T n θ σ ∈ C and, as T is infinite, let τ ∈ T havelength n . Then θ τ → φ for every φ ∈ C because if φ = (cid:87) σ ∈ T m θ σ ∈ C , then m ≤ n , θ τ (cid:150) m is a disjunct of φ , and θ τ → θ τ (cid:150) m . Therefore C is satisfiable by the truth assignment thatsatisfies θ τ . By SAT there exists a valid assignment ν for C . Let P be { i ∈ (cid:78) : ν ( x i ) = T } .We show that P is a path through T . Given n ∈ (cid:78) , let σ ≺ P be such that | σ | = n . Bydefinition of P , ( ∀ i < n )( σ ( i ) = ↔ ν ( x i ) = T ) , so ν ( θ σ ) = T , from which it follows that σ ∈ T n . (cid:3) RWKL , a weakening of
WKL , has been recently introduced by Flood in [5]. Givenan infinite binary tree, the principle does not assert the existence of a path, but rather ofan infinite subset of a path through the tree. Initially called RKL , it has been renamedto
RWKL in [2] to give a consistent R prefix to Ramseyan principles. This principle hasbeen shown to be strictly weaker than SRT and WKL by Flood, and strictly strongerthan DNR by Bienvenu & al. in [2]. By analogy with
RWKL , we formulate Ramsey-typeversions of satisfiability problems.
Definition 1.3.
Let C be a sequence of Boolean formulas over an infinite set of variables V .A set H is homogeneous for C if there is a truth value c ∈ (cid:66) such that every conjunction of afinite set of formulas in C is satisfiable by a truth assignment ν such that ( ∀ a ∈ H )( ν ( a ) = c ) . Definition 1.4.
LRSAT is the statement “Let C be a finitely satisfiable set of Booleanformulas over an infinite set of variables V . For every infinite set L ⊆ V there exists aninfinite set H ⊆ L homogeneous for C .” The corresponding instance of LRSAT is the tuple
HE COMPLEXITY OF SATISFACTION PROBLEMS IN REVERSE MATHEMATICS 3 ( V , C , L ) . RSAT is obtained by restricting
LRSAT to L = V . Then an instance of RSAT isan ordered pair ( V , C ) .The equivalence between WKL and SAT over
RCA extends to their Ramseyan ver-sion. The proof is relatively easy and directly adaptable from proof of Theorem 1.2. Theorem 1.5 (Bienvenu & al. [2]) . RCA (cid:96) RWKL ↔ RSAT ↔ LRSAT
Definitions and notations.
Some classes of Boolean formulas – bijunctive, affine,horn, ... – have been extensively studied in complexity theory, leading to the well-knowndichotomy theorem due to Schaefer. We give a precise definition of those classes in orderto state our dichotomy theorems.
Definition 1.6. A literal is either a Boolean variable (positive literal), or its negation (neg-ative literal). A clause is a disjunction of literals. A clause is horn if it has at most onepositive literal, co-horn if it has at most one negative literal and bijunctive if it has at most 2literals. If we number Boolean variables, we can associate to each Boolean formula ϕ withBoolean variables x , . . . , x n a relation [ ϕ ] ⊆ (cid:66) n such that (cid:126) a ∈ [ ϕ ] iff ϕ ( (cid:126) a ) holds. If S is a setof relations, an S-formula over a set of variables V is a formula of the form R ( y , . . . , y n ) for some R ∈ S and y , . . . , y n ∈ V . Example 1.7.
Let S = {→} . ( x → y ) is an S-formula but ( x → ¬ y ) is not. Neither is ( x → y ) ∧ ( y → z ) . The formula ( x → y ) is equivalent to the horn clause ( ¬ x ∨ y ) where theliterals are ¬ x and y. Definition 1.8.
A formula ϕ is i-valid for i ∈ (cid:66) if ϕ ( i , . . . , i ) holds. It is horn (resp. co-horn , bijunctive ) if it is a conjunction of horn (resp. co-horn, bijunctive) clauses. A formulais affine if it is a conjunction of formulas of the form x ⊕ · · · ⊕ x n = i for i ∈ (cid:66) where ⊕ isthe exclusive or.A relation R ⊆ { , } n is bijunctive (resp. horn , co-horn , affine , i-valid ) if there isbijunctive (resp. horn, co-horn, affine, i -valid) formula ϕ such that R = [ ϕ ] . A relation R is i-default for i ∈ (cid:66) if for every (cid:126) r ∈ R and every j < | (cid:126) r | , the vector (cid:126) s defined by (cid:126) s ( j ) = i and (cid:126) s ( k ) = (cid:126) r ( k ) otherwise, is also in R . In particular every i -default relation is i -valid. Wedenote by ISAT ( S ) the class of satisfiable conjunctions of S -formulas.1.2. Dichotomies.
We first state the celebrated dichotomy theorem from Schaefer. Inter-estingly, the corresponding dichotomies in reverse mathematics are not based on the sameclasses of relations as the ones from Schaefer.
Theorem 1.9 (Schaefer’s dichotomy [10]) . Let S be a finite set of Boolean relations. If Ssatisfies one of the conditions ( a ) − ( f ) below, then ISAT ( S ) is polynomial-time decidable.Otherwise, ISAT ( S ) is log-complete in NP. (a) Every relation in S is F -valid. (b) Every relation in S is T -valid. (c) Every relation in S is horn (d)
Every relation in S is co-horn (e)
Every relation in S is affine. (f)
Every relation in S is bijunctive.
In the remainder of this paper, S will be a – possibly infinite – class of Boolean relations.Note that there is no effectiveness requirement on S . Definition 1.10.
SAT ( S ) is the statement “For every finitely satisfiable set C of S -formulasover an infinite set of variables V , there exists an infinite assignment ν : V → (cid:66) satisfying C ”.We will prove the following theorem based on Schaefer’s theorem. LUDOVIC PATEY
Theorem 1.11.
If S satisfies one of the conditions ( a ) − ( d ) below, then SAT ( S ) is provableover RCA . Otherwise SAT ( S ) is equivalent to WKL over RCA . (a) Every relation in S is F -valid. (b) Every relation in S is T -valid. (c) If R ∈ S is not F -default then R = [ x ] . (d) If R ∈ S is not T -default then R = [ ¬ x ] . SAT ( S ) principles are not fully satisfactory as these are not robust notions: if we define SAT ( S ) in terms of satisfiable sets of conjunctions of S -formulas, this yields a differentdichotomy theorems. In particular, RCA (cid:96) SAT ([ x ] , [ ¬ y ]) whereas RCA (cid:96) SAT ([ x ∧¬ y ]) ↔ WKL . Ramseyan versions of satisfaction problems have better properties. Definition 1.12.
RSAT ( S ) is the statement “For every finitely satisfiable set C of S -formulasover an infinite set of variables V , there exists an infinite set H ⊆ V homogeneous for C ”.Usual reductions between satisfiability problems involve fresh variable introductions.This is why it is natural to define a localized version of those principles, i.e. where thehomogeneous set has to lie within a pre-specified set. Definition 1.13.
LRSAT ( S ) is the statement “For every finitely satisfiable set C of S -formulas over an infinite set of variables V and every infinite set X ⊆ V , there exists aninfinite set H ⊆ X homogeneous for C ”.In particular, we define LRSAT ( F -valid ) (resp. LRSAT ( T -valid ) , LRSAT ( Horn ) , LRSAT ( CoHorn ) , LRSAT ( Bijunctive ) or LRSAT ( Affine ) ) to denote LRSAT ( S ) where S is the set of all F -valid (resp. T -valid, horn, co-horn, bijunctive or affine) relations. We will prove the fol-lowing dichotomy theorem. Theorem 1.14.
