Pool Resolution and Its Relation to Regular Resolution and DPLL with Clause Learning

Abstract

Pool Resolution for propositional CNF formulas is introduced. Its relationship to state-of-the-art satisfiability solvers is explained. Every regular-resolution derivation is also a pool-resolution derivation. It is shown that a certain family of formulas, called NT^**(n) has polynomial sized pool-resolution refutations, whereas the shortest regular refutations have an exponential lower bound. This family is a variant of the GT(n) family analyzed by Bonet and Galesi (FOCS 1999), and the GT’n family shown to require exponential-length regular-resolution refutations by Alekhnovitch, Johannsen, Pitassi and Urquhart (STOC 2002). Thus, Pool Resolution is exponentially stronger than Regular Resolution. Roughly speaking a general-resolution derivation is a pool-resolution derivation if its directed acyclic graph (DAG) has a depth-first search tree that satisfies the regularity restriction: on any path in this tree no resolution variable is repeated. In other words, once a clause is derived at a node and used by its tree parent, its derivation is forgotten, and subsequent uses of that clause treat it as though it were an input clause. This policy is closely related to DPLL search with recording of so-called conflict clauses. Variations of DPLL plus conflict analysis currently dominate the field of high-performance satisfiability solving. The power of Pool Resolution might provide some theoretical explanation for their success.