The Complexity of Propositional Proofs

Abstract-The relativized weak pigeonhole principle states that if at least 2n out of n 2 pigeons fly into n holes, then some hole must be doubly occupied. We prove that every DNF-refutation of the CNF encoding of this principle requires size 2 (log n) 3/2− for every > 0 and every sufficiently large n. For its proof we need to discuss the existence of unbalanced low-degree bipartite expanders satisfying a certain robustness condition.

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“…Recall (cf. [20], [21]), this formula is obtained from PHP 2n n by zeroing out all P u,v with (u, v) not an edge of G.…”

Section: Proof Outline and Comparison To Previous Workmentioning

confidence: 99%

Lower Bounds for DNF-Refutations of a Relativized Weak Pigeonhole Principle

2015

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“…Rather than trying to give a full account on all developments in QBF proof complexity, we therefore briefly sketch the situation on conceptual techniques for QBF calculi. It is interesting to compare this situation to the classical case for which we refer the reader to [19,40].…”

Section: O Beyersdorff Et Al / Journal Of Computer and System Scienmentioning

confidence: 99%

“…There are exponential separations known between tree-like and dag-like Resolution in the classical case (cf. [40]), and these easily carry over to an exponential separation between tree-like and dag-like Q-Resolution. Given a Resolution proof graph, we can think of the tree-like Resolution as the collection of paths connecting the empty clause and the original matrix clauses in the graph.…”

Section: Preliminariesmentioning

confidence: 99%

A Game Characterisation of Tree-like Q-resolution Size

Beyersdorff

Chew

Sreenivasaiah

2015

Language and Automata Theory and Applications

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We provide a characterisation for the size of proofs in tree-like Q-Resolution and tree-like QU-Resolution by a Prover-Delayer game, which is inspired by a similar characterisation for the proof size in classical tree-like Resolution. This gives one of the first successful transfers of one of the lower bound techniques for classical proof systems to QBF proof systems. We apply our technique to show the hardness of three classes of formulas for tree-like Q-Resolution. In particular, we give a proof of the hardness of the parity formulas from [10] for tree-like Q-Resolution and of the formulas of Kleine Büning et al. (1995) [29] for tree-like QU-Resolution.

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“…In particular Frege systems currently form a strong barrier [BBP95], and all current lower bound methods seem to be insufficient for these strong systems. A detailed survey of recent advances in propositional proof complexity is contained in [Seg07].…”

Section: Propositional Proof Complexitymentioning

confidence: 99%

Theory and Applications of Models of Computation

2010

Lecture Notes in Computer Science

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Abstract. Proof complexity is an interdisciplinary area of research utilizing techniques from logic, complexity, and combinatorics towards the main aim of understanding the complexity of theorem proving procedures. Traditionally, propositional proofs have been the main object of investigation in proof complexity. Due their richer expressivity and numerous applications within computer science, also non-classical logics have been intensively studied from a proof complexity perspective in the last decade, and a number of impressive results have been obtained. In this paper we give the first survey of this field concentrating on recent developments in proof complexity of non-classical logics. Propositional Proof ComplexityOne of the starting points of propositional proof complexity is the seminal paper of Cook and Reckhow [CR79] where they formalized propositional proof systems as polynomial-time computable functions which have as their range the set of all propositional tautologies. In that paper, Cook and Reckhow also observed a fundamental connection between lengths of proofs and the separation of complexity classes: they showed that there exists a propositional proof system which has polynomial-size proofs for all tautologies (a polynomially bounded proof system) if and only if the class NP is closed under complementation. From this observation the so called Cook-Reckhow programme was derived which serves as one of the major motivations for propositional proof complexity: to separate NP from coNP (and hence P from NP) it suffices to show super-polynomial lower bounds to the size of proofs in all propositional proof systems.Although the first super-polynomial lower bound to the lengths of proofs had already been shown by Tseitin in the late 60's for a sub-system of resolution [Tse68], the first major achievement in this programme was made by Haken in 1985 when he showed an exponential lower bound to the proof size in Resolution for a sequence of propositional formulas describing the pigeonhole principle [Hak85]. In the last two decades these lower bounds were extended to a number of further propositional systems such as the

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The Complexity of Propositional Proofs

Cited by 173 publications

References 138 publications

Lower Bounds for DNF-Refutations of a Relativized Weak Pigeonhole Principle

Lower Bounds for DNF-Refutations of a Relativized Weak Pigeonhole Principle

A Game Characterisation of Tree-like Q-resolution Size

Theory and Applications of Models of Computation

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