254 research outputs found

    Resolution Trees with Lemmas: Resolution Refinements that Characterize DLL Algorithms with Clause Learning

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    Resolution refinements called w-resolution trees with lemmas (WRTL) and with input lemmas (WRTI) are introduced. Dag-like resolution is equivalent to both WRTL and WRTI when there is no regularity condition. For regular proofs, an exponential separation between regular dag-like resolution and both regular WRTL and regular WRTI is given. It is proved that DLL proof search algorithms that use clause learning based on unit propagation can be polynomially simulated by regular WRTI. More generally, non-greedy DLL algorithms with learning by unit propagation are equivalent to regular WRTI. A general form of clause learning, called DLL-Learn, is defined that is equivalent to regular WRTL. A variable extension method is used to give simulations of resolution by regular WRTI, using a simplified form of proof trace extensions. DLL-Learn and non-greedy DLL algorithms with learning by unit propagation can use variable extensions to simulate general resolution without doing restarts. Finally, an exponential lower bound for WRTL where the lemmas are restricted to short clauses is shown

    Bounded-Depth Frege Lower Bounds for Weaker Pigeonhole Principles

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    Circuit complexity, proof complexity, and polynomial identity testing

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    We introduce a new algebraic proof system, which has tight connections to (algebraic) circuit complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic circuits (VNP is not equal to VP). As a corollary to the proof, we also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems, and begins to shed light on why proof complexity lower bounds for various proof systems have been so much harder than lower bounds on the corresponding circuit classes. In doing so, we highlight the importance of polynomial identity testing (PIT) for understanding proof complexity. More specifically, we introduce certain propositional axioms satisfied by any Boolean circuit computing PIT. We use these PIT axioms to shed light on AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. We show that either: a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not have polynomial-size circuits of depth d - a notoriously open question for d at least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we have a lower bound on AC^0[p]-Frege. Using the algebraic structure of our proof system, we propose a novel way to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity

    Efficiently Simulating Higher-Order Arithmetic by a First-Order Theory Modulo

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    In deduction modulo, a theory is not represented by a set of axioms but by a congruence on propositions modulo which the inference rules of standard deductive systems---such as for instance natural deduction---are applied. Therefore, the reasoning that is intrinsic of the theory does not appear in the length of proofs. In general, the congruence is defined through a rewrite system over terms and propositions. We define a rigorous framework to study proof lengths in deduction modulo, where the congruence must be computed in polynomial time. We show that even very simple rewrite systems lead to arbitrary proof-length speed-ups in deduction modulo, compared to using axioms. As higher-order logic can be encoded as a first-order theory in deduction modulo, we also study how to reinterpret, thanks to deduction modulo, the speed-ups between higher-order and first-order arithmetics that were stated by G\"odel. We define a first-order rewrite system with a congruence decidable in polynomial time such that proofs of higher-order arithmetic can be linearly translated into first-order arithmetic modulo that system. We also present the whole higher-order arithmetic as a first-order system without resorting to any axiom, where proofs have the same length as in the axiomatic presentation

    Proceedings of SAT Competition 2021 : Solver and Benchmark Descriptions

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    On the proof complexity of logics of bounded branching

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    We investigate the proof complexity of extended Frege (EF) systems for basic transitive modal logics (K4, S4, GL, ...) augmented with the bounded branching axioms BBk\mathbf{BB}_k. First, we study feasibility of the disjunction property and more general extension rules in EF systems for these logics: we show that the corresponding decision problems reduce to total coNP search problems (or equivalently, disjoint NP pairs, in the binary case); more precisely, the decision problem for extension rules is equivalent to a certain special case of interpolation for the classical EF system. Next, we use this characterization to prove superpolynomial (or even exponential, with stronger hypotheses) separations between EF and substitution Frege (SF) systems for all transitive logics contained in S4.2GrzBB2\mathbf{S4.2GrzBB_2} or GL.2BB2\mathbf{GL.2BB_2} under some assumptions weaker than PSPACENP\mathrm{PSPACE \ne NP}. We also prove analogous results for superintuitionistic logics: we characterize the decision complexity of multi-conclusion Visser's rules in EF systems for Gabbay--de Jongh logics Tk\mathbf T_k, and we show conditional separations between EF and SF for all intermediate logics contained in T2+KC\mathbf{T_2 + KC}.Comment: 58 page
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