Characterizing the NP-PSPACE gap in the satisfiability problem for modal logic

Authors:
Joseph Y. Halpern;Leandro Chaves Rêgo
Affiliations:
Computer Science Department, Cornell University;Statistics Department, Federal University of Pernambuco, Brazil
Venue:
IJCAI'07 Proceedings of the 20th international joint conference on Artifical intelligence
Year:
2007

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Abstract

There has been a great deal of work on characterizing the complexity of the satisfiability and validity problem for modal logics. In particular, Ladner showed that the satisfiability problem for all logics between K and S4 is PSPACE-hard, while for S5 it is NP-complete. We show that it is negative introspection, the axiom ¬Kp ⇒ K¬Kp, that causes the gap: if we add this axiom to any modal logic between K and S4, then the satisfiability problem becomes NP-complete. Indeed, the satisfiability problem is NP-complete for any modal logic that includes the negative introspection axiom.