Relativization makes contradictions harder for Resolution

Dantchev, Stefan; Martin, Barnaby

doi:10.1016/j.apal.2013.10.009

Cited by 3 publications

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References 24 publications

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“…For the ordering principle, we can encode that there exists a non-empty ordered subset S ⊆ T of arbitrary size such that it is possible for all elements in S to find a smaller element inside S. This relativization step transforms the previously very easy ordering principle formulas into relativized versions that are exponentially hard for resolution [Dan06,DM14]. For the PHP formulas, we specify that we have a set of M ≫ m pigeons mapped into into n < m holes such that there exists a subset of m pigeons that are mapped injectively.…”

Section: Any Sums-of-squares Refutation Of K-clique(g) Requires Degre...mentioning

confidence: 99%

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

Lauria

Nordström²

2017

comput. complex.

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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 Ω(d) for values of d = d(n) from constant all the way up to n δ for some universal constant δ. 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 NP-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in [Krajíček '04] and [Dantchev and Riis '03], and then applying a restriction argument as in [Atserias, Müller, and Oliva '13] and [Atserias, Lauria, and Nordström '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. * This is the full-length version of the paper with the same title to appear in Proceedings of the 30th Annual Computational Complexity Conference (CCC '15). 1 All proofs for systems of polynomial equations or for formulas in conjunctive normal form (CNF) in this paper will be proofs of unsatisfiability, and we will therefore use the two terms "proof" and "refutation" interchangeably.2 This is sometimes also referred to as the "rank," but we will stick to the term "degree" in this paper.

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Section: Any Sums-of-squares Refutation Of K-clique(g) Requires Degre...mentioning

confidence: 99%

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

Lauria

Nordström²

2017

comput. complex.

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The treewidth of proofs

Müller

Szeider

2017

Information and Computation

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Narrow Proofs May Be Maximally Long

Atserias

Lauria

Nordström

2016

ACM Trans. Comput. Logic

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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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Relativization makes contradictions harder for Resolution

Cited by 3 publications

References 24 publications

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

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

The treewidth of proofs

Narrow Proofs May Be Maximally Long

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