Journal of Logic and Computation, Volume 11, Issue 4, pp. 609-622: Abstract.
The Complexity of Poor Man's Logic
Department of Computer Science, Rochester Institute of Technology, 102 Lomb Memorial Drive, Rochester, NY 14623-5608, USA. E-mail: email@example.com
Motivated by description logics, we investigate what happens to the complexity of modal satisfiability problems if we only allow formulas built from literals, [and], [loz], and [square]. Previously, the only known result was that the complexity of the satisfiability problem for K drops from PSPACE-complete to coNP-complete. In this paper we show that not all modal logics behave like K. In particular, we show that the complexity of the satisfiability problem with respect to frames in which each world has at least one successor drops from PSPACE-complete to P, but that in contrast the satisfiability problem with respect to the class of frames in which each world has at most two successors remains PSPACE-complete. As a corollary of the latter result, we also solve the open problem from the complexity classification of description logics. In the last section, we classify the complexity of the satisfiability problem for K for all other restrictions on the set of operators.
Keywords: Modal logic, computational complexity, description logic