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

Atserias, Albert; Müller, Moritz; Oliva, Sergi

doi:10.1109/ccc.2013.20

Cited by 2 publications

References 26 publications

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Narrow Proofs May Be Maximally Long

Atserias

Lauria

Nordström

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n Ω(w) . This shows that the simple counting argument that any formula refutable in width w must have a proof in size n O(w) is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. Our results do not extend all the way to Lasserre, however, where the formulas we study have proofs of constant rank and size polynomial in both n and w. IntroductionProof complexity studies how hard it is to prove that propositional logic formulas are tautologies. While the original motivation for this line of research, as discussed in [CR79], was to prove superpolynomial lower bounds on proof size for increasingly stronger proof systems as a way towards establishing NP = co-NP (and hence P = NP), it is probably fair to say that most current research in proof complexity is driven by other concerns.One such concern is the connection to SAT solving. By a standard transformation any propositional logic formula can be converted to another formula in conjunctive normal form (CNF) that has the same size up to constant factors and is unsatisfiable if and only if the original formula is a tautology. Any algorithm for solving SAT defines a proof system in the sense that the execution trace of the algorithm constitutes a polynomial-time verifiable witness of unsatisfiability. 1 In fact, most modern-day SAT solvers can be seen to search for proofs in systems at fairly low levels in the proof complexity hierarchy, and upper and lower bounds for these proof systems hence give information about the potential and limitations of the corresponding SAT solvers. In this work, we focus on such proof systems. BackgroundThe dominant strategy in applied SAT solving today is so-called conflict-driven clause learning (CDCL) [BS97, MS99, MMZ + 01], which is ultimately based on the resolution proof system [Bla37]. The most studied complexity measure for resolution is size (also referred to as * This is the full-length version of the paper [ALN14], which appeared in Proceedings of the 29th Annual IEEE Conference on Computational Complexity (CCC '14).1 Such a witness is often referred to as a refutation rather than a proof , and these two terms are sometimes used interchangeably.

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