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A Branching Time Logic with Two Types of Probability Operators

  • Zoran Ognjanović
  • Dragan Doder
  • Zoran Marković
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6929)

Abstract

We introduce a propositional logic whose formulas are built using the language of CTL *, enriched by two types of probability operators: one speaking about probabilities on branches, and one speaking about probabilities of sets of branches with the same initial state. An infinitary axiomatization for the logic, which is shown to be sound and strongly complete with respect to the corresponding class of models, is proposed.

Keywords

Temporal Logic Inference Rule Probability Operator Axiomatic System Time Logic 
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.

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References

  1. 1.
    Abadi, M.: The power of temporal proofs. Theoretical Computer Science 65, 35–83 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aziz, A., Singhal, V., Balarin, F., Brayton, R.K., Sangiovanni-Vincentelli, A.L.: It usually works: The temporal logic of stochastic systems. In: Wolper, P. (ed.) CAV 1995. LNCS, vol. 939, Springer, Heidelberg (1995)Google Scholar
  3. 3.
    Bianco, A., de Alfaro, L.: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026, pp. 499–512. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  4. 4.
    Burgess, J.: Axioms for tense logic. I. “Since” and “until”. Notre Dame Journal of Formal Logic 23(4), 367–374 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burgess, J.: Logic and time. The Journal of Symbolic Logic 44(4), 566–582 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Burgess, J.: Basic tense logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. II, pp. 89–133. D. Reidel Publishing Compaany, Dordrecht (1984); Kopetz, H., Kakuda, Y. (eds.) Dependable Computing and Fault-Tolerant Systems. Responsive Computer Systems, Vol. 7 pp. 30–52. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  7. 7.
    Doder, D., Ognjanović, Z., Marković, Z.: An Axiomatization of a First-order Branching Time Temporal Logic. Journal of Universal Computer Science 16(11), 1439–1451 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Doder, D., Marković, Z., Ognjanović, Z., Perović, A., Rašković, M.: A Probabilistic Temporal Logic That Can Model Reasoning about Evidence. In: Link, S., Prade, H. (eds.) FoIKS 2010. LNCS, vol. 5956, pp. 9–24. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Doder, D., Marinković, B., Maksimović, P., Perović, A.: A Logic with Conditional Probability Operators. Publications de l’Institut Mathématique, Nouvelle Série, Beograd 87(101), 85–96 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Emerson, E., Clarke, E.: Using branching time logic to synthesize synchronization skeletons. Sci. Comput. Program. 2, 241–266 (1982)CrossRefzbMATHGoogle Scholar
  11. 11.
    Emerson, E.: Temporal and Modal Logic. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics, pp. 995–1072. North-Holland Pub. Co./MIT Press (1990)Google Scholar
  12. 12.
    Fagin, R., Halpern, J., Megiddo, N.: A logic for reasoning about probabilities. Information and Computation 87(1–2), 78–128 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Feldman, Y.: A decidable propositional dynamic logic with explicit probabilities. Information and Control 63, 11–38 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guelev, D.P.: A propositional dynamic logic with qualitative probabilities. Journal of Philosophical Logic 28(6), 575–605 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Guelev, D.P.: Probabilistic neighbourhood logic. In: Joseph, M. (ed.) FTRTFT 2000. LNCS, vol. 1926, pp. 264–275. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  16. 16.
    Guelev, D.P.: Probabilistic Interval Temporal Logic and Duration Calculus with Infinite Intervals: Complete Proof Systems. Logical Methods in Computer Science 3(3), 1–43 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspect of Computing 6(5), 512–535 (1994)CrossRefzbMATHGoogle Scholar
  18. 18.
    Hart, S., Sharir, M.: Probabilistic temporal logics for finite and bounded models. In: 16th ACM Symposium on Theory of Computing, pp. 1–13. ACM, New York (1984): Extended version In: Information and Control 70 (2/3), 97-155 (1986)Google Scholar
  19. 19.
    Hung, D.V., Chaochen, Z.: Probabilistic Duration Calculus for Continuous Time. Formal Aspects of Computing 11(1), 21–44 (1999)CrossRefzbMATHGoogle Scholar
  20. 20.
    Kozen, D.: A probabilistic PDL. Journal of Computer and System Sciences 30, 162–178 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lehmann, D., Shelah, S.: Reasoning with time and chance. Information and Control 53, 165–198 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Liu, Z., Ravn, A.P., Sorensen, E.V., Chaochen, Z.: A Probabilistic Duration Calculus. In: Kopetz, H., Kakuda, Y. (eds.) Dependable Computing and Fault-Tolerant Systems. Responsive Computer Systems, vol. 7, pp. 30–52. Springer, Heidelberg (1993)Google Scholar
  23. 23.
    Marković, Z., Ognjanović, Z., Rašković, M.: A Probabilistic Extension of Intuitionistic Logic. Mathematical Logic Quarterly 49, 415–424 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ognjanović, Z., Rašković, M.: Some probability logics with new types of probability operators. Journal of Logic and Computation 9(2), 181–195 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ognjanović, Z., Rašković, M.: Some first-order probability logics. Theoretical Computer Science 247(1-2), 191–212 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ognjanović, Z.: Discrete Linear-time Probabilistic Logics: Completeness, Decidability and Complexity. Journal of Logic Computation 16(2), 257–285 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ognjanović, Z., Perović, A., Rašković, M.: Logics with the Qualitative Probability Operator. Logic Journal of IGPL 16(2), 105–120 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Perović, A., Ognjanović, Z., Rašković, M., Marković, Z.: A Probabilistic Logic with Polynomial Weight Formulas. In: Hartmann, S., Kern-Isberner, G. (eds.) FoIKS 2008. LNCS, vol. 4932, pp. 239–252. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  29. 29.
    Pnueli, A.: The Temporal Logic of Programs. In: Proceedings of the 18th IEEE Symposium Foundations of Computer Science (FOCS 1977), pp. 46–57 (1977)Google Scholar
  30. 30.
    Prior, A.: Time and Modality. Oxford University Press, Oxford (1957)zbMATHGoogle Scholar
  31. 31.
    Rašković, M., Marković, Z., Ognjanović, Z.: A Logic with Approximate Conditional Probabilities that can Model Default Reasoning. International Journal of Approximate Reasoning 49(1), 52–66 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Reynolds, M.: An axiomatization of full computation tree logic. The Journal of Symbolic Logic 66(3), 1011–1057 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Segerberg, K.: Qualitative probability in a modal setting. In: Fenstad, J.E. (ed.) Proceedings of the Second Scandinavian Logic Symposium. North-Holland, Amsterdam (1971)Google Scholar
  34. 34.
    Stirling, C.: Modal and temporal logic. Handbook of Logic in Computer Science 2, 477–563 (1992)MathSciNetGoogle Scholar
  35. 35.
    Tzanis, E., Hirsch, R.: Probabilistic Logic over Paths. Electronic Notes in Theoretical Computer Science 220(3), 79–96 (2008)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Zoran Ognjanović
    • 1
  • Dragan Doder
    • 2
  • Zoran Marković
    • 1
  1. 1.Matematički institut SANUBeogradSerbia
  2. 2.Mašinski fakultetBeogradSerbia

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