Resolution and Pebbling Games

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.