Quantifier Reordering for QBF

Abstract

State-of-the-art procedures for evaluating quantified Boolean formulas often expect input formulas in prenex conjunctive normal form (PCNF). We study dependency schemes as a means of reordering the quantifier prefix of a PCNF formula while preserving its truth value. Dependency schemes associate each formula with a binary relation on its variables (the dependency relation) that imposes constraints on certain operations manipulating the formula's quantifier prefix. We prove that known dependency schemes support a stronger reordering operation than was previously known. We present an algorithm that, given a formula and its dependency relation, computes a compatible reordering with a minimum number of quantifier alternations. In combination with a dependency scheme that can be computed in polynomial time, this yields a polynomial time heuristic for reducing the number of quantifier alternations of an input formula. The resolution-path dependency scheme is the most general dependency scheme introduced so far. Using an interpretation of resolution paths as directed paths in a formula's implication graph, we prove that the resolution-path dependency relation can be computed in polynomial time.