A satisfiability procedure for quantified Boolean formulae

Abstract

We present a satisfiability tester Q SAT for quantified Boolean formulae and a restriction Q SAT CNF of Q SAT to unquantified conjunctive normal form formulae. Q SAT makes use of procedures which replace subformulae of a formula by equivalent formulae. By a sequence of such replacements, the original formula can be simplified to true or false . It may also be necessary to transform the original formula to generate a subformula to replace. Q SAT CNF eliminates collections of variables from an unquantified clause form formula until all variables have been eliminated. Q SAT and Q SAT CNF can be applied to hardware verification and symbolic model checking. Results of an implementation of Q SAT CNF are described, as well as some complexity results for Q SAT and Q SAT CNF . Q SAT runs in linear time on a class of quantified Boolean formulae related to symbolic model checking. We present the class of “long and thin” unquantified formulae and give evidence that this class is common in applications. We also give theoretical and empirical evidence that Q SAT CNF is often faster than Davis and Putnam-type satisfiability checkers and ordered binary decision diagrams (OBDDs) on this class of formulae. We give an example where Q SAT CNF is exponentially faster than BDDs.