Modal satisifiability in a constraint logic environment

Abstract

The modal satisfiability problem has to date been solved using either a specifically designed algorithm, or by translating the modal logic formula into a different class of problem, such as a first-order logic, a propositional satisfiability problem or a constraint satisfaction problem. These approaches and the solvers developed to support them are surveyed and a synthesis thereof is presented. The translation of a modal K formula into a constraint satisfaction problem, as developed by Brand et al. [18], is further enhanced. The modal formula, which must be in conjunctive normal form, is translated into layered propositional formulae. Each of these layers is translated into a constraint satisfaction problem and solved using the constraint solver ECLiPSe. I extend this translation to deal with reflexive and transitive accessibility relations, thereby providing for the modal logics KT and S4. Two of the difficulties that arise when these accessibility relations are added are that the resultant formula increases considerably in complexity, and that it is no longer in conjunctive normal form (CNF). I eliminate the need for the conversion of the formula to CNF and deal instead with formulae that are in negation normal form (NNF). I apply a number of enhancements to the formula at each modal layer before it is translated into a constraint satisfaction problem. These include extensive simplification, the assignment of a single value to propositional variables that occur only positively or only negatively, and caching the status of the formula at each node of the search tree. All of these significantly prune the search space. The final results I achieve compare favorably with those obtained by other solvers.