Either
RCA (cid:96) LRSAT ( S ) or LRSAT ( S ) is equivalent to one of the fol-lowing principles over RCA : LRSAT LRSAT ([ x (cid:54) = y ]) LRSAT ( Affine ) LRSAT ( Bijunctive ) As we will see in Theorem 4.1, each of those principles are equivalent to their nonlocalized version. As well,
LRSAT ([ x (cid:54) = y ]) coincides with an already existing princi-ple about bipartite graphs [2] called RCOLOR and LRSAT is equivalent to
RWKL over
RCA . Hence LRSAT ( S ) is either provable over RCA , or equivalent to one of RCOLOR , RSAT ( Affine ) , RSAT ( Bijunctive ) and RWKL over
RCA .2. Schaefer’s dichotomy theorem Definition 2.1.
Let S be a set of Boolean relations and V be a set of variables. Let ϕ be an S -formula over V . We denote by Var ( ϕ ) the set of variables occurring in ϕ . An assignment for ϕ is a function ν : Var ( ϕ ) → { T , F } . An assignment can be naturally extended to afunction over formulas by the natural interpretation rules for logical connectives. Thenan assignment ν satisfies ϕ if ν ( ϕ ) = T . The set of assignments satisfying ϕ is written Assign ( ϕ ) . Variable substitution is defined in the usual way and is written ϕ [ y / x ] , meaningthat all occurrences of x in ϕ are replaced by y . We will also write ϕ [ y / X ] where X is aset of variables to denote substitution of all occurrences of a variable of X in ϕ by y . A constant is either F or T . Definition 2.2.
Let S be a set of Boolean relations. The class of existentially quantified S -formulas – i.e. of the form ( ∃ (cid:126) x ) R [ (cid:126) x ,(cid:126) y ] with R ∈ S – is denoted by Gen ∗ NC ( S ) . We also define HE COMPLEXITY OF SATISFACTION PROBLEMS IN REVERSE MATHEMATICS 5
Rep ∗ NC ( S ) = (cid:8) [ R ] : R ∈ Gen ∗ NC ( S ) (cid:9) , ie. the relations represented by existentially quantified S -formula. By abuse of notation, we may use Rep ∗ NC ( R ) when R is a relation to denote Rep ∗ NC ( { R } ) .Given some set of Boolean realtions S , the set Rep ∗ NC ( S ) might not exist over RCA .However, we shall not use it as a set, but within relations of the form [ R ] ∈ Rep ∗ NC ( S ) ,which can be seen as an abbreviation for an arithmetical statement using only R and S asparameters. Also note that the definition of Gen ∗ NC ( S ) and Rep ∗ NC ( S ) differ from Schae-fer’s definition of Gen NC ( S ) and Rep NC ( S ) in that the latter are closed under conjunction.Therefore, reusing Schaefer’s lemmas must be done with some precautions, checking thathis proofs do not use conjunction. This is the case of the following lemma: Lemma 2.3 (Schaefer in [10, 4.3]) . RCA proves that at least one of the following holds: (a) Every relation in S is F -valid. (b) Every relation in S is T -valid. (c) [ x ] and [ ¬ x ] are contained in Rep ∗ NC ( S ) . (d) [ x (cid:54) = y ] ∈ Rep ∗ NC ( S ) . One easily sees that if every relation in S is F -valid (resp. T -valid) then RCA (cid:96) SAT ( S ) as the assignment always equal to F (resp. T ) is a valid assignment and is computable. Wewill now see that problems parameterized by relations either F -default or [ x ] (resp. T -default or [ ¬ x ] ) are also solvable over RCA .The proof of the following lemma justifies the name F -default (resp. T -default) byusing a strategy for solving an instance ( V , C ) of SAT ( S ) consists in defining an assignmentwhich given a variable x will give it the default value F (resp. T ) unless it finds the clause ( x ) ∈ C , where ( x ) is the clause with x as the unique literal. Lemma 2.4.
RCA proves that if the only relation in S which is not F -default is [ x ] or theonly relation which is not T -default is [ ¬ x ] then SAT ( S ) holds.Proof. Assume [ x ] is the only relation of S which is not F -default. Given an instance ( V , C ) of SAT ( S ) , define the assignment ν : V → { F , T } as follows: ν ( x ) = T iff ( x ) ∈ C . Theassignment ν exists by ∆ -comprehension. Suppose for the sake of contradiction that thereis a formula ϕ ∈ C such that ν ( ϕ ) = F . If ϕ = ( x ) for some variable x , then by definitionof ν , ν ( x ) = T hence ν ( ϕ ) = T . So suppose ϕ = R ( x , . . . , x n ) for some n ∈ (cid:78) , where R isa F -default relation. Let I = { i < n : ( x i ) ∈ C } . As C is finitely satisfiable, so is ϕ (cid:86) i ∈ I ( x i ) .Let µ be an assignment satisfying ϕ (cid:86) i ∈ I ( x i ) . In particular µ ( x i ) = T for each i ∈ I and µ satisfies ϕ . By F -defaultness of R , the vector (cid:126) r defined by (cid:126) r ( i ) = T for i ∈ I and (cid:126) r ( i ) = F otherwise is in R . But by definition of ν , ν ( x i ) = T iff i ∈ I , hence (cid:126) r = ν ( x ) . . . ν ( x n ) ∈ R and ν ( ϕ ) = T . So ν is a valid assignment and the proof can easily be formalized over RCA . Hence RCA (cid:96) SAT ( S ) . The same reasoning holds whenever the only relation of S which is not T -default is [ ¬ x ] . (cid:3) The following lemma simply reflects the fact that
SAT ([ x (cid:54) = y ]) can be seen as a refor-mulation of COLOR which is equivalent to WKL over RCA [6]. Lemma 2.5.
RCA proves that if [ x (cid:54) = y ] ∈ Rep ∗ NC ( S ) , then WKL ↔ SAT ( S ) .Proof. As RCA (cid:96) WKL → SAT , it suffices to prove that
RCA (cid:96) SAT ( S ) → WKL to obtain desired equivalence. Fix an infinite, locally bipartite, computable graph G =( V , E ) and let θ ∈ Gen ∗ NC ( S ) be such that [ θ ] = [ x (cid:54) = y ] . By definition, θ = ( ∃ (cid:126) z ) R ( x , y ,(cid:126) z ) for some R ∈ S . Take an infinite set W of fresh variables disjoint from V and define LUDOVIC PATEY an instance ( V ∪ W , C ) of SAT ( S ) by taking C = { R ( x , y ,(cid:126) z ) : x < y ∧ { x , y } ∈ E ∧ ( (cid:126) z ∈ W has not yet been used ) } . The set C is finitely satisfiable because G is locally bipartite.Let ν : V ∪ W → (cid:66) be an assignment satisfying C and let P = { x ∈ V : ν ( x ) = F } and P = { x ∈ V : ν ( x ) = T } . We claim that P , P is a bipartition of G . Suppose for the sakeof absurd that the exists an i < x < y ∈ P i such that { x , y } ∈ E . Thenthere exists fresh variables (cid:126) z ∈ W such that R ( x , y ,(cid:126) z ) ∈ C . In particular, ν satisfies R ( x , y ,(cid:126) z ) ,hence the formula θ ( x , y ) so ν ( x ) (cid:54) = ν ( y ) , contradicting the assumption that x , y ∈ P i . Hence RCA (cid:96) SAT ( S ) → COLOR . (cid:3) Theorem 1.11 is proven by a case analysis using Lemma 2.3, by noticing that when weare not in cases already handled by Lemma 2.4 and Lemma 2.5, we can find n -ary formulasencoding [ x ] and [ ¬ x ] with n ≥
