Enhanced Unsatisfiable Cores for QBF: Weakening Universal to Existential Quantifiers

Abstract

We propose an enhanced notion of unsatisfiable cores for QBF in prenex CNF that weakens universal to existential quantifiers in addition to the traditional removal of clauses. We can thus obtain unsatisfiable cores that are semantically different from those obtained by the traditional notion; this gives rise to explanations - and, via hitting set duality, diagnoses - of unsatisfiability that are not provided by traditional unsatisfiable cores. We use a source-to-source transformation on QBF that reduces the weakening of universal to existential quantifiers to the removal of clauses. This enables any tool or method that can compute unsatisfiable cores of the traditional notion to also compute unsatisfiable cores of our enhanced notion. We implement our approach in the QBF solver DepQBF, and we experimentally evaluate it on a subset of QBFLIB. Several case studies illustrate that interesting information can be learned from our enhanced notion of unsatisfiable cores.