Certified DQBF Solving by Definition Extraction

Abstract

We propose a new decision procedure for dependency quantified Boolean formulas (DQBFs) that uses interpolation-based definition extraction to compute Skolem functions in a counter-example guided inductive synthesis (CEGIS) loop. In each iteration, a family of candidate Skolem functions is tested for correctness using a SAT solver, which either determines that a model has been found, or returns an assignment of the universal variables as a counterexample. Fixing a counterexample generally involves changing candidates of multiple existential variables with incomparable dependency sets. Our procedure introduces auxiliary variables-which we call arbiter variables-that each represent the value of an existential variable for a particular assignment of its dependency set. Possible repairs are expressed as clauses on these variables, and a SAT solver is invoked to find an assignment that deals with all previously seen counterexamples. Arbiter variables define the values of Skolem functions for assignments where they were previously undefined, and may lead to the detection of further Skolem functions by definition extraction.A key feature of the proposed procedure is that it is certifying by design: for true DQBF, models can be returned at minimal overhead. Towards certification of false formulas, we prove that clauses can be derived in an expansion-based proof system for DQBF. In an experimental evaluation on standard benchmark sets, a prototype implementation was able to match (and in some cases, surpass) the performance of state-of-the-art-solvers. Moreover, models could be extracted and validated for all true instances that were solved.