2. Thus diagonalizing against some values becomes a Σ event. Proof of Theorem 1.11.
We reason by case analysis. Cases where every relation in S is F -valid (resp. T -valid) are trivial. Cases where the only relation in S which is not F -default(resp. T -default) is [ x ] (resp. [ ¬ x ] ), and whenever [ x (cid:54) = y ] ∈ Rep ∗ NC ( S ) are already handledby Lemma 2.4 and Lemma 2.5.In the remaining case, by Lemma 2.3, [ x ] and [ ¬ x ] ∈ Rep ∗ NC ( S ) . First we show that itsuffices to find two relations R , R ∈ S together with two formulas ψ , ψ ∈ Gen ∗ NC ( S ) such that x (cid:54)∈ Var ( ψ ) ∪ Var ( ψ ) and the following holds [( ∃ (cid:126) z ) R ( x ,(cid:126) z ) ∧ ψ ( (cid:126) z )] = [ x ] and [( ∃ (cid:126) z ) R ( x ,(cid:126) z ) ∧ ψ ( (cid:126) z )] = [ ¬ x ] to prove the existence of a path through an infinite binary tree. Then, we show that suchrelations exist. Note that the difference with the assumption that [ x ] and [ ¬ x ] ∈ Rep ∗ NC ( S ) is that the relations R and R have arity greater than 1, hence the relations R and R maybe added arbitrarily late to the set of formulas with fresh variables. Fix two disjoint setsof variables: V = (cid:8) x σ : σ ∈ < (cid:78) (cid:9) and W = { y , . . . } . Let T ⊆ < (cid:78) be an infinite tree. Wedefine an instance ( V ∪ W , C ) of SAT ( S ) such that every satisfying assignment computesan infinite path through T . We define the set C by stages C = /0 ⊆ C ⊆ . . . Assume thatat stage s , the existence of each S -formula over variables (cid:8) x σ , y i : σ ∈ i , i < s (cid:9) has beendecided. Given some string σ ∈ < (cid:78) , we denote by T [ σ ] s the set of strings τ ∈ T of length s such that τ (cid:23) σ .1. If T [ σ (cid:95) ] s is empty but not T [ σ (cid:95) ] s for some σ ∈ < s , then add R ( x σ ,(cid:126) y ) and ψ ( (cid:126) y ) to C s for some fresh variables (cid:126) y ∈ W (cid:114) { y i : i < s } .2. If T [ σ (cid:95) ] s is empty but not T [ σ (cid:95) ] s for some σ ∈ < s , then add R ( x σ ,(cid:126) y ) and ψ ( (cid:126) y ) to C s for some fresh variables (cid:126) y ∈ W (cid:114) { y i : i < s } .This finishes the construction. We have ensured that for any satisfying assignment ν for C and any string σ ∈ T inducing an infinite subtree, σ (cid:95) ν ( x σ ) also induces an infinite subtree.Define the strictly increasing sequence of strings σ = ε ≺ σ ≺ . . . by σ s + = σ (cid:95) s ν ( x σ s ) .The set P = (cid:83) s σ s is an infinite path through T . This proof can easily be formalized in RCA . Hence RCA (cid:96) SAT ( S ) → WKL .We now find the relations R , R ∈ S and define the formulas ψ and ψ ∈ Gen ∗ NC ( S ) .Suppose there exists a relation R ∈ S which is not F -valid and is different from [ x ] . De-fine the formula ϕ = R ( x , . . . ) and let ν ∈ Assign ( ϕ ) be such that ∀ U ⊆ ν − ( { T } ) , theassignment which coincides with ν except for U does not satisfy ϕ . Because R is not F -valid, ν − ( { T } ) (cid:54) = /0. Suppose w.l.o.g. that x ∈ ν − ( { T } ) . Then the following holds HE COMPLEXITY OF SATISFACTION PROBLEMS IN REVERSE MATHEMATICS 7 for some constants i , i , . . . [ ϕ (cid:94) x ∈ ν − ( { T } ) (cid:114) { x } ( x ) (cid:94) x ∈ ν − ( { F } ) ( ¬ x )] = [ x ∧ ( x = i ) ∧ ( x = i ) . . . ] Suppose now the only non F -valid relation in S is [ x ] , in which case there is a F -validrelation R ∈ S which is not F -default. Thus there is a non-empty finite set I ⊂ ω anda vector (cid:126) r ∈ R such that (cid:126) r ( i ) = T for each i ∈ I , but for every such (cid:126) r ∈ R , ∃ j (cid:54)∈ I suchthat (cid:126) r ( j ) = T . Consider a minimal (in pointwise natural order) such (cid:126) r . Define the formula ϕ = R ( x , . . . ) . Suppose without loss of generality that 1 (cid:54)∈ I and (cid:126) r ( ) = T . Then thefollowing holds for some constants i , i , . . . [ ϕ (cid:94) i ∈ I ( x i ) (cid:94) (cid:126) r ( i )= ( ¬ x i )] = [ x ∧ ( x = i ) ∧ ( x = i ) . . . ] Similarly we can take any relation R of S which is not T -valid and is different from [ ¬ x ] or which is T -valid but not T -default to construct an S -formula ψ ∈ Gen ∗ NC ( S ) with y (cid:54)∈ Var ( ψ ) and constants i , i , . . . such that [ R ( x , . . . ) ∧ ψ ] = [ ¬ x ∧ ( x = i ) ∧ ( x = i ) . . . ] . This finishes the proof. (cid:3)
3. Ramsey-type Schaefer’s dichotomy theoremThe proof of Theorem 1.14 can be split into four steps, each of them being dichotomiesthemselves. The first one, Theorem 3.4, states the existence of a gap between provabil-ity in
RCA and implying LRSAT ([ x (cid:54) = y ]) over RCA . Then we focus successively ontwo classes of boolean formulas: bijunctive formulas (Theorem 3.12) and affine formu-las (Theorem 3.16) whose corresponding principles happen to be either a consequence of LRSAT ([ x (cid:54) = y ]) or equivalent to the full class of bijunctive (resp. affine) formulas. Re-maining cases are handled by Theorem 3.17. We first state a trivial relation between asatisfaction principle and its Ramseyan version. Lemma 3.1.
RCA (cid:96) SAT ( S ) → LRSAT ( S ) Proof.
Let ( V , C , L ) be an instance of LRSAT ( S ) . Let ν : V → (cid:66) be a satisfying assignmentfor C . Then either { x ∈ L : ν ( x ) = T } or { x ∈ L : ν ( x ) = F } is infinite, and both sets existby ∆ -comprehension. (cid:3) Definition 3.2.
Let S be a set of relations over Booleans. The class of existentiallyquantified S -formulas with constants and closed under conjunction – i.e. of the form ( ∃ (cid:126) x ) (cid:86) i < n R i [ (cid:126) x ,(cid:126) y , T , F ] with R i ∈ S – is denoted by Gen ( S ) . We also define Rep ( S ) = { [ R ] : R ∈ Gen ( S ) } , ie. the relations represented by existentially quantified S -formula withconstants and closed under conjunction. By abuse of notation, we may use Rep ( R ) when R is a relation to denote Rep ( { R } ) . We can also define similar relations without constants,denoted by Gen NC and Rep NC . Lemma 3.3.
