On Satisfiability in ATL with Strategy Contexts

  • Nicolas Troquard
  • Dirk Walther
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7519)

Abstract

This paper is a study of Brihaye et al.’s ATL with strategy contexts. We focus on memory-less strategies and establish that the resulting logic is undecidable. An immediate corollary follows that the problem of satisfiability checking of every variant of ATL with strategy context introduced by Brihaye et al. is undecidable. We also relate ATL\(_{\textit{sc}}\) with memory-less strategies with ATL with explicit strategies, providing a decidable fragment.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ågotnes, T., Goranko, V., Jamroga, W.: Alternating-time temporal logics with irrevocable strategies. In: Proceedings of TARK 2007, pp. 15–24. ACM (2007)Google Scholar
  2. 2.
    Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. The Journal of the ACM 49(5), 672–713 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. In: Perspectives Mathematical Logic. Springer (1997)Google Scholar
  4. 4.
    Brihaye, T., Da Costa, A., Laroussinie, F., Markey, N.: ATL with Strategy Contexts and Bounded Memory. In: Artemov, S., Nerode, A. (eds.) LFCS 2009. LNCS, vol. 5407, pp. 92–106. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Broersen, J., Herzig, A., Troquard, N.: A STIT-Extension of ATL. In: Fisher, M., van der Hoek, W., Konev, B., Lisitsa, A. (eds.) JELIA 2006. LNCS (LNAI), vol. 4160, pp. 69–81. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Da Costa, A., Laroussinie, F., Markey, N.: ATL with strategy contexts: Expressiveness and model checking. In: Proceedings of FSTTCS 2010. LIPIcs, vol. 8, pp. 120–132. Schloss Dagstuhl – Leibniz-Zentrum fuer Informatik (2010)Google Scholar
  7. 7.
    Gabbay, D., Kurucz, A., Wolter, F., Zakharyaschev, M.: Many-dimensional modal logics: theory and applications. Studies in Logic, vol. 148. Elsevier Science (2003)Google Scholar
  8. 8.
    Gurevich, Y., Lewis, H.R.: The word problem for cancellation semigroups with zero. The Journal of Symbolic Logic 49(1), 184–191 (1984)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Halmos, P.: Algebraic logic, IV. Transactions of the AMS 86, 1–27 (1957)Google Scholar
  10. 10.
    Herzig, A., Schwarzentruber, F.: Properties of logics of individual and group agency. In: Proceedings of AiML 2008, pp. 133–149. College Publications (2008)Google Scholar
  11. 11.
    Horty, J.F.: Agency and Deontic Logic. Oxford University Press, Oxford (2001)CrossRefMATHGoogle Scholar
  12. 12.
    Johnson, J.: Nonfinitizability of classes of representable polyadic algebras. The Journal of Symbolic Logic 34(3), 344–352 (1969)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kurucz, A.: S5 x S5 x S5 lacks the finite model property. In: Proceedings of AiML 2002, pp. 321–327. World Scientific (2002)Google Scholar
  14. 14.
    Laroussinie, F., Markey, N., Oreiby, G.: On the expressiveness and complexity of ATL. Logical Methods in Computer Science 4(2) (2008)Google Scholar
  15. 15.
    Maddux, R.: The equational theory of CA3 is undecidable. The Journal of Symbolic Logic 45(2), 311–316 (1980)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Marx, M.: Complexity of products of modal logics. Journal of Logic and Computation 9(2), 197–214 (1999)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mortimer, M.: On languages with two variables. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 21, 135–140 (1975)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Troquard, N., Walther, D.: ATL with contexts: agency and explicit strategies. In: Proceedings of LAMAS@AAMAS 2012 (2012)Google Scholar
  19. 19.
    Walther, D., van der Hoek, W., Wooldridge, M.: Alternating-time temporal logic with explicit strategies. In: Proceedings of TARK 2007, pp. 269–278. ACM (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Nicolas Troquard
    • 1
  • Dirk Walther
    • 2
  1. 1.Laboratory for Applied Ontology (ISTC-CNR)TrentoItaly
  2. 2.Universidad Politécnica de MadridSpain

Personalised recommendations