Computing Smallest MUSes of Quantified Boolean Formulas

Andreas Niskanen, Jere Mustonen, Jeremias Berg, Matti Järvisalo

Abstract: Computing small (subset-minimal or smallest) explanations is a computationally challenging task for various logics and non-monotonic formalisms. Arguably the most progress in practical algorithms for computing explanations has been made for propositional logic in terms of minimal unsatisfiable subsets (MUSes) of conjunctive normal form formulas. In this work, we propose an approach to computing smallest MUSes of quantified Boolean formulas (QBFs), building on the so-called implicit hitting set approach and modern QBF solving techniques. Connecting to non-monotonic formalisms, our approach finds applications in the realm of abstract argumentation in computing smallest strong explanations of acceptance and rejection. Justifying our approach, we pinpoint the complexity of deciding the existence of small MUSes for QBFs with any fixed number of quantifier alternations. We empirically evaluate the approach on computing strong explanations in abstract argumentation frameworks as well as benchmarks from recent QBF Evaluations.