A Combination of Explicit and Deductive Knowledge with Branching Time: Completeness and Decidability Results

  • Alessio Lomuscio
  • Bożena Woźna
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3904)


Logics for knowledge and time comprise logic combinations between epistemic logic S5 n for n agents and temporal logic. In this paper we examine a logic combination of Computational Tree Logic and an epistemic logic augmented to include an additional epistemic operator representing explicit knowledge. We show the resulting system enjoys the finite model property, decidability and is finitely axiomatisable. It is further shown that the expressivity of the resulting system enables us to represent a non-standard notion of deductive knowledge which seems promising for applications.


Model Check Modal Logic Temporal Logic Directed Acyclic Graph Multiagent System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge (2001)CrossRefMATHGoogle Scholar
  2. 2.
    Catach, L.: Normal multimodal logics. In: Proceedings of the 7th National Conference on Artificial Intelligence (AAAI 1988), pp. 491–495. Morgan Kaufmann, San Francisco (1988)Google Scholar
  3. 3.
    Clarke, E., Emerson, E.: Design and synthesis of synchronization skeletons for branching-time temporal logic. In: Kozen, D. (ed.) Logic of Programs 1981. LNCS, vol. 131, pp. 52–71. Springer, Heidelberg (1982)CrossRefGoogle Scholar
  4. 4.
    Dolev, D., Yao, A.: On the security of public key protocols. IEEE Trans. Inf. Theory 29(2), 198–208 (1983)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Emerson, E.A.: Temporal and modal logic. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, ch. 16, pp. 996–1071. Elsevier Science Publishers, Amsterdam (1990)Google Scholar
  6. 6.
    Emerson, E.A., Clarke, E.M.: Using branching-time temporal logic to synthesize synchronization skeletons. Science of Computer Programming 2(3), 241–266 (1982)CrossRefMATHGoogle Scholar
  7. 7.
    Emerson, E.A., Halpern, J.Y.: Decision procedures and expressiveness in the temporal logic of branching time. Journal of Computer and System Sciences 30(1), 1–24 (1985)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fagin, R., Halpern, J.Y.: Belief, awareness, and limited reasoning. Artificial Intelligence 34(1), 39–76 (1988)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about Knowledge. MIT Press, Cambridge (1995)MATHGoogle Scholar
  10. 10.
    Fagin, R., Halpern, J.Y., Vardi, M.: A nonstandard approach to the logical omniscience problem. Artificial Intelligence 79 (1995)Google Scholar
  11. 11.
    Fagin, R., Halpern, J.Y., Vardi, M.Y.: A model-theoretic analysis of knowledge. Journal of ACM 91, 382–428 (1991)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fischer, M.J., Ladner, R.E.: Propositional dynamic logic of regular programs. Journal of Computer and System Sciences 18(2), 194–211 (1979)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gammie, P., van der Meyden, R.: MCK: Model checking the logic of knowledge. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 479–483. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Halpern, J., van der Meyden, R., Vardi, M.Y.: Complete axiomatisations for reasoning about knowledge and time. SIAM Journal on Computing 33(3), 674–703 (2003)CrossRefMATHGoogle Scholar
  15. 15.
    Halpern, J.Y., Moses, Y., Vardi, M.Y.: Algorithmic knowledge. In: Theoretical Aspects of Reasoning About Knowledge. In: Proceedings of the 5th Conference (TARK 1994), pp. 255–266. Morgan Kaufmann Publishers, San Francisco (1994)Google Scholar
  16. 16.
    Hintikka, J.: Knowledge and Belief, An Introduction to the Logic of the Two Notions. Cornell University Press, Ithaca (NY) (1962)Google Scholar
  17. 17.
    van der Hoek, W.: Systems for knowledge and belief. Journal of Logic and Computation 3(2), 173–195 (1993)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    van der Hoek, W., Wooldridge, M.: Model checking knowledge and time. In: Proceedings of the 9th International SPIN Workshop on Model Checking of Software (2002)Google Scholar
  19. 19.
    Konolige, K.: A Deduction Model of Belief. Brown University Press (1986)Google Scholar
  20. 20.
    Kracht, M., Wolter, F.: Properties of independently axiomatizable bimodal logics. Journal of Symbolic Logic 56(4), 1469–1485 (1991)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lehman, D.: Knowledge, common knowledge, and related puzzles. In: Proceedings of the 3rd ACM Symposium on Principles of Distributed Computing, pp. 62–67 (1984)Google Scholar
  22. 22.
    Lomuscio, A., van der Meyden, R., Ryan, M.: Knowledge in multi-agent systems: Initial configurations and broadcast. ACM Transactions of Computational Logic 1(2) (2000)Google Scholar
  23. 23.
    Lomuscio, A., Sergot, M.: Deontic interpreted systems. Studia Logica 75(1), 63–92 (2003)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lomuscio, A., Woźna, B.: A complete and decidable security-specialised logic and its application to the tesla protocol. In: Proceedings of the 5th International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS 2006). ACM Press, New York (2006) (to appear)Google Scholar
  25. 25.
    van der Meyden, R.: Axioms for knowledge and time in distributed systems with perfect recall. In: Proceedings of the 9th Annual IEEE Symposium on Logic in Computer Science, pp. 448–457. IEEE Computer Society Press, Los Alamitos (1994)CrossRefGoogle Scholar
  26. 26.
    van der Meyden, R., Wong, K.: Complete axiomatizations for reasoning about knowledge and branching time. Studia Logica 75(1), 93–123 (2003)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Penczek, W., Lomuscio, A.: Verifying epistemic properties of multi-agent systems via bounded model checking. Fundamenta Informaticae 55(2), 167–185 (2003)MathSciNetMATHGoogle Scholar
  28. 28.
    Perrig, A., Canetti, R., Tygar, J.D., Song, D.X.: Efficient authentication and signing of multicast streams over lossy channels. In: IEEE Symposium on Security and Privacy, pp. 56–73 (May 2000)Google Scholar
  29. 29.
    Pucella, R.: Deductive Algorithmic Knowledge. In: Proceedings of the 8th International Symposium on Artificial Intelligence and Mathematics (SAIM 2004), Online Proceedings: AI&M 22-2004 (2004)Google Scholar
  30. 30.
    Raimondi, F., Lomuscio, A.: Verification of multiagent systems via ordered binary decision diagrams: an algorithm and its implementation. In: Proceedings of the 3rd International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS 2004), vol. II. ACM, New York (2004)Google Scholar
  31. 31.
    Sato, J.M.: A study of Kripke style methods for some modal logic by Gentzen’s sequential method. Technical report, Publication Research Institute for Mathematical Science (1977)Google Scholar
  32. 32.
    Spann, E.: Nextime is not necessary. In: Proceedings of the 3rd Conference on Theoretical Aspects of Reasoning about Knowledge, pp. 241–256 (1990)Google Scholar
  33. 33.
    van der Hoek, W., Wooldridge, M.: Cooperation, knowledge, and time: Alternating-time temporal epistemic logic and its applications. Studia Logica 75(1), 125–157 (2003)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    van der Meyden, R., Shilov, H.: Model checking knowledge and time in systems with perfect recall. In: Pandu Rangan, C., Raman, V., Sarukkai, S. (eds.) FST TCS 1999, vol. 1738, pp. 432–445. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  35. 35.
    van der Meyden, R., Su, K.: Symbolic model checking the knowledge of the dining cryptographers. In: Proceedings of the 17th IEEE Computer Security Foundations Workshop (CSFW 2004), pp. 280–291. IEEE Computer Society, Los Alamitos (2004)CrossRefGoogle Scholar
  36. 36.
    Woźna, B., Lomuscio, A., Penczek, W.: Bounded model checking for knowledge over real time. In: Proceedings of the 4st International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS 2005), vol. I, pp. 165–172. ACM Press, New York (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alessio Lomuscio
    • 1
  • Bożena Woźna
    • 1
  1. 1.Department of Computer ScienceUniversity College LondonLondonUnited Kingdom

Personalised recommendations