RCA proves: If T is a sequence of Boolean relations such that [ x (cid:54) = y ] ∈ Rep NC ( T ) , and S is a sequence of relations in Rep NC ( T ) , then LRSAT ( T ) → LRSAT ( S ) .Proof. Let ( V , C , L ) be an instance of LRSAT ( S ) . Say V = { x , x , . . . } and C = { ϕ , ϕ , . . . } .Define an instance ( V ∪ F , D , L ) of LRSAT ( T ) where F = { y , y , . . . } is a set of freshvariables disjoint from V , and D is a set of formulas defined by stages as follows. Atstage 0, D = /0. In order to make D computable, we will ensure that after stage s , noformula over { x i , y i : i < s } will be added to D . At stage s , we want to add constraintsof ϕ s to D . Because S ⊆ Rep NC ( T ) and T is c.e., we can effectively find a formula LUDOVIC PATEY ψ ∈ Gen NC ( T ) equivalent to ϕ s and translate it into a finite set of formulas ψ ∗ as fol-lows: ( ∃ z . ψ ) ∗ (cid:39) ( ψ [ y / z ]) ∗ where y ∈ F is a fresh variable, ( ψ ∧ ψ ) ∗ (cid:39) ψ ∗ ∪ ψ ∗ , R ( x i , . . . , x i n ) ∗ (cid:39) { R ( y j , . . . , y j n ) , x i = y j , . . . , x i n = y j n } where y j k are fresh variables of F and x = y is a notation for the composition of ( ∃ z ) x (cid:54) = z ∧ z (cid:54) = y . Add ψ ∗ to D . It is easyto check that any solution to ( V ∪ F ∪ { c , c } , D , L ) is a solution to ( V , C , L ) . (cid:3) From provability to
LRSAT ([ x (cid:54) = y ]) . Our first dichotomy for Ramseyan principlesis between
RCA and LRSAT ([ x (cid:54) = y ]) . Theorem 3.4.
If S satisfies one of the conditions (a)-(d) below then
RCA (cid:96) LRSAT ( S ) .Otherwise RCA (cid:96) LRSAT ( S ) → LRSAT ([ x (cid:54) = y ]) . (a) Every relation in S is F -valid. (b) Every relation in S is T -valid. (c) Every relation in S is horn. (d)
Every relation in S is co-horn.
The proof of Theorem 3.4 follows Theorem 3.10.
Lemma 3.5 (Schaefer in [10, 3.2.1]) . RCA proves: If S contains some relation which isnot horn and some relation which is not co-horn, then [ x (cid:54) = y ] ∈ Rep ( S ) . Lemma 3.6.
RCA proves that at least one of the following holds: (a) Every relation in S is F -valid. (b) Every relation in S is T -valid. (c) Every relation in S is horn. (d)
Every relation in S is co-horn. (e) [ x (cid:54) = y ] ∈ Rep NC ( S ) .Proof. Assume none of cases (a), (b) and (e) holds. Then by Lemma 2.3, [ x ] and [ ¬ x ] arecontained in Rep NC ( S ) , hence Rep NC ( S ) = Rep ( S ) . So by Lemma 3.5, either every relationin S is horn, or every relation in S is co-horn. (cid:3) It is easy to see that
LRSAT ( F -valid ) and LRSAT ( T -valid ) both hold over RCA . Wewill now prove that so do LRSAT ( Horn ) and LRSAT ( CoHorn ) , but first we must introducethe powerful tool of closure under functions . Definition 3.7.
We say that a relation R ⊆ (cid:66) n is closed or invariant under an m -ary function f and that f is a polymorphism of R if for every m -tuple (cid:104) v , . . . , v m (cid:105) of vectors of R , (cid:126) f ( v , . . . , v m ) ∈ R where (cid:126) f is the coordinate-wise application of the function f .We denote the set of all polymorphisms of R by Pol ( R ) , and for a set Γ of Booleanrelations we define Pol ( Γ ) = { f : f ∈ Pol ( R ) for every R ∈ Γ } . Similarly for a set B ofBoolean functions, Inv ( B ) = { R : B ⊆ Pol ( R ) } is the set of invariants of B . One easily seesthat the projection functions are polymorphism of every Boolean relation R . In particular,the identity function is a polymorphism of R . As well, the composition of polymorphismsof R form again a polymorphism of R . So given a set of Boolean relations S , Pol ( S ) contains all projection functions and is closed under composition. The sets of functionssatisfying those closure properties have been studied in universal algebra under the nameof clones . We have seen that for every set of Boolean relations S , Pol ( S ) is a clone. Post [9]studied the lattice of clones of Boolean functions and proved that they admit a finite basis.The lattice structure of the Boolean clones has connections with the complexity of sat-ifiability problems. Indeed, if some clone A is a subset of another clone B , then Inv ( A ) ⊇ Inv ( B ) . But then trivially LRSAT ( Inv ( A )) → LRSAT ( Inv ( B )) . As well, we shall seethat as soon as [ x = y ] ∈ Rep NC ( S ) , the sets Rep NC ( S ) and Inv ( Pol ( S )) coincide. There-fore, assuming that the equality relation is representable in S , the study of the strengthof LRSAT ( S ) can be reduced to the study of the strength of LRSAT ( Inv ( A )) for everyclone in Post’s lattice. HE COMPLEXITY OF SATISFACTION PROBLEMS IN REVERSE MATHEMATICS 9
Definition 3.8.
The conjunction function conj : (cid:66) → (cid:66) is defined by conj ( a , b ) = a ∧ b ,the disjunction function disj : (cid:66) → (cid:66) by disj ( a , b ) = a ∨ b , the affine function aff : (cid:66) → (cid:66) by aff ( a , b , c ) = a ⊕ b ⊕ c = T and the majority function maj : (cid:66) → (cid:66) by maj ( a , b , c ) =( a ∧ b ) ∨ ( a ∧ c ) ∨ ( b ∧ c ) .The following theorem due to Schaefer characterizes relations in terms of closure undersome functions. The proof is relativizable and involves finite objects. Hence it can beeasily proven to hold over RCA . Theorem 3.9 (Schaefer [10]) . A relation is (1) horn iff it is closed under conjunction function (2) co-horn iff it is closed under disjunction function (3) affine iff it is closed under affine function (4) bijunctive iff it is closed under majority function
In other words, using Post’s lattice, a relation R is horn iff E ⊆ Pol ( R ) , co-horn iff V ⊆ Pol ( R ) , affine iff L ⊆ Pol ( R ) and bijunctive iff D ⊆ Pol ( R ) . In the case of horn andco-horn relations, we will use the closure of the valid assignments under the conjunctionand disjunction functions to prove that LRSAT ( Horn ) and LRSAT ( CoHorn ) both holdover RCA . Theorem 3.10.
If every relation in S is horn (resp. co-horn) then
RCA (cid:96) LRSAT ( S ) .Proof. We will prove it over
RCA for the horn case. The proof for co-horn relationsis similar. Let ( V , C , L ) be an instance of LRSAT ( Horn ) and F ⊆ L be the collection ofvariables x ∈ L such that there exists a finite set C fin ⊆ C for which every valid assignment ν satisfies ν ( x ) = T .Case 1: F is infinite. Because F is Σ , we can find an infinite ∆ subset H of F . The set H is homogeneous for C with color T .Case 2: F is finite. We take H = L (cid:114) F and claim that H is homogeneous for C withcolor F . If H is not homogeneous for C , then there exists a finite set C fin ⊆ C witnessingit. Let H fin = Var ( C fin ) ∩ H . By definition of not being homogeneous with color F , forevery assignment ν satisfying C fin , there exists a variable x ∈ H fin such that ν ( x ) = T .By definition of H , for each variable x ∈ H there exists a valid assignment ν x such that ν x ( x ) = F . By Theorem 3.9, the class valid assignments of a finite horn formula is closedunder conjunction. So ν = (cid:86) x ∈ H fin ν x is a valid assignment for C fin such that ν ( x ) = F foreach x ∈ H fin . Contradiction. (cid:3) Proof of Theorem 3.4.
