Studia Logica

, Volume 75, Issue 1, pp 93–123

Complete Axiomatizations for Reasoning about Knowledge and Branching Time

  • Ron van der Meyden
  • Ka-shu Wong
Article

Abstract

Sound and complete axiomatizations are provided for a number of different logics involving modalities for the knowledge of multiple agents and operators for branching time, extending previous work of Halpern, van der Meyden and Vardi [to appear, SIAM Journal on Computing] for logics of knowledge and linear time. The paper considers the system constraints of synchrony, perfect recall and unique initial states, which give rise to interaction axioms. The language is based on the temporal logic CTL*, interpreted with respect to a version of the bundle semantics.

modal logic epistemic logic logic of knowledge temporal logic branching time multi-agent systems perfect recall synchrony 

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References

  1. [Bar88]
    Barwise, J., ‘Three views of common knowledge’, in M. Y. Vardi, (ed.), Proc. Second Conference on Theoretical Aspects of Reasoning about Knowledge, pp. 365-379, Morgan Kaufmann, San Francisco, Calif., 1988.Google Scholar
  2. [BHWZ02]
    Bauer, S., I. Hodkinson, F. Wolter, and M. Zakharyaschev, ‘On non-local and weak monodic quantified CTL*'. submitted for publication, 2002.Google Scholar
  3. [Bur79]
    Burgess, J., ‘Logic and time’, Journal of Symbolic Logic, 44:556-582, 1979.Google Scholar
  4. [DFW98]
    Dixon, C., M. Fisher, and M. Wooldridge, ‘Resolution for temporal logics of knowledge’, Journal of Logic and Computation, 8(3):345-372, 1998.Google Scholar
  5. [EH85]
    Emerson, E. A., and J. Y. Halpern, ‘Decision procedures and expressiveness in the temporal logic of branching time’, Journal of Computer and System Sciences, 30(1):1-24, 1985.Google Scholar
  6. [FHMV95]
    Fagin, R., J. Y. Halpern, Y. Moses, and M. Y. Vardi, Reasoning about Knowledge, MIT Press, Cambridge, Mass., 1995.Google Scholar
  7. [FHV91]
    Fagin, R., J. Y. Halpern, and M. Y. Vardi, ‘A model-theoretic analysis of knowledge’, Journal of the ACM, 91(2):382-428, 1991. A preliminary version appeared in Proc. 25th IEEE Symposium on Foundations of Computer Science, 1984.Google Scholar
  8. [GPSS80]
    Gabbay, D., A. Pnueli, S. Shelah, and J. Stavi, ‘On the temporal analysis of fairness’, in Proc. 7th ACM Symp. on Principles of Programming Languages, pp. 163-173, 1980.Google Scholar
  9. [Hin62]
    Hintikka, J., Knowledge and Belief, Cornell University Press, Ithaca, N.Y., 1962.Google Scholar
  10. [HM92]
    Halpern, J. Y., and Y. Moses, ‘A guide to completeness and complexity for modal logics of knowledge and belief’, Artificial Intelligence, 54:319-379, 1992.Google Scholar
  11. [HV86]
    Halpern, J. Y., and M. Y. Vardi, ‘The complexity of reasoning about knowledge and time’, in Proc. 18th ACM Symp. on Theory of Computing, pp. 304-315, 1986.Google Scholar
  12. [HV88a]
    Halpern, J. Y., and M. Y. Vardi, ‘The complexity of reasoning about knowledge and time in asynchronous systems’, in Proc. 20th ACM Symp. on Theory of Computing, pp. 53-65, 1988.Google Scholar
  13. [HV88b]
    Halpern, J. Y., and M. Y. Vardi, ‘The complexity of reasoning about knowledge and time: synchronous systems’, Research Report RJ 6097, IBM, 1988.Google Scholar
  14. [HV89]
    Halpern, J. Y., and M. Y. Vardi, ‘The complexity of reasoning about knowledge and time, I: lower bounds’, Journal of Computer and System Sciences, 38(1):195-237, 1989.Google Scholar
  15. [HvdMVar]
    Halpern, J. Y., R. Van Der Meyden, and M. Y. Vardi, ‘Complete axiomatizations for reasoning about knowledge and time’, SIAM Journal on Computing, to appear.Google Scholar
  16. [KP81]
    Kozen, D., and R. Parikh, ‘An elementary proof of the completeness of PDL’, Theoretical Computer Science, 14(1):113-118, 1981.Google Scholar
  17. [Leh84]
    Lehmann, D., ‘Knowledge, common knowledge, and related puzzles’, in Proc. 3rd ACM Symp. on Principles of Distributed Computing, pp. 62-67, 1984.Google Scholar
  18. [LR86]
    Ladner, R. E., and J. H. Reif, ‘The logic of distributed protocols (preliminary report)’, in J. Y. Halpern, (ed.), Theoretical Aspects of Reasoning about Knowledge: Proc. 1986 Conference, pp. 207-222. Morgan Kaufmann, San Francisco, Calif., 1986.Google Scholar
  19. [Mey94]
    Van Der Meyden, R., ‘Axioms for knowledge and time in distributed systems with perfect recall’, in Proc. 9th IEEE Symp. on Logic in Computer Science, pp. 448-457. 1994.Google Scholar
  20. [PR85]
    Parikh, R., and R. Ramanujam, ‘Distributed processing and the logic of knowledge’, in R. Parikh, (ed.), Proc. Workshop on Logics of Programs, pp. 256-268, 1985.Google Scholar
  21. [Rey01]
    Reynolds, M., ‘An axiomatization of full computation tree logic’, Journal of Symbolic Logic, 66(3):1011-1057, Sept 2001.Google Scholar
  22. [Sat77]
    Sato, M., ‘A study of Kripke-style methods for some modal logics by Gentzen's sequential method’, Publications Research Institute for Mathematical Sciences, Kyoto University, 13(2), 1977.Google Scholar
  23. [Spa90]
    Spaan, E., ‘Nexttime is not necessary’, in R. J. Parikh, (ed.), Theoretical Aspects of Reasoning about Knowledge: Proc. Third Conference, pp. 241-256. Morgan Kaufmann, San Francisco, Calif., 1990.Google Scholar
  24. [Sti92]
    Stirling, C. ‘Modal and temporal logics’, in S. Abramsky, Dov M. Gabbay, and T. S. E. Maibaum, (eds.), Handbook of Logic in Computer Science Volume 2: Background: Computational Structures, pp. 477-563. Clarendon Press, Oxford, 1992.Google Scholar
  25. [WDF98]
    Wooldridge, M., C. Dixon, and M. Fisher, ‘A tableau-based proof method for temporal logics of knowledge and belief’, Journal of Applied Non-Classical Logics, 8(3), 1998.Google Scholar
  26. [Zan85]
    Zanardo, A., ‘A finite axiomatization of the set of strongly valid Ockhamist formulas’, Journal of Philosophical Logic, 14(447–468), 1985.Google Scholar
  27. [Zan96]
    Zanardo, A., ‘Branching-time logic with quantification over branches: the point of view of modal logic’, Journal of Symbolic Logic, 61(1):1-39, March 1996.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Ron van der Meyden
    • 1
  • Ka-shu Wong
    • 1
  1. 1.School of Computer Science and EngineeringUniversity of New South WalesSydneyAustralia

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