Tableaux with Dynamic Filtration for Layered Modal Logics

  • Authors:

  • Olivier Gasquet;Bilal Said

  • Affiliations:

  • Université Paul Sabatier, IRIT - LILaC, 118 route de Narbonne, F-31062 Toulouse cedex 9, France;Université Paul Sabatier, IRIT - LILaC, 118 route de Narbonne, F-31062 Toulouse cedex 9, France

  • Venue:
  • TABLEAUX '07 Proceedings of the 16th international conference on Automated Reasoning with Analytic Tableaux and Related Methods
  • Year:
  • 2007

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Abstract

In this paper we prove that the satisfiability problem for the class of what we call layered modal logics (LML) is in NEXPTIME, and hence, is decidable. Roughly, LML are logics characterized by semantical properties only stating the existence of possible worlds that are in some sense "further" than the other. Typically, they include various confluence-like properties, while they do not include density-like properties. Such properties are of interest for formalizing the interaction between dynamic and epistemic modalities for rational agents for example. That these logics are decidable may be not very surprising, but we show that they are all in NEXPTIME, some of them being known to be NEXPTIME-complete. For this, we give a sound and complete tableau calculus and prove that open tableaux are of exponential size. This cannot be done by using usual filtration which cannot cope with confluence. We introduce here a new technique we call dynamic filtrationthat allows to filtrate worlds one layer at a time keeping the total number of nodes within an exponential bound.