If every relation in S is F -valid (resp. T -valid) then LRSAT ( S ) holds obviously over RCA . If every relation in S is horn (resp. co-horn) then by The-orem 3.10, LRSAT ( S ) holds also over RCA . By Lemma 3.6, the only remaining caseis where [ x (cid:54) = y ] ∈ Rep NC ( S ) . There exists a finite (hence c.e.) subset T ⊆ S such that [ x (cid:54) = y ] ∈ Rep NC ( T ) . By Lemma 3.3, RCA (cid:96) LRSAT ( T ) → LRSAT ([ x (cid:54) = y ]) , hence RCA (cid:96) LRSAT ( S ) → LRSAT ([ x (cid:54) = y ]) . (cid:3) The following technical lemma will be very useful for the remainder of the paper.
Lemma 3.11.
RCA proves the following: Suppose T is a sequence of Boolean relationssuch that T contains a relation which is not F -valid T contains a relation which is not T -valid [ x (cid:54) = y ] ∈ Rep NC ( T ) If S is a sequence such that S ⊆ Rep NC ( T ∪ { [ x ] , [ ¬ x ] } ) then LRSAT ( T ) → LRSAT ( S ) .Proof. We reason by case analysis. Suppose that [ x ] and [ ¬ x ] are both in Rep NC ( T ) . Then S ⊆ Rep NC ( T ) , so by Lemma 3.3, RCA (cid:96) LRSAT ( T ) → LRSAT ( S ) .Suppose now that either [ x ] or [ ¬ x ] is not in Rep NC ( T ) . Then by Lemma 4.3 of [10], ev-ery relation in T is complementive, that is, if (cid:126) r ∈ R for some R ∈ T , then the pointwise nega-tion of (cid:126) r is also in R . By Lemma 3.3, it suffices to ensure that RCA (cid:96) LRSAT ( Rep NC ( T )) → LRSAT ( T ∪ { [ x ] , [ ¬ x ] } ) to conclude, as RCA (cid:96) LRSAT ( T ) → LRSAT ( Rep NC ( T )) . Let ( V , C , L ) be an instance of LRSAT ( T ∪{ [ x ] , [ ¬ x ] } ) . Say V = { x , x , . . . } and C = { ϕ , ϕ , . . . } .Define an instance ( V ∪ { c , c } , D , L ) of LRSAT ( Rep NC ( T )) such that c , c (cid:54)∈ V and withthe set of formulas D = { c (cid:54) = c } ∪ { R ( (cid:126) x ) ∈ C : R (cid:54) = [ x ] ∧ R (cid:54) = [ ¬ x ] } ∪ { x = c : ( ¬ x ) ∈ C } ∪ { x = c : ( x ) ∈ C } Note that [ x = y ] ∈ Rep NC ( T ) as [ x = y ] = [( ∃ z ) x (cid:54) = z ∧ z (cid:54) = y ] and [ x (cid:54) = y ] ∈ Rep NC ( T ) . Theinstance ( V ∪ { c , c } , D , L ) is obviously finitely satisfiable as every valid assignment ν of ( V , C , L ) induces an assignment of ( V ∪ { c , c } , D , L ) by setting ν ( c ) = F and ν ( c ) = T . Conversely, we prove that for every assignment ν satisfying ( V ∪ { c , c } , D , L ) , theassignment µ defined to be ν if ν ( c ) = F and the pointwise negation of ν if ν ( c ) = T satisfies ( V , C , L ) . Suppose there exists a finite subset E ⊂ C such that µ ( (cid:86) E ) = F . Forevery formula ( ¬ x ) ∈ E , µ ( x ) = µ ( c ) = F and for every ( x ) ∈ E , µ ( x ) = µ ( c ) = T . Sothere must exist a relation R ∈ T such that R ( (cid:126) x ) ∈ E and µ ( R ( (cid:126) x )) = F . By complementationof R , ν ( R ( (cid:126) x )) = F , but R ( (cid:126) x ) ∈ D , contradicting the assumption that ν satisfies D . Therefore,every infinite set H ⊆ L homogeneous for D is homogeneous for C . (cid:3) Bijunctive satisfiability.
Our second dichotomy theorem concerns bijunctive rela-tions. Either the related principle is a consequence of
LRSAT ([ x (cid:54) = y ]) over RCA , or ithas full strength of LRSAT ( Bijunctive ) . In the remainder of this subsection, we will makethe following assumptions and denote them by the shorthand in the right column of thetable: (i) S contains only bijunctive relations ( D ⊆ Pol ( S ) )(ii) S contains a relation which is not F -valid ( I (cid:54)⊆ Pol ( S ) )(iii) S contains a relation which is not T -valid ( I (cid:54)⊆ Pol ( S ) )(iv) [ x (cid:54) = y ] ∈ Rep NC ( S ) ( Pol ( S ) ⊆ D ) Theorem 3.12.
If S contains only affine relations then
RCA (cid:96) LRSAT ([ x (cid:54) = y ]) → LRSAT ( S ) .Otherwise RCA (cid:96) LRSAT ( S ) ↔ LRSAT ( Bijunctive ) . The proof of Theorem 3.12 follows Lemma 3.15.
Definition 3.13.
For any set S of relations, the co-clone of S is the closure of S by existen-tial quantification, equality and conjunction. We denote it by (cid:104) S (cid:105) .Remark that in general, Rep NC ( S ) may be different from (cid:104) S (cid:105) if [ x = y ] (cid:54)∈ Rep NC ( S ) .However in our case, we assume that [ x (cid:54) = y ] ∈ Rep NC ( S ) , hence [ x = y ] ∈ Rep NC ( S ) and Rep NC ( S ) = (cid:104) S (cid:105) . The following property will happen to be very useful for proving that arelation R ∈ Rep NC ( S ) . Lemma 3.14 (Folklore) . Inv ( Pol ( S )) = (cid:104) S (cid:105) Lemma 3.15.
One of the following holds: (a)
Rep NC ( S ) contains all bijunctive relations. (b) S ⊆ Rep NC ( { [ x ] , [ x (cid:54) = y ] } ) . HE COMPLEXITY OF SATISFACTION PROBLEMS IN REVERSE MATHEMATICS 11
Proof.
By the blanket assumption of the subsection, D ⊆ Pol ( S ) ⊆ D . Either D ⊆ Pol ( S ) or Pol ( S ) = D . If D ⊆ Pol ( S ) , then every relation in S is affine, so S ⊆ Inv ( D ) = Rep NC ( { [ x ] , [ x (cid:54) = y ] } ) . If Pol ( S ) = D then Rep NC ( S ) = (cid:104) S (cid:105) = Inv ( Pol ( S )) = Inv ( D ) which is the set of all bijunctive relations. (cid:3) Proof of Theorem 3.12.
By Lemma 3.15, either
Rep NC ( S ) contains all bijunctive relationsor S ⊆ Rep NC ( { [ x ] , [ x (cid:54) = y ] } ) . In the latter case, by Lemma 3.11 LRSAT ([ x (cid:54) = y ]) implies LRSAT ( S ) over RCA . In the former case, there exists a finite basis S ⊆ S such that Rep NC ( S ) contains all bijunctive relations. In particular S is a c.e. set, so RCA (cid:96) LRSAT ( S ) → LRSAT ( Bijunctive ) . Any instance of LRSAT ( S ) being an instance of LRSAT ( S ) , RCA (cid:96) LRSAT ( S ) → LRSAT ( Bijunctive ) . The reverse implication followsdirectly from the assumption that every relation in S is bijunctive. So RCA (cid:96) LRSAT ( S ) ↔ LRSAT ( Bijunctive ) . (cid:3) Affine satisfiability.
