A Symbolic Search Based Approach for Quantified Boolean Formulas

Abstract

Solving Quantified Boolean Formulas (QBF) has become an important and attractive research area, since several problem classes might be formulated efficiently as QBF instances (e.g. planning, non monotonic reasoning, two-player games, model checking, etc). Many QBF solvers has been proposed, most of them perform decision tree search using the DPLL-like techniques. To set free the variable ordering heuristics that are traditionally constrained by the static order of the QBF quantifiers, a new symbolic search based approach (QBdd(Sat)) is proposed. It makes an original use of binary decision diagram to represent the set of models (or prime implicants) of the boolean formula found using search-based satisfiability solver. Our approach is enhanced with two interesting extensions. First, powerful reduction operators are introduced in order to dynamically reduce the BDD size and to answer the validity of the QBF. Second, useful cuts are achieved on the search tree thanks to the nogoods generated from the BDD representation. Using DPLL-likes (resp. local search) techniques, our approach gives rise to a complete QBdd(DPLL) (resp. incomplete QBdd(LS)) solver. Our preliminary experimental results show that on some classes of instances from the QBF evaluation, QBdd(DPLL) and QBdd(LS) are competitive with state-of-the-art QBF solvers.