An algorithm to evaluate quantified Boolean formulae and its experimental evaluation

The high computational complexity of advanced reasoning tasks such as reasoning about knowledge and planning calls for efficient and reliable algorithms for reasoning problems harder than NP. In this paper we propose Evaluate, an algorithm for evaluating quantified Boolean formulae (QBFs). Algorithms for evaluation of QBFs are suitable for experimental analysis of problems that belong to a wide range of complexity classes, a property not easily found in other formalisms. Evaluate is a generalization of the Davis-Putnam procedure for SAT and is guaranteed to work in polynomial space. Before presenting the algorithm, we discuss several abstract properties of QBFs that we singled out to make it more efficient. We also discuss various options that were investigated about heuristics and data structures and report the main results of the experimental analysis. In particular, Evaluate is orders of magnitude more efficient than a nested backtracking procedure that resorts to a Davis-Putnam algorithm for handling the innermost set of quantifiers. Moreover, experiments show that randomly generated QBFs exhibit regular patterns such as phase transition and easy-hard-easy distribution.