In this section, we will prove that if S satisfies none of the previ-ous cases and contains only affine relations, then the corresponding Ramseyan satisfactionproblem is equivalent to LRSAT ( Affine ) over RCA . So suppose that(i) S contains only affine relations ( L ⊆ Pol ( S ) )(ii) S contains a relation which is not bijunctive ( D (cid:54)⊆ Pol ( S ) )(iii) S contains a relation which is not F -valid ( I (cid:54)⊆ Pol ( S ) )(iv) S contains a relation which is not T -valid ( I (cid:54)⊆ Pol ( S ) )(v) [ x (cid:54) = y ] ∈ Rep NC ( S ) ( Pol ( S ) ⊆ D )In particular, Pol ( S ) (cid:40) D . Theorem 3.16.
RCA (cid:96) LRSAT ( S ) ↔ LRSAT ( Affine ) Proof.
By assumption, every relation in S is affine. Hence RCA (cid:96) LRSAT ( Affine ) → LRSAT ( S ) . As L ⊆ Pol ( S ) (cid:40) D , Pol ( S ) is either L or L . In particular, Pol ( S ∪{ [ x ] , [ ¬ x ] } ) = L Considering the corresponding invariants,
Inv ( L ) ⊆ Inv ( Pol ( S ∪ { [ x ] , [ ¬ x ] } )) = (cid:104) S ∪ { [ x ] , [ ¬ x ] }(cid:105) = Rep NC ( S ∪ { [ x ] , [ ¬ x ] } ) There exists a finite basis S such that Rep NC ( S ) contains all affine relations. Inv ( L ) being the set of affine relations, S ⊂ Rep NC ( S ∪ { [ x ] , [ ¬ x ] } ) . There exists a finite (hencec.e.) subset T of S such that S ⊆ Rep NC ( T ∪ { [ x ] , [ ¬ x ] } ) . In particular, { R : R is affine } ⊆ Rep NC ( S ) ⊆ Rep NC ( T ∪ { [ x ] , [ ¬ x ] } ) By Lemma 3.11,
RCA (cid:96) LRSAT ( T ) → LRSAT ( Affine ) , hence RCA (cid:96) LRSAT ( S ) → LRSAT ( Affine ) . (cid:3) Remaining cases.
Based on Post’s lattice, the only remaining cases are
Pol ( S ) = N or Pol ( S ) = I . Theorem 3.17. If Pol ( S ) ⊆ N then RCA (cid:96) LRSAT ( S ) ↔ LRSAT .Proof.
The direction
RCA (cid:96) LRSAT → LRSAT ( S ) is obvious. We will prove the con-verse. Because Pol ( S ) ⊆ N , Pol ( S ∪ { [ x ] } ) = I . Rep NC ( S ∪ { [ x ] } ) = (cid:104) S ∪ { [ x ] }(cid:105) = Inv ( Pol ( S ∪ { [ x ] } )) ⊇ Inv ( I ) Note that
Inv ( I ) is the set of all Boolean relations. As Inv ( I ) has a finite basis, there existsa finite S ⊆ S such that Rep NC ( S ∪ { [ x ] } ) contains all Boolean relations. By Lemma 3.11, RCA (cid:96) LRSAT ( S ) → LRSAT . Hence
RCA (cid:96) LRSAT ( S ) ↔ LRSAT . (cid:3) Proof of Theorem 1.14.
By case analysis over
Pol ( S ) . If I , I , E and V are included in Pol ( S ) (that is, if S contains only F -valid, T -valid, horn or co-horn relations) then by Theo-rem 3.4, RCA (cid:96) LRSAT ( S ) . If D ⊆ Pol ( S ) ⊆ D then RCA (cid:96) LRSAT ( S ) ↔ LRSAT ([ x (cid:54) = y ]) by Theorem 3.12. By the same theorem, if Pol ( S ) = D then RCA (cid:96) LRSAT ( S ) ↔ LRSAT ( Bijunctive ) . If L ⊆ Pol ( S ) ⊆ L then by Theorem 3.16, RCA (cid:96) LRSAT ( S ) ↔ LRSAT ( Affine ) . Otherwise, I ⊆ Pol ( S ) ⊆ N in which case RCA (cid:96) LRSAT ( S ) ↔ LRSAT by Theorem 3.17. (cid:3)
The principle
LRSAT ([ x (cid:54) = y ]) coincides with an already existing principle about bi-partite graphs. For k ∈ (cid:78) , we say that a graph G = ( V , E ) is k-colorable if there is afunction f : V → k such that ( ∀ ( x , y ) ∈ E )( f ( x ) (cid:54) = f ( y )) , and we say that a graph is finitelyk-colorable if every finite induced subgraph is k -colorable. Definition 3.18.
Let G = ( V , E ) be a graph. A set H ⊆ V is homogeneous for G if everyfinite V ⊆ V induces a subgraph that is k -colorable by a coloring that colors every v ∈ V ∩ H color 0. LRCOLOR k is the statement “For every infinite, finitely k -colorable graph G =( V , E ) and every infinite L ⊆ V there exists an infinite H ⊆ L that is homogeneous for G ”. RCOLOR k is the restriction of LRCOLOR k with L = V . An instance of LRCOLOR k is apair ( G , L ) . For RCOLOR k , it is simply the graph G . Theorem 3.19.
RCA (cid:96) RCOLOR ↔ LRSAT ([ x (cid:54) = y ]) Proof.
See [2] for a proof of
RCA (cid:96) RCOLOR ↔ LRCOLOR . There exists a di-rect mapping between an instance ( V , C , L ) of LRSAT ([ x (cid:54) = y ]) and an instance ( G , L ) of LRCOLOR where G = ( V , E ) by taking E = {{ x , y } : x (cid:54) = y ∈ C } . (cid:3)
4. The strength of satisfiabilityLocalized principles are relatively easy to manipulate as they can express relations de-fined using existential quantifier by restricting the localized set L to the variables not cap-tured by any quantifier. However we will see that when the set of relations has some goodclosure properties, the unlocalized version of the principle is as expressive as its localizedone. Theorem 4.1.
RCA proves that if S be a Σ co-clone then RSAT ( S ) ↔ LRSAT ( S ) Proof.
The implication
LRSAT ( S ) → RSAT ( S ) is obvious. To prove the converse, let ( V , C , L ) be an instance of LRSAT ( S ) with V = { x i : i ∈ ω } and C = { θ i : i ∈ ω } . Let C L be a computable enumeration of formulas φ ( (cid:126) x ) = R ( (cid:126) x ) with R ∈ S and (cid:126) x ⊂ L such that thereexists a finite subset C fin of C for which every valid truth assignment ν over C fin satisfies ν ( φ ) = T .If C L is finite, then there is a bound m such that if φ ∈ C L then max ( i : x i ∈ Var ( φ )) ≤ m .Then take H = { x i ∈ L : i > m } . H ⊆ L and is infinite because L is infinite. Claim.
