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Journal of Logic and Computation 2007 17(4):795-806; doi:10.1093/logcom/exm029
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© The Author, 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Articles

Characterizing the NP-PSPACE Gap in the Satisfiability Problem for Modal Logic

Joseph Y. Halpern

Computer Science Department, Cornell University, USA. E-mail: halpern{at}cs.cornell.edu

Leandro Chaves Rêgo

Statistics Department, Federal University of Pernambuco, Brazil. E-mail: leandro{at}de.ufpe.br

Received 23 February 2007.


  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 {Rightarrow} 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.

Keywords: Euclidean Property; Negative Introspection; K5; S5; PSPACE; NP; Knowledge Representation


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