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A (Biased) Proof Complexity Survey for SAT Practitioners
"... This talk is intended as a selective survey of proof complexity, focusing on some comparatively weak proof systems that are of particular interest in connection with SAT solving. We will review resolution, polynomial calculus, and cutting planes (related to conflictdriven clause learning, Gröbne ..."
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This talk is intended as a selective survey of proof complexity, focusing on some comparatively weak proof systems that are of particular interest in connection with SAT solving. We will review resolution, polynomial calculus, and cutting planes (related to conflictdriven clause learning, Gröbner basis computations, and pseudoBoolean solvers, respectively) and some proof complexity measures that have been studied for these proof systems. We will also briefly discuss if and how these proof complexity measures could provide insights into SAT solver performance.
From Small Space to Small Width in Resolution
"... In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of formulas is always an upper bound on the width needed to refute them. Their proof is beautiful but somewhat mysterious in that it relies heavily on tools from fi ..."
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In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of formulas is always an upper bound on the width needed to refute them. Their proof is beautiful but somewhat mysterious in that it relies heavily on tools from finite model theory. We give an alternative, completely elementary, proof that works by simple syntactic manipulations of resolution refutations. As a byproduct, we develop a “blackbox ” technique for proving space lower bounds via a “static ” complexity measure that works against any resolution refutation—previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similar methods.
An Ultimate TradeOff in Propositional Proof Complexity
, 2015
"... We exhibit an unusually strong tradeoff between resolution proof width and treelike proof size. Namely, we show that for any parameter k = k(n) there are unsatisfiable kCNFs that possess refutations of width O(k), but such that any treelike refutation of width n1−/k must necessarily have double ..."
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We exhibit an unusually strong tradeoff between resolution proof width and treelike proof size. Namely, we show that for any parameter k = k(n) there are unsatisfiable kCNFs that possess refutations of width O(k), but such that any treelike refutation of width n1−/k must necessarily have double exponential size exp(nΩ(k)). Conceptually, this means that there exist contradictions that allow narrow refutations, but so that in order to keep the size of a narrow refutation even within a single exponent, it must necessarily use a high degree of parallelism. Viewed differently, every treelike narrow refutation is exponentially worse not only than wide refutations of the same contradiction, but of any other contradiction with the same number of variables. This seems to significantly deviate from the established pattern of most, if not all, tradeoff results in complexity theory. Our construction and proof methods combine, in a nontrivial way, two previously known techniques: the hardness escalation method based on substitution formulas and expansion. This combination results in a hardness compression approach that strives to preserve hardness of a contradiction while significantly decreasing the number of its variables.
On the Interplay Between Proof Complexity and SAT Solving
, 2015
"... This paper is intended as an informal and accessible survey of proof complexity for nonexperts, focusing on some comparatively weak proof systems of particular interest in connection with SAT solving. We review resolution, polynomial calculus, and cutting planes (related to conflictdriven clause ..."
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This paper is intended as an informal and accessible survey of proof complexity for nonexperts, focusing on some comparatively weak proof systems of particular interest in connection with SAT solving. We review resolution, polynomial calculus, and cutting planes (related to conflictdriven clause learning, Gröbner basis computations, and pseudoBoolean solving, respectively) and some complexity measures that have been studied for these proof systems. We also discuss briefly to what extent proof complexity could provide insights into SAT solver performance, and how concerns related to applied SAT solving can give rise to interesting complexitytheoretic questions. Along the way, we highlight a number of current research challenges.