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 filtration that allows to filtrate worlds one layer at a time keeping the total number of nodes within an exponential bound.
This work has been partially supported by the project ARROWS of the French Agence Nationale de la Recherche.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Baldoni, M.: Normal Multimodal Logics With Interaction Axioms. In: Basin, D., D’Agostino, M., Gabbay, D.M., Matthews, S., Viganò, L. (eds.) Labelled Deduction. Applied Logic Series, vol. 17, pp. 33–53. Kluwer Academic Publisher, Boston (2000)
Chellas, B.F.: Modal Logic: an introduction. Cambridge Univ. Press, Cambridge (1980)
Castilho, M.A., del Farinas Cerro, L., Gasquet, O., Herzig, A.: Modal Tableaux with Propagation Rules and Structural Rules. Fundamenta Informaticae 32(3-4), 281–297 (1997)
del Fariñas Cerro, L., Gasquet, O.: A general framework for pattern-driven modal tableaux. Logic Journal of the IGPL 10(1), 51–84 (2002)
Del Fariñas Cerro, L., Gasquet, O.: Tableaux Based Decision Procedures for Modal Logics of Confluence and Density. Fundamenta Informaticae 40(4), 317–333 (1999)
Gabbay, D.M.: Selective filtration in modal logic. Theoria 30, 323–330 (1970)
Gasquet, O.: On the influence of confluence in modal logics. Fundamenta Informaticae 70(3) (2005)
Goré, R.: Tableau methods for modal and temporal logics. In: D’Agostino, M., Gabbay, D., Hähnle, R., Posegga, J. (eds.) Handbook of Tableau Methods, Kluwer Academic Publishers, Boston (1999)
Ladner, R.: The computational complexity of provability in systems of modal logic. SIAM Journal on Computing 6, 467–480 (1977)
Lemmon, E.J., Scott, D.S.: An introduction to modal logic. Oxford, Blackwell (1977)
Massacci, F.: Single step tableaux for modal logics: methodology, computations, algorithms. Journal of Automated Reasoning 24 (2000)
Shapirovsky, I.: On PSPACE-decidability in transitive modal logics. In: Proc. AiML 2005 (2005)
Shethman, V.: Filtration via bisimulation. In: Proc. AiML 2005 (2005)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gasquet, O., Said, B. (2007). Tableaux with Dynamic Filtration for Layered Modal Logics. In: Olivetti, N. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2007. Lecture Notes in Computer Science(), vol 4548. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73099-6_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-73099-6_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73098-9
Online ISBN: 978-3-540-73099-6
eBook Packages: Computer ScienceComputer Science (R0)