For every c ∈ (cid:66) , H is homogeneous for C with color c.Proof of claim. If not then there exists a finite subset C fin of C such that H is nothomogeneous for C fin with color c . Let (cid:126) y = Var ( C fin ) (cid:114) L . Because S is a co-clone, itis closed under finite conjunction and projection, hence ( ∃ (cid:126) y ) (cid:86) C fin is equivalent to an S -formula, say ϕ . In particular Var ( ϕ ) ⊆ Var ( C fin ) ∩ L and ϕ ∈ C L . For every assignment ν satisfying ϕ , there is a variable x ∈ H such that ν ( x ) = ¬ c . Then Var ( ϕ ) ∩ H (cid:54) = /0. However ϕ ∈ C L , so Var ( ϕ ) ∩ H = /0 by definition of H . Contradiction. This finishes the proof ofthe claim. HE COMPLEXITY OF SATISFACTION PROBLEMS IN REVERSE MATHEMATICS 13
So suppose instead C L = { φ i : i ∈ (cid:78) } is infinite, and suppose each φ i is unique. Weconstruct an instance ( V (cid:48) , C (cid:48) ) of RSAT ( S ) by taking V (cid:48) = L ∪ { y n : n ∈ (cid:78) } and constructing C (cid:48) by stages as follows: At stage 0, C (cid:48) = /0. At stage s +
1, look at φ s = R ( x , . . . , x m ) and let x i be the greatest variable in lexicographic order among x , . . . , x m . Add the formula x i = y s and the formula R ( x , . . . , x i − , y s , x i + , . . . , x m ) to C (cid:48) . Then go to next stage. This finishesthe construction. Note that C (cid:48) is satisfiable, otherwise by definition there would be a finiteunsatisfiable subset C fin ⊂ C L from which we could extract an unsatisfiable subset of C .Also note that, by assuming that φ i is unique and x i is the greatest variable in lexicographicorder, the number of stages s such that the formula x = y s is added to C (cid:48) is finite for eachvariable x .Let H (cid:48) be an infinite set homogeneous for C (cid:48) with color c . We can extract from H (cid:48) an in-finite subset of L homogeneous for C (cid:48) with color c because either L ∩ H (cid:48) or { x ∈ L : ( x = y n ) ∈ C (cid:48) and y n ∈ H (cid:48) } is infinite and both are homogeneous for C (cid:48) with color c . So fix H ⊆ L , an infinite set ho-mogeneous for C (cid:48) (and for C L ) with color c . Claim.
H is homogeneous for C with color c.Proof of claim.
By the same argument as previous claim, suppose there is a finite subset C fin of C such that H is not homogeneous for C fin with color c . Let ϕ be the S -formulaequivalent to ( ∃ (cid:126) y ) (cid:86) C fin where (cid:126) y = Var ( C fin ) (cid:114) L . For every valid assignment ν for ϕ ,there is a variable x ∈ H such that ν ( x ) = ¬ c . But ϕ ∈ C L and hence H is homogeneousfor ϕ with color c . Contradiction. This last claims finishes the proof of Theorem 4.1. (cid:3) Noticing that affine (resp. bijunctive) relations form a co-clone, we immediately deducethe following corollary.
Corollary 4.2.
RSAT ( Affine ) and RSAT ( Bijunctive ) are equivalent to their local versionover RCA . A useful principle below
WKL for studying the strength of a statement is the notion of diagonally non-computable function . Definition 4.3.
A total function f is diagonally non-computable if ( ∀ e ) f ( e ) (cid:54) = Φ e ( e ) . DNR is the corresponding principle, i.e. for every X , there exists a function d.n.c. rel-ative to X . DNR is known to coincide with the restriction of
RWKL to trees of positive measure([5, 2]). On the other side, there exists an ω -model of DNR which is not a model of
RCOLOR ([2]). We will now prove that we can compute a diagonally non-computablefunction from any infinite set homogeneous for a particular set of affine formulas. As RSAT implies
LRSAT ( Affine ) over RCA , it gives another proof of RCA (cid:96) RWKL → DNR . Theorem 4.4.
RCA (cid:96) RSAT ( Affine ) → DNR .Proof.
We construct a computable set C of affine formulas over a computable set V ofvariables such that every infinite set homogeneous for C computes a diagonally non-computable function. Relativization is straightforward. Let t : (cid:78) → (cid:78) be the computablefunction defined by t ( ) = t ( e + ) = + ∑ ei = t ( i ) . Note that every image by t iseven. For every e ∈ (cid:78) , let (cid:10) D e , j : j ∈ (cid:78) (cid:11) denote the canonical enumeration of all finite setsof size t ( e ) . We fix a countable set of variables V = { x , x , . . . } a define a set of formulas C satisfying the following requirements: (cid:82) e : Φ e ( e ) ↓⇒ D e , Φ e ( e ) is not homogeneous for C We first show how to construct a d.n.c. function from an infinite set H homogeneousfor C , assuming that each requirement is satisfied. Let g ( · ) be such that D e , g ( e ) are theleast t ( e ) elements of H . We claim that g is a d.n.c. function: If Φ e ( e ) ↑ then obviously g ( e ) (cid:54) = Φ e ( e ) . If Φ e ( e ) ↓ then because of requirement (cid:82) e , D e , Φ e ( e ) ∩ ¯ H (cid:54) = /0, hence D e , g ( e ) (cid:54) = D e , Φ e ( e ) so g ( e ) (cid:54) = Φ e ( e ) .We define C by stages. At stage 0, C = /0. To make C computable, we will not add to C any formula over { x i : i ≤ s } after stage s . Suppose at stage s Φ e , s ( e ) ↓ for some e < s – wecan assume w.l.o.g. that at most one e halts at each stage –. Then add x s ⊕ x s (cid:76) D e , Φ e , s ( e ) to C . This finishes stage s . One easily check that each requirements is satisfied as x s ⊕ x s (cid:76) D e , Φ e , s is logically equivalent to (cid:76) D e , Φ e , s ( e ) , and as D e , Φ e , s ( e ) has even size, so therelation is neither F -valid nor T -valid, hence D e , Φ e , s ( e ) is not homogeneous for C . Claim.
The resulting instance is satisfiable.Proof of claim.
If not, there exists a finite C fin ⊂ C which is not satisfiable. For agiven Turing index e , define C e to be the set of formulas added in some stage s at which Φ i , s ( i ) ↓ for some i < e . There exists an e max such that C fin ⊆ C e max . We will define a validassignment ν e of C e by Σ -induction over e .If e =
0, then C = /0 and ν = /0 is a valid assignment. Suppose we have a validassignment ν e for some C e . We will construct a valid assignment ν e + for C e + . If Φ e ( e ) ↑ then C e + = C e and ν e is a valid assignment for C e + . Otherwise Φ e ( e ) ↓ . C e + = C e ∪ (cid:110) x s ⊕ x s (cid:76) D e , Φ e ( e ) (cid:111) . Var ( C e ) has at most ∑ e − i = elements, hence D e , Φ e ( e ) (cid:114) Var ( C e ) is notempty. We can hence easily extend our valuation ν e to D e , Φ e ( e ) such that the resultingvaluation satisfies C e + . This claim finishes the proof of Theorem 4.4. (cid:3)
5. Conclusions and questionsSatisfaction principles happen to collapse in the case of a full assignment existencestatement. The definition is not robust and the conditions of the corresponding dichotomytheorem evolves if we make the slight modification of allowing conjunctions in our defini-tion of formulas.However, the proposed Ramseyan version leads to a much more robust dichotomy the-orem with four main subsystems. The conditions of “tractability” – here provability over
RCA – differ from those of Schaefer dichotomy theorem but the considered classes ofrelations remain the same. We obtain the surprising result that infinite versions of hornand co-horn satisfaction problems are provable over RCA and strictly weaker than bijunc-tive and affine corresponding principles, whereas the complexity classification of [1] hasshown that horn satisfiability was P-complete under AC reduction, hence at least as strongas bijunctive satisfiability which is NL-complete.5.1. Summary of principles considered.
