The Computational Complexity of the Satisfiability of Modal Horn Clauses for Modal Propositional Logics

This paper presents complexity results about the satisfiability ofmodal Horn clauses for several modal propositional logics. Almost all these results are negative in the sense that restricting the input formula to modal Horn clauses does not decrease the inherent complexity of the satisfiability problem. We first show that, when restricted to modal Horn clauses, the satisfiability problem for any modal logic between K and S4 or between K and B is PSPACE-hard. As a result, the satisfiability of modal Horn clauses as well as the satisfiability of unrestricted formulas for any of K, T, B and S4 is PSPACEcomplete. This result refutes the expectation (Fariiias de1 Cerro and Penttonen 1987) of getting a polynomial-time algorithm for the satisfiability of modal Horn clauses for these logics as long as P # PSPACE. Next, we consider S4.3 and extensions of K5 including K5, KD5, K45, KD45 and S5, the satisfiability problem for each of which in general is known to be NP-complete, and show that for each extension of K5, a polynomial-time algorithm for the satisfiability of modal Horn clauses can be obtained; but for S4.3, together with some linear tense logics closely related to S4.3 like CL, SL and PL, the satisfiability of modal Horn clauses still remains NP-complete.