Improved Conflict-Clause Minimization Leads to Improved Propositional Proof Traces

Recent empirical results show that recursive, or expensive, conflict-clause minimization is quite beneficial on industrial-style propositional satisfiability problems. The details of this procedure appear to be unpublished to date, but may be found in the open-source code of MiniSat 2.0, for example. Biere reports that proof traces are made more complicated when conflict-clause minimization is used because some clauses need to be resolved upon multiple times during the minimization procedure as found in MiniSat 2.0. Biere proposes a proof-trace format in which the set of clause numbers needed for a certain derivation is given, but their order is not specified. This paper presents a new procedure for conflict-clause minimization that is slightly more efficient and, more importantly, discovers a correct order so that each clause used for the derivation is resolved upon only once. This permits the proof trace to specify the order in which to use the clauses, greatly reducing the burden on software that processes the proof trace. The method is validated on the unsatisfiable formulas used for industrial benchmarks in the verified-unsatisfiable track of the SAT 2007 competition.