The following diagram summarizes the knownrelations between the principles considered here. Single arrows express implication over
RCA . Double arrows mean that implications are strict. A crossed arrow denotes a non-implication over ω -models.Localized and non-localized principles coincide for the main principles because of The-orem 4.1. By [2], there exists an ω -model of DNR – and even
WWKL –which is not amodel of RCOLOR . The missing arrows are all unknown.5.2. Open questions.
Very few relations are known between the four main subsystemsstudied in this paper:
RSAT , RSAT ( Affine ) , RSAT ( Bijunctive ) and RCOLOR . Theo-rem 1.14 states that LRSAT ( S ) is equivalent to one of the above mentioned principles, or HE COMPLEXITY OF SATISFACTION PROBLEMS IN REVERSE MATHEMATICS 15
Figure 1. Summary of principlesis provable over
RCA . However those principles are not known to be pairwise distinct. Inparticular the principle RCOLOR introduced in [2] is not even known to be strictly below RWKL . Question 5.1.
What are the relations between
RSAT , RSAT ( Affine ) , RSAT ( Bijunctive ) and RCOLOR ? Question 5.2.
Does
RCOLOR imply DNR over
RCA ? Does it imply RWKL ? Acknowledgements . The author is thankful to Laurent Bienvenu and Paul Shafer for theiravailability during the different steps giving birth to the paper, and for their useful sugges-tions. The author is funded by the John Templeton Foundation (‘Structure and Random-ness in the Theory of Computation’ project). The opinions expressed in this publicationare those of the author(s) and do not necessarily reflect the views of the John TempletonFoundation. References [1] Eric Allender, Michael Bauland, Neil Immerman, Henning Schnoor, and Heribert Vollmer. The complexityof satisfiability problems: Refining schaefer’s theorem. In
Mathematical Foundations of Computer Science2005 , pages 71–82. Springer, 2005.[2] Laurent Bienvenu, Ludovic Patey, and Paul Shafer. A Ramsey-Type König’s lemma and its variants. inpreparation.[3] Andrei A Bulatov. A dichotomy theorem for constraints on a three-element set. In
Proceedings of the 43rdAnnual IEEE Symposium on Foundations of Computer Science (FOCS’02) , pages 649–658. IEEE, 2002.[4] Nadia Creignou and Miki Hermann. Complexity of generalized satisfiability counting problems.
Informa-tion and Computation , 125(1):1–12, 1996. [5] Stephen Flood. Reverse mathematics and a Ramsey-type König’s Lemma.
Journal of Symbolic Logic ,77(4):1272–1280, 2012.[6] Jeffry L. Hirst. Marriage theorems and reverse mathematics. In
Logic and computation (Pittsburgh, PA,1987) , volume 106 of
Contemp. Math. , pages 181–196. Amer. Math. Soc., Providence, RI, 1990.[7] Sanjeev Khanna and Madhu Sudan. The optimization complexity of constraint satisfaction problems. In
Electonic Colloquium on Computational Complexity . Citeseer, 1996.[8] Victor W Marek and Jeffrey B Remmel. The complexity of recursive constraint satisfaction problems.
An-nals of Pure and Applied Logic , 161(3):447–457, 2009.[9] Emil L Post.
The two-valued iterative systems of mathematical logic . Number 5. Princeton University Press,1942.[10] Thomas J. Schaefer. The complexity of satisfiability problems. In
Conference Record of the Tenth AnnualACM Symposium on Theory of Computing (San Diego, Calif., 1978) , pages 216–226. ACM, New York,1978.[11] Stephen G. Simpson.
Subsystems of second order arithmetic . Perspectives in Logic. Cambridge UniversityPress, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY, second edition, 2009.
HE COMPLEXITY OF SATISFACTION PROBLEMS IN REVERSE MATHEMATICS 17
Appendix A. Post’s lattice R R BFR MM M M S S S S S S S S S S S S S S S S S S S S S S S S DD D LL L L L VV V V E E E E II I I N N Laboratoire PPS, Université Paris Diderot, Paris, FRANCE
E-mail address : [email protected] HE COMPLEXITY OF SATISFACTION PROBLEMS IN REVERSE MATHEMATICS 19
Class Definition Base(s) BF all Boolean functions {∧ , ¬} R { f ∈ BF | f is 0-reproducing } {∧ , ⊕} R { f ∈ BF | f is 1-reproducing } {∨ , x ⊕ y ⊕ R R ∩ R {∨ , x ∧ ( y ⊕ z ⊕ ) } M { f ∈ BF | f is monotonic } {∧ , ∨ , , M M ∩ R {∧ , ∨ , M M ∩ R {∧ , ∨ , M M ∩ R {∧ , ∨} S n { f ∈ BF | f is 0-separating of degree n } {→ , dual ( t n ) } S { f ∈ BF | f is 0-separating } {→} S n { f ∈ BF | f is 1-separating of degree n } { x ∧ y , t n } S { f ∈ BF | f is 1-separating } { x ∧ y } S n S n ∩ R { x ∨ ( y ∧ z ) , dual ( t n ) } S S ∩ R { x ∨ ( y ∧ z ) } S n S n ∩ M { dual ( t n ) , } S S ∩ M { x ∨ ( y ∧ z ) , } S n S n ∩ R ∩ M { x ∨ ( y ∧ z ) , dual ( t n ) } S S ∩ R ∩ M { x ∨ ( y ∧ z ) } S n S n ∩ R { x ∧ ( y ∨ z ) , t n } S S ∩ R { x ∧ ( y ∨ z ) } S n S n ∩ M { t n , } S S ∩ M { x ∧ ( y ∨ z ) , } S n S n ∩ R ∩ M { x ∧ ( y ∨ z ) , t n } S S ∩ R ∩ M { x ∧ ( y ∨ z ) } D { f | f is self-dual } { ( x ∧ y ) ∨ ( x ∧ z ) ∨ ( y ∧ z ) } D D ∩ R { ( x ∧ y ) ∨ ( x ∧ z ) ∨ ( y ∧ z ) } D D ∩ M { ( x ∧ y ) ∨ ( y ∧ z ) ∨ ( x ∧ z ) } L { f | f is linear} {⊕ , } L L ∩ R {⊕} L L ∩ R {↔} L L ∩ R { x ⊕ y ⊕ z } L L ∩ D { x ⊕ y ⊕ z ⊕ } V { f | f is an ∨ -function or a constant function} {∨ , , } V [ {∨} ] ∪ [ { } ] {∨ , } V [ {∨} ] ∪ [ { } ] {∨ , } V [ {∨} ] {∨} E { f | f is an ∧ -function or a constant function} {∧ , , } E [ {∧} ] ∪ [ { } ] {∧ , } E [ {∧} ] ∪ [ { } ] {∧ , } E [ {∧} ] {∧} N [ {¬} ] ∪ [ { } ] ∪ [ { } ] {¬ , } , {¬ , } N [ {¬} ] {¬} I [ { id } ] ∪ [ { } ] ∪ [ { } ] { id , , } I [ { id } ] ∪ [ { } ] { id , } I [ { id } ] ∪ [ { } ] { id , } I [ { id } ] { id } Figure 2. The list of all Boolean clones with definitions and bases, where t n : = (cid:87) n + i = (cid:86) n + j = , j (cid:54) = i x j and dual ( f )( a , . . . , a n ) = ¬ f ( ¬ a . . . , ¬ a n ))