A satisfiability procedure for quantified Boolean formulae Original Research Article

We present a satisfiability tester QSAT for quantified Boolean formulae and a restriction QSATCNF of QSAT to unquantified conjunctive normal form formulae. QSAT 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. QSATCNF eliminates collections of variables from an unquantified clause form formula until all variables have been eliminated. QSAT and QSATCNF can be applied to hardware verification and symbolic model checking. Results of an implementation of QSATCNF are described, as well as some complexity results for QSAT and QSATCNF. QSAT 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 QSATCNF 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 QSATCNF is exponentially faster than BDDs.