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An Expressive Temporal Logic for Real Time

  • Yoram Hirshfeld
  • Alexander Rabinovich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)

Abstract

We add to the standard temporal logic with the modalities ”Until” and ”Since”, a sequence of “counting modalities”: For each n the modality C n (X), which says that X will be true at least at n points in the next unit of time, and its past counterpart \({\overleftarrow{C}_n}\), which says that X has happened at least n times in the last unit of time. We prove that this temporal logic is as expressive as can be hoped for; all the modalities that can be expressed in a strong natural decidable predicate logic framework, are expressible in this temporal logic.

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References

  1. 1.
    Alur, R., Feder, T., Henzinger, T.A.: The Benefits of Relaxing Punctuality. Journal of the ACM 43, 116–146 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alur, R., Henzinger, T.A.: Logics and Models of Real Time: a survey. In: Huizing, C., de Bakker, J.W., Rozenberg, G., de Roever, W.-P. (eds.) REX 1991. LNCS, vol. 600, pp. 74–106. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  3. 3.
    Brzozowski, J.A., Knast, R.: The dot depth hierarchy of star free languages is infinite. J. of Computing Systems Science 16, 37–55 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Barringer, B.H., Kuiper, R., Pnueli, A.: A really abstract concurrent model and its temporal logic. In: Proceedings of the 13th annual symposium on principles of programing languages, pp. 173–183 (1986)Google Scholar
  5. 5.
    Ebbinghaus, H.D., Flum, J.: Finite Model Theory. Perspectives in mathematical logic. Springer, Heidelberg (1991)Google Scholar
  6. 6.
    Ehrenfeucht, A.: An application of games to the completeness problem for formalized theories. Fundamenta Mathematicae 49, 129–141 (1961)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Feferman, S., Vaught, R.L.: The first-order properties of products of algebraic systems. Fundamenta Mathematicae 47, 57–103 (1959)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Gabbay, D.M., Hodkinson, I., Reynolds, M.: Temporal Logics, vol. 1. Clarendon Press, Oxford (1994)CrossRefGoogle Scholar
  9. 9.
    Gabbay, D.M., Pnueli, A., Shelah, S., Stavi, J.: On the Temporal Analysis of Fairness. In: 7th ACM Symposium on Principles of Programming Languages, Las Vegas, pp. 163–173 (1980)Google Scholar
  10. 10.
    Henzinger, T.A.: It’s about time: Real-time logics reviewed. In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 439–454. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  11. 11.
    Gurevich, Y.: Monadic second-order theories. In: Barwise, J., Feferman, S. (eds.) Model-Theoretic Logics, pp. 479–506. Springer, Heidelberg (1985)Google Scholar
  12. 12.
    Hirshfeld, Y., Rabinovich, A.: A Framework for Decidable Metrical Logics. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 422–432. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  13. 13.
    Hirshfeld, Y., Rabinovich, A.: Quantitative Temporal Logic. In: Flum, J., Rodríguez-Artalejo, M. (eds.) CSL 1999. LNCS, vol. 1683, pp. 172–187. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  14. 14.
    Hirshfeld, Y., Rabinovich, A.: Logics for Real Time: Decidability and Complexity. Fundam. Inform. 62(1), 1–28 (2004)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Hirshfeld, Y., Rabinovich, A.: Timer formulas and decidable metric temporal logic. Information and Computation 198(2), 148–178 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hirshfeld, Y., Rabinovich, A.: Expressiveness of Metric Modalities for Continuous Time. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 211–220. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Hirshfeld, Y., Rabinovich, A.: On The complexity of the temporal logic with counting modalities. Manuscript (2006)Google Scholar
  18. 18.
    Kamp, H.: Tense Logic and the Theory of Linear Order. Ph.D. thesis, University of California L.A. (1968)Google Scholar
  19. 19.
    Manna, Z., Pnueli, A.: Models for reactivity. Acta informatica 30, 609–678 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Makowsky, J.A.: Algorithmic aspects of the Feferman-Vaught theorem. Annals of Pure and Applied Logic 126(1-3), 159–213 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Rabinovich, A.: Expressive Power of Temporal Logics. In: Brim, L., Jančar, P., Křetínský, M., Kucera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 57–75. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  22. 22.
    Shelah, S.: The monadic theory of order. Ann. of Math. 102, 349–419 (1975)CrossRefGoogle Scholar
  23. 23.
    Thomas, W.: Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds.) Structures in Logic and Computer Science. LNCS, vol. 1261, pp. 118–143. Springer, Heidelberg (1997)Google Scholar
  24. 24.
    Wilke, T.: Specifying Time State Sequences in Powerful Decidable Logics and Time Automata. In: Langmaack, H., de Roever, W.-P., Vytopil, J. (eds.) FTRTFT 1994 and ProCoS 1994. LNCS, vol. 863, pp. 694–715. Springer, Heidelberg (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Yoram Hirshfeld
    • 1
  • Alexander Rabinovich
    • 1
  1. 1.School of Computer Science, Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael

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