On the pigeonhole and related principles in deep inference and monotone systems

International audienceWe construct quasipolynomial-size proofs of the propositional pigeonhole principle in the deep inference system KS, addressing an open problem raised in previous works and matching the best known upper bound for the more general class of monotone proofs. We make significant use of monotone formulae computing boolean threshold functions, an idea previously considered in works of Atserias et al. The main construction, monotone proofs witnessing the symmetry of such functions, involves an implementation of merge-sort in the design of proofs in order to tame the structural behaviour of atoms, and so the complexity of normalization. Proof transformations from previous work on atomic flows are then employed to yield appropriate KS proofs. As further results we show that our constructions can be applied to provide quasipolynomial-size KS proofs of the parity principle and the generalized pigeonhole principle. These bounds are inherited for the class of monotone proofs, and we are further able to construct n^O(log log n) -size monotone proofs of the weak pigeonhole principle with (1 + ε)n pigeons and n holes for ε = 1/ polylog n, thereby also improving the best known bounds for monotone proofs

Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs

We prove that there is a constant K such that Tseitin formulas for an undirected graph G requires proofs of size 2tw(G)Ω(1/d) in depth-d Frege systems for d < (Formula presented.) where tw(G) is the treewidth of G. This extends Håstad recent lower bound for the grid graph to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2tw(G)O(1/d)poly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution

Bounded-depth Frege complexity of Tseitin formulas for all graphs

We prove that there is a constant K such that Tseitin formulas for a connected graph G requires proofs of size 2tw(G)javax.xml.bind.JAXBElement@531a834b in depth-d Frege systems for [Formula presented], where tw(G) is the treewidth of G. This extends Håstad's recent lower bound from grid graphs to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2tw(G)javax.xml.bind.JAXBElement@25a4b51fpoly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution

Tight Size-Degree Bounds for Sums-of-Squares Proofs

We exhibit families of $4$-CNF formulas over $n$ variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) $d$ but require SOS proofs of size $n^{\Omega(d)}$ for values of $d = d(n)$ from constant all the way up to $n^{\delta}$ for some universal constant$\delta$. This shows that the $n^{O(d)}$ running time obtained by using the Lasserre semidefinite programming relaxations to find degree-$d$ SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining $\mathsf{NP}$-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and Riis'03], and then applying a restriction argument as in [Atserias, M\"uller, and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a generic method of amplifying SOS degree lower bounds to size lower bounds, and also generalizes the approach in [ALN14] to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively

Exponential Separations Using Guarded Extension Variables

We study the complexity of proof systems augmenting resolution with inference rules that allow, given a formula ? in conjunctive normal form, deriving clauses that are not necessarily logically implied by ? but whose addition to ? preserves satisfiability. When the derived clauses are allowed to introduce variables not occurring in ?, the systems we consider become equivalent to extended resolution. We are concerned with the versions of these systems without new variables. They are called BC?, RAT?, SBC?, and GER?, denoting respectively blocked clauses, resolution asymmetric tautologies, set-blocked clauses, and generalized extended resolution. Each of these systems formalizes some restricted version of the ability to make assumptions that hold "without loss of generality," which is commonly used informally to simplify or shorten proofs. Except for SBC?, these systems are known to be exponentially weaker than extended resolution. They are, however, all equivalent to it under a relaxed notion of simulation that allows the translation of the formula along with the proof when moving between proof systems. By taking advantage of this fact, we construct formulas that separate RAT? from GER? and vice versa. With the same strategy, we also separate SBC? from RAT?. Additionally, we give polynomial-size SBC? proofs of the pigeonhole principle, which separates SBC? from GER? by a previously known lower bound. These results also separate the three systems from BC? since they all simulate it. We thus give an almost complete picture of their relative strengths

Algebraic Proofs over Noncommutative Formulas

AbstractWe study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege, yielding a semantic way to define a Cook–Reckhow (i.e., polynomially verifiable) algebraic analog of Frege proofs, different from that given in Buss et al. (1997) and Grigoriev and Hirsch (2003). We then turn to an apparently weaker system, namely, polynomial calculus (PC) where polynomials are written as ordered formulas (PC over ordered formulas, for short). Given some fixed linear order on variables, an arithmetic formula is ordered if for each of its product gates the left subformula contains only variables that are less-than or equal, according to the linear order, than the variables in the right subformula of the gate. We show that PC over ordered formulas (when the base field is of zero characteristic) is strictly stronger than resolution, polynomial calculus and polynomial calculus with resolution (PCR), and admits polynomial-size refutations for the pigeonhole principle and Tseitinʼs formulas. We conclude by proposing an approach for establishing lower bounds on PC over ordered formulas proofs, and related systems, based on properties of lower bounds on noncommutative formulas (Nisan, 1991).The motivation behind this work is developing techniques incorporating rank arguments (similar to those used in arithmetic circuit complexity) for establishing lower bounds on propositional proofs

A Complexity Gap for Tree-Resolution

It is shown that any sequence  psi_n of tautologies which expresses thevalidity of a fixed combinatorial principle either is "easy" i.e. has polynomialsize tree-resolution proofs or is "difficult" i.e requires exponentialsize tree-resolution proofs. It is shown that the class of tautologies whichare hard (for tree-resolution) is identical to the class of tautologies whichare based on combinatorial principles which are violated for infinite sets.Actually it is shown that the gap-phenomena is valid for tautologies basedon infinite mathematical theories (i.e. not just based on a single proposition).We clarify the link between translating combinatorial principles (ormore general statements from predicate logic) and the recent idea of using the symmetrical group to generate problems of propositional logic.Finally, we show that it is undecidable whether a sequence  psi_n (of thekind we consider) has polynomial size tree-resolution proofs or requiresexponential size tree-resolution proofs. Also we show that the degree ofthe polynomial in the polynomial size (in case it exists) is non-recursive,but semi-decidable.Keywords: Logical aspects of Complexity, Propositional proof complexity,Resolution proofs.

Automating Resolution is NP-Hard

We show that the problem of finding a Resolution refutation that is at most polynomially longer than a shortest one is NP-hard. In the parlance of proof complexity, Resolution is not automatizable unless P = NP. Indeed, we show it is NP-hard to distinguish between formulas that have Resolution refutations of polynomial length and those that do not have subexponential length refutations. This also implies that Resolution is not automatizable in subexponential time or quasi-polynomial time unless NP is included in SUBEXP or QP, respectively

The symmetry rule in propositional logic

AbstractThe addition of the symmetry rule to the resolution system sometimes allows considerable shortening in the length of refutations. We prove exponential lower bounds on the size of resolution refutations using two forms of a global symmetry rule. The paper also discusses the relationship of symmetry rules to the extension rule that allows the use of abbreviative definitions in proofs