Multi-agent Logic with Distances Based on Linear Temporal Frames

  • Vladimir Rybakov
  • Sergey Babenyshev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6114)


The paper investigates a new temporal logic \(\mathcal{T\!L}^M_{Dist}\), which combines temporal operations with the operations of localised agent’s knowledge and operations responsible for measuring distances. The main goal is to construct a logical framework for modelling logical laws, which describe interactions between such operations. We consider issues of satisfiability and decidability for \(\mathcal{T\!L}^M_{Dist}\). Our principal result is the algorithm which recognizes theorems of \(\mathcal{T\!L}^M_{Dist}\), which implies that \(\mathcal{T\!L}^M_{Dist}\) is decidable, and the satisfiability problem for \(\mathcal{T\!L}^M_{Dist}\) is solvable.


multi-agent systems multi-modal logics decision algorithms satisfiability Kripke semantics distance measuring 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Pnueli, A.: The temporal logic of programs. In: Proc. of the 18th Annual Symp. on Foundations of Computer Science, pp. 46–57. IEEE, Los Alamitos (1977)Google Scholar
  2. 2.
    Manna, Z., Pnueli, A.: Temporal Verification of Reactive Systems: Safety. Springer, Heidelberg (1995)Google Scholar
  3. 3.
    Barringer, H., Fisher, M., Gabbay, D., Gough, G.: Advances in Temporal Logic. Applied logic series, vol. 16. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  4. 4.
    Vardi, M.Y.: Reasoning about the past with two-way automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 628–641. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  5. 5.
    van Benthem, J.: The Logic of Time. Kluwer, Dordrecht (1991)zbMATHGoogle Scholar
  6. 6.
    van Benthem, J., Bergstra, J.: Logic of transition systems. Journal of Logic, Language and Information 3(4), 247–283 (1994)CrossRefGoogle Scholar
  7. 7.
    Gabbay, D., Hodkinson, I.: An axiomatisation of the temporal logic with until and since over the real numbers. Journal of Logic and Computation 1(2), 229–260 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hodkinson, I.: Temporal logic and automata, chapter ii of temporal logic. In: Gabbay, D.M., Reynolds, M.A., Finger, M. (eds.) Mathematical Foundations and Computational Aspects, vol. 2, pp. 30–72. Clarendon Press, Oxford (2000)Google Scholar
  9. 9.
    de Jongh, D., Veltman, F., Verbrugge, R.: Completeness by construction for tense logics of linear time. In: Troelstra, A., Visser, A., van Benthem, J., Veltman, F. (eds.) Liber Amicorum for Dick de Jongh. Institute of Logic, Language and Computation, Amsterdam (2004)Google Scholar
  10. 10.
    Kacprzak, M.: Undecidability of a multi-agent logic. Fundamenta Informaticae 45(2-3), 213–220 (2003)MathSciNetGoogle Scholar
  11. 11.
    Fagin, R., Halpern, J., Moses, Y., Vardi, M.: Reasoning About Knowledge. MIT Press, Cambridge (1995)zbMATHGoogle Scholar
  12. 12.
    Rybakov, V.: Logic of discovery in uncertain situations– deciding algorithms. In: Apolloni, B., Howlett, R.J., Jain, L. (eds.) KES 2007, Part II. LNCS (LNAI), vol. 4693, pp. 950–958. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Rybakov, V.: Multi-agent logics with interacting agents based on linear temporal logic: Deciding algorithms. In: Rutkowski, L., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2008. LNCS (LNAI), vol. 5097, pp. 1243–1253. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Rybakov, V.: Rules of inference with parameters for intuitionistic logic. Journal of Symbolic Logic 57(3), 912–923 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rybakov, V.: Linear temporal logic with until and next, logical consecutions. Annals of Pure and Applied Logic 155(1), 32–45 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Rybakov, V.: Logical consecutions in discrete linear temporal logic. Journal of Symbolic Logic 70(4), 1137–1149 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Rybakov, V.: Logical consecutions in intransitive temporal linear logic of finite intervals. Journal of Logic and Computation 15(5), 663–678 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Rybakov, V.: Admissible Logical Inference Rules. Studies in Logic and the Foundations of Mathematics, vol. 136. Elsevier Sci. Publ., North-Holland, Amsterdam (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Vladimir Rybakov
    • 1
  • Sergey Babenyshev
    • 1
  1. 1.Department of Computing and MathematicsManchester Metropolitan UniversityManchesterU.K.

Personalised recommendations