Propositional proof systems based on maximum satisfiability

Abstract The paper describes the use of dual-rail MaxSAT systems to solve Boolean satisfiability (SAT), namely to determine if a set of clauses is satisfiable. The MaxSAT problem is the problem of satisfying the maximum number of clauses in an instance of SAT. The dual-rail encoding adds extra variables for the complements of variables, and allows encoding an instance of SAT as a Horn MaxSAT problem. We discuss three implementations of dual-rail MaxSAT: core-guided systems, minimal hitting set (MaxHS) systems, and MaxSAT resolution inference systems. All three of these can be more efficient than resolution and thus than conflict-driven clause learning (CDCL). All three systems can give polynomial size refutations for the pigeonhole principle, the doubled pigeonhole principle and the mutilated chessboard principles. The dual-rail MaxHS MaxSat system can give polynomial size proofs of the parity principle. However, dual-rail MaxSAT resolution requires exponential size proofs for the parity principle; this is proved by showing that constant depth Frege augmented with the pigeonhole principle can polynomially simulate dual-rail MaxSAT resolution. Consequently, dual-rail MaxSAT resolution does not simulate cutting planes. We further show that core-guided dual-rail MaxSAT and weighted dual-rail MaxSAT resolution polynomially simulate resolution. Finally, we report the results of experiments with core-guided dual-rail MaxSAT and MaxHS dual-rail MaxSAT showing strong performance by these systems.

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