Conference Paper

A Combinatorial Characterization of ResolutionWidth.

Univ. Politecnica de Catalunya, Barcelona, Spain
Conference: 18th Annual IEEE Conference on Computational Complexity (Complexity 2003), 7-10 July 2003, Aarhus, Denmark
Source: DBLP


Abstract We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result is a surprising proof that the minimum,space of refuting a 3-CNF formula is always bounded,from below by the minimum width of refuting it (minus 3). This solves a well-known open problem. The second application is the unification of several width lower bound arguments, and a new width lower bound for the dense linear order principle. Since we also show that this principle has resolution refutations of polynomial size, this provides yet another example showing that the relationship between size and width cannot be made subpolynomial.

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Available from: Víctor Dalmau, Dec 11, 2014
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    ABSTRACT: We define a collection of Prover-Delayer games to characterise some subsystems of propositional resolution. We give some natural criteria for the games which guarantee lower bounds on the resolution width. By an adaptation of the size-width tradeoff for resolution of [10] this result also gives lower bounds on proof size. We also use games to give upper bounds on proof size, and in particular describe a good strategy for the Prover in a certain game which yields a short refutation of the Linear Ordering principle. Using previous ideas we devise a new algorithm to automatically generate resolution refutations. On bounded width formulas, our algorithm is as least as good as the width based algorithm of [10]. Moreover, it finds short proofs of the Linear Ordering principle when the variables respect a given order. Finally we approach the question of proving that a formula F is hard to refute if and only if is “almost” satisfiable. We prove results in both directions when “almost satisfiable” means that it is hard to distuinguish F from a satisfiable formula using limited pebbling games.
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