Hardness measures and resolution lower bounds

Various “hardness” measures have been studied for resolution, providing theoretical insight into the proof complexity of resolution and its fragments, as well as explanations for the hardness of instances in SAT solving. In this report we aim at a unified view of a number of hardness measures, including different measures of width, space and size of resolution proofs. We also extend these measures to all clause-sets (possibly satisfiable). One main contribution is a unified game-theoretic characterisation of these measures. We obtain new relations between the different hardness measures. In particular, we prove a generalised version of Atserias and Dalmau’s result on the relation between resolution width and space from [3]. As an application, we study hardness of PHP and variations, considering also satisfiable PHP. Especially we consider EPHP, the extension of PHP by Cook ([19]) which yields polynomial-size resolution refutations. Another application is to XOR-principles.