Decidability and Complexity via Mosaics of the Temporal Logic of the Lexicographic Products of Unbounded Dense Linear Orders

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8152)


This article considers the temporal logic of the lexicographic products of unbounded dense linear orders and provides via mosaics a complete decision procedure in nondeterministic polynomial time for the satisfiability problem it gives rise to.


Linear temporal logic lexicographic product satisfiability problem decidability complexity mosaic method decision procedure 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balbiani, P.: Time representation and temporal reasoning from the perspective of non-standard analysis. In: Brewka, G., Lang, J. (eds.) Eleventh International Conference on Principles of Knowledge Representation and Reasoning, pp. 695–704. AAAI (2008)Google Scholar
  2. 2.
    Balbiani, P.: Axiomatization and completeness of lexicographic products of modal logics. In: Ghilardi, S., Sebastiani, R. (eds.) FroCoS 2009. LNCS, vol. 5749, pp. 165–180. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Balbiani, P.: Axiomatizing the temporal logic defined over the class of all lexicographic products of dense linear orders without endpoints. In: Markey, N., Wijsen, J. (eds.) Temporal Representation and Reasoning, pp. 19–26. IEEE (2010)Google Scholar
  4. 4.
    Van Benthem, J.: The Logic of Time. Kluwer (1991)Google Scholar
  5. 5.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press (2001)Google Scholar
  6. 6.
    Bull, R.: That all normal extensions of S4.3 have the finite model property. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 12, 314–344 (1966)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Caleiro, C., Viganò, L., Volpe, M.: On the mosaic method for many-dimensional modal logics: a case study combining tense and modal operators. Logica Universalis 7, 33–69 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Euzenat, J., Montanari, A.: Time granularity. In: Fisher, M., Gabbay, D., Vila, L. (eds.) Handbook of Temporal Reasoning in Artificial Intelligence, pp. 59–118. Elsevier (2005)Google Scholar
  9. 9.
    Fine, K.: The logics containing S4.3. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 17, 371–376 (1971)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gabbay, D., Kurucz, A., Wolter, F., Zakharyaschev, M.: Many-Dimensional Modal Logics: Theory and Applications. Elsevier (2003)Google Scholar
  11. 11.
    Gabbay, D., Shehtman, V.: Products of modal logics, part 1. Logic Journal of the IGPL 6, 73–146 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gagné, J.-R., Plaice, J.: A nonstandard temporal deductive database system. Journal of Symbolic Computation 22, 649–664 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kracht, M., Wolter, F.: Properties of independently axiomatizable bimodal logics. Journal of Symbolic Logic 56, 1469–1485 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kurucz, A.: Combining modal logics. In: Blackburn, P., van Benthem, J., Wolter, F. (eds.) Handbook of Modal Logic, pp. 869–924. Elsevier (2007)Google Scholar
  15. 15.
    Litak, T., Wolter, F.: All finitely axiomatizable tense logics of linear time flows are CoNP-complete. Studia Logica 81, 153–165 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Marx, M., Mikulás, S., Reynolds, M.: The mosaic method for temporal logics. In: Dyckhoff, R. (ed.) TABLEAUX 2000. LNCS (LNAI), vol. 1847, pp. 324–340. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  17. 17.
    Marx, M., Venema, Y.: Local variations on a loose theme: modal logic and decidability. In: Grädel, E., Kolaitis, P., Libkin, L., Marx, M., Spencer, J., Vardi, M., Venema, Y., Weinstein, S. (eds.) Finite Model Theory and its Applications, pp. 371–429. Springer (2007)Google Scholar
  18. 18.
    Nakamura, K., Fusaoka, A.: Reasoning about hybrid systems based on a nonstandard model. In: Orgun, M.A., Thornton, J. (eds.) AI 2007. LNCS (LNAI), vol. 4830, pp. 749–754. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Németi, I.: Decidable versions of first order logic and cylindric-relativized set algebras. In: Csirmaz, L., Gabbay, D., de Rijke, M. (eds.) Logic Colloquium 1992, pp. 171–241. CSLI Publications (1995)Google Scholar
  20. 20.
    Reynolds, M.: A decidable temporal logic of parallelism. Notre Dame Journal of Formal Logic 38, 419–436 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Reynolds, M., Zakharyaschev, M.: On the products of linear modal logics. Journal of Logic and Compution 11, 909–931 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wolter, F.: Fusions of modal logics revisited. In: Kracht, M., de Rijke, M., Wansing, H., Zakharyaschev, M. (eds.) Advances in Modal Logic, pp. 361–379. CSLI Publications (1998)Google Scholar
  23. 23.
    Zakharyaschev, M., Alekseev, A.: All finitely axiomatizable normal extensions of K4.3 are decidable. Mathematical Logic Quarterly 41, 15–23 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institut de recherche en informatique de ToulouseCNRS — University of ToulouseToulouse Cedex 9France
  2. 2.Department of Computer Science and Information SystemsBirbeck — University of LondonLondonUK

Personalised recommendations