Q-Resolution with Generalized Axioms

Abstract

Abstract. Q-resolution is a proof system for quantified Boolean formulas (QBFs) in prenex conjunctive normal form (PCNF) which underlies search-based QBF solvers with clause and cube learning (QCDCL). With the aim to derive and learn stronger clauses and cubes earlier in the search, we generalize the axioms of the Q-resolution calculus resulting in an exponentially more powerful proof system. The generalized axioms introduce an interface of Q-resolution to any other QBF proof system allowing for the direct combination of orthogonal solving techniques. We implemented a variant of the Q-resolution calculus with generalized axioms in the QBF solver DepQBF. As two case studies, we apply integrated SAT solving and resource-bounded QBF preprocessing during the search to heuristically detect potential axiom applications. Experiments with application benchmarks indicate a substantial performance improvement.

DepQBF 6.0: A Search-Based QBF Solver Beyond Traditional QCDCL

“…DepQBF 6.0 implements a variant of QCDCL that is based on a generalization of the Q-resolution calculus (QRES). The generalization is achieved by equipping QRES with generalized clause and cube axioms to be used in clause and cube learning [31]. The generalized axioms provide an extensible framework of interfaces for the integration of arbitrary QBF proof systems in QRES, and hence in QCDCL.…”

“…1 In contrast to traditional QCDCL based on QRES [14,21,24,48], DepQBF 6.0 implements a variant of QCDCL that relies on a generalization of QRES. This generalization is due to a set of new axioms added to QRES [31]. In practice, derivations made by the added axioms in QCDCL are based on arbitrary QBF proof systems.…”

A Duality-Aware Calculus for Quantified Boolean Formulas

“…While there is some work on relating proof systems (e.g., [18]), the actual search strategies have not been compared yet. Especially when different techniques are integrated as currently proposed in [19], a better understanding of the individual solving techniques is indispensable. APPENDIX Example 1.…”

Shortening QBF Proofs with Dependency Schemes

“…In this way, the system shares similarities with 'Q-resolution with generalised axioms' [20]. Regarding proof complexity, a drawback of that calculus is that every false formula may be refuted in a single step.…”

“…In line with [20], we could have allowed assignments due to unit propagation and pure literal elimination in dyn-Q(D)-Res. This would allow additional existential literals to be included in a D-assignment provided that they are valid assignments under Boolean constraint propagation.…”