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

Polylogarithmic Cuts in Models of V^0

We study initial cuts of models of weak two-sorted Bounded Arithmetics with respect to the strength of their theories and show that these theories are stronger than the original one. More explicitly we will see that polylogarithmic cuts of models of $\mathbf{V}^0$ are models of $\mathbf{VNC}^1$ by formalizing a proof of Nepomnjascij's Theorem in such cuts. This is a strengthening of a result by Paris and Wilkie. We can then exploit our result in Proof Complexity to observe that Frege proof systems can be sub exponentially simulated by bounded depth Frege proof systems. This result has recently been obtained by Filmus, Pitassi and Santhanam in a direct proof. As an interesting observation we also obtain an average case separation of Resolution from AC0-Frege by applying a recent result with Tzameret.Comment: 16 page

Disjoint NP-pairs from propositional proof systems

For a proof system P we introduce the complexity class DNPP(P) of all disjoint NP-pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make the previously defined canonical NP-pairs of these proof systems hard or complete for DNPP(P). Moreover we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for DNPP(P) and the reductions between the canonical pairs exist

Classes of representable disjoint NP-pairs

For a propositional proof system P we introduce the complexity class of all disjoint -pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make canonical -pairs associated with these proof systems hard or complete for . Moreover, we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for and the reductions between the canonical pairs exist

On extracting computations from propositional proofs (a survey)

This paper describes a project that aims at showing that propositional proofs of certain tautologies in weak proof system give upper bounds on the computational complexity of functions associated with the tautologies. Such bounds can potentially be used to prove (conditional or unconditional) lower bounds on the lengths of proofs of these tautologies and show separations of some weak proof systems. The prototype are the results showing the feasible interpolation property for resolution. In order to prove similar results for systems stronger than resolution one needs to define suitable generalizations of boolean circuits. We will survey the known results concerning this project and sketch in which direction we want to generalize them

Sparser Random 3SAT Refutation Algorithms and the Interpolation Problem:Extended Abstract

We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek [12], as a family of unsatisfiable propositional formulas for which refutations of small size in any propo-sitional proof system that possesses the feasible interpolation property imply an efficient deterministic refutation algorithm for random 3SAT with n variables and Ω(n1.4) clauses. Such small size refutations would improve the state of the art (with respect to the clause density) efficient refutation algorithm, which works only for Ω(n1.5) many clauses [13]. We demonstrate polynomial-size refutations of the 3XOR principle in resolution operating with disjunctions of quadratic equations with small integer coefficients, denoted R(quad); this is a weak extension of cutting planes with small coefficients. We show that R(quad) is weakly autom-atizable iff R(lin) is weakly automatizable, where R(lin) is similar to R(quad) but with linear instead of quadratic equations (introduced in [25]). This reduces the problem of refuting random 3CNF with n vari-ables and Ω(n1.4) clauses to the interpolation problem of R(quad) and to the weak automatizability of R(lin)

Nondeterministic functions and the existence of optimal proof systems

We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class is equivalent to the question whether the standard proof system for SAT is p-optimal, and to the assumption that every optimal proof system is p-optimal. Assuming only the existence of a p-optimal proof system for SAT, we show that every set with an optimal proof system has a p-optimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class . An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint -pairs and its generalizations to tuples of sets from and with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class . Question Q1 for is equivalent to the question of whether every disjoint -pair is easy to separate. In addition, we characterize this problem by the question of whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint -pairs, and we show how different interpolation properties can be modeled by -pairs associated with the underlying proof system

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 that 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.Peer ReviewedPostprint (author's final draft

Feasible Interpolation for Polynomial Calculus and Sums-Of-Squares

We prove that both Polynomial Calculus and Sums-of-Squares proof systems admit a strong form of feasible interpolation property for sets of polynomial equality constraints. Precisely, given two sets P(x,z) and Q(y,z) of equality constraints, a refutation ? of P(x,z) ? Q(y,z), and any assignment a to the variables z, one can find a refutation of P(x,a) or a refutation of Q(y,a) in time polynomial in the length of the bit-string encoding the refutation ?. For Sums-of-Squares we rely on the use of Boolean axioms, but for Polynomial Calculus we do not assume their presence

Feasible Interpolation for QBF Resolution Calculi

In sharp contrast to classical proof complexity we are currently short of lower bound techniques for QBF proof systems. In this paper we establish the feasible interpolation technique for all resolution-based QBF systems, whether modelling CDCL or expansion-based solving. This both provides the first general lower bound method for QBF proof systems as well as largely extends the scope of classical feasible interpolation. We apply our technique to obtain new exponential lower bounds to all resolution-based QBF systems for a new class of QBF formulas based on the clique problem. Finally, we show how feasible interpolation relates to the recently established lower bound method based on strategy extraction