A New System and Methodology for Generating Random Modal Formulae

Previous methods for generating random modal formulae (for the multi-modal logic K (m)) result either in awed test sets or for-mulae that are too hard for current modal decision procedures and, also, unnatural. We present here a new system and generation methodology which results in unnawed test sets and more-natural formulae that are better suited for current decision procedures. Most empirical testing of decision procedures for propositional modal logics, usually for the multi-modal logic K (m) , employs randomly generated formulae. This style of testing was initially proposed by Giunchiglia et al 3] and later improved by them and also by Hustadt and Schmidt 6, 1]. Other kinds of randomly generated formulae have been proposed by Massacci 7]. Randomly generated formulae have been used with all the recent, highly optimised modal decision procedures, including DLP 8], FaCT 4], KSatC 1], *SAT 2], and TA 6], and have been used in several comparisons of these systems 7]. The basic idea behind the random generator in 3, 6, 1] is to generate a number of clauses. Each clause has the same number of disjuncts. As most testing as been on clauses with three disjuncts, the generated formulae are often called 3CNF 2m formulae. Each disjunct in each clause is independently either a possibly negated propositional variable or a modal formula consisting of a possibly negated box over a clause. The embedded clauses are generated in the same way as the top-level clauses, except that at some maximum modal depth all the disjuncts are possibly negated propositional variables. Care is taken to not repeat disjuncts in a clause, nor to have complementary disjuncts in a clause. Testing consists of setting various parameters, usually the maximum modal depth, d, the probability that a disjunct is a propositional literal, p, and the number of propositional variables, N. Most tests use standard values for the other parameters, namely three disjuncts in a clause, C, and only one kind of modal box, m. Then several values are picked for the number of top-level clauses in a formula, L, and for each value for L many (usually about 100) formulae are generated and tested for satissability. This testing methodology as described above provides a decent test for current modal decision procedures, allowing good comparisons of their behaviour.