The Complexity of Poor Man's Logic

Details The Complexity of Poor Man's Logic The Complexity of Poor Man's Logic

1999-11-28

arxiv.org/abs/cs/9911014v2

Journal of Logic and Computation, 11(4), 609--622, 2001. Extended abstract in STACS 2000

Computer Science

Logic in Computer Science

Corrected version of the journal version. The changes are in Section 6, where Theorem 6.3(2) was added to handle two missing c

Scientific paper

Motivated by description logics, we investigate what happens to the complexity of modal satisfiability problems if we only allow formulas built from literals, $\wedge$, $\Diamond$, and $\Box$. Previously, the only known result was that the complexity of the satisfiability problem for K dropped from PSPACE-complete to coNP-complete (Schmidt-Schauss and Smolka, 1991 and Donini et al., 1992). 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 Donini et al.'s complexity classification of description logics (Donini et al., 1997). In the last section, we classify the complexity of the satisfiability problem for K for all other restrictions on the set of operators.

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