Pool resolution is NP-hard to recognize

A pool resolution proof is a dag-like resolution proof which admits a depth-first traversal tree in which no variable is used as a resolution variable twice on any branch. The problem of determining whether a given dag-like resolution proof is a valid pool resolution proof is shown to be NP-complete. Propositional resolution has been the foundational method for reasoning in propositional logic, especially for forming refutations of satisfiability of set of clauses. In recent years, the most successful satisfiability testers have used the DPLL (Davis-Putnam-Logeman-Loveland) algorithm combined with clause learning, backtracking, restarts, and other techniques. (See Beame, Kautz and Sabharwal [4] for an overview of clause learning.) Pool resolution was introduced by Van Gelder [11] as an resolution-based refutation system that provides a good theoretical model for the proofs produced by real-world satisfiability testing algorithms that incorporate clause learning and backtracking. Van Gelder proved that pool resolution is exponentially stronger than regular resolution. Bacchus, Hertel, Pitassi and Van Gelder [3], building on techniques from [4], proved that pool resolution can “effectively p-simulate” full resolution; and Buss, Hoffmann, and Johannsen [6, Th. 19] gave an effective p-simulation for a system similar to pool resolution. However, it is open whether pool resolution can directly p-simulate full resolution. ∗Supported in part by NSF grant DMS-0700533.