Effective Characterizations of Simple Fragments of Temporal Logic Using Prophetic Automata

  • Sebastian Preugschat
  • Thomas Wilke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7213)


We present a framework for obtaining effective characterizations of simple fragments of future temporal logic (LTL) with the natural numbers as time domain. The framework is based on prophetic automata (also known as complete unambiguous Büchi automata), which enjoy strong structural properties, in particular, they separate the “finitary fraction” of a regular language of infinite words from its “infinitary fraction” in a natural fashion. Within our framework, we provide characterizations of all natural fragments of temporal logic, where, in some cases, no effective characterization had been known previously, and give lower and upper bounds for their computational complexity.


Temporal Logic Regular Language Linear Temporal Logic Negation Normal Form Left Congruence 
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  1. 1.
    Brzozowski, J.A., Simon, I.: Characterizations of locally testable events. Discrete Math. 4(3), 243–271 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Carton, O., Michel, M.: Unambiguous Büchi Automata. In: Gonnet, G.H., Panario, D., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 407–416. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Carton, O., Michel, M.: Unambiguous Büchi automata. Theor. Comput. Sci. 297, 37–81 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cohen, J., Perrin, D., Pin, J.-É.: On the expressive power of temporal logic. J. Comput. System Sci. 46(3), 271–294 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Diekert, V., Kufleitner, M.: Fragments of first-order logic over infinite words. Theory Comput. Syst. 48(3), 486–516 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Allen Emerson, E.: Temporal and modal logic. In: Handbook of Theoretical Computer Science, vol. B, pp. 995–1072. Elsevier, Amsterdam (1990)Google Scholar
  7. 7.
    Etessami, K., Wilke, T.: An until hierarchy and other applications of an Ehrenfeucht-Fraïssé game for temporal logic. Inf. Comput. