Partial Implicit Unfolding in the Davis-Putnam Procedure for Quantified Boolean Formulae

Abstract

Quantified Boolean formulae offer a means of representing many propositional formula exponentially more compactly than propositional logic. Recent work on automating reasoning with QBF has concentrated on extending the Davis-Putnam procedure to handle QBF. Although the resulting procedures make it possible to evaluate QBF that could not be efficiently reduced to propositional logic (requiring worst-case exponential space), its efficiency often lags much behind the reductive approach when the reduction is possible. We attribute this inefficiency to the fact that many of the unit resolution steps possible in the reduced (propositional logic) formula are not performed in the corresponding QBF. To combine the conciseness of the QBF representation and the stronger inferences available in the unquantified representation, we introduce a stronger propagation algorithm for QBF which could be seen as partially unfolding the universal quantification. The algorithm runs in worst-case exponential time, like the reduction of QBF to propositional logic, but needs only polynomial space. By restricting the algorithm the exponential behavior can be avoided while still preserving many of the useful inferences.