Algorithms & Complexity Results for Input & Unit Resolution

In this paper we explore the complexity of various problems pertaining to Input Resolution, a very restricted form of Resolution which requires that at least one input to every instance of the resolution rule be an input clause. This proof system is of practical interest because Input Resolution is equivalent to Unit Resolution, a popular subroutine used in many areas of computer science, especially SAT-solvers. We first prove a number of tractability results for Input Resolution, which leads us to algorithms showing its automatizability, as well as the NP-Completeness of the Input Resolution size problem. However, our main result is an exact equivalence between the pure black pebbling number of any DAG G and the Input Resolution total space of the pebbling contradiction formula Peb(G). This equivalence allows us to prove the PSPACE-Completeness of various forms of the Input Resolution derivation total space problem, and as a corollary yield the PSPACE-Completeness of the Input derivation width problem as well as a massive size / total space tradeoff for Input Resolution. These PSPACE-Completeness and tradeoff results are somewhat surprising when juxtaposed with our earlier theorems; on one hand, Input Resolution seems to be a completely tractable and even trivial Resolution refinement, but on the other hand some of its other characteristics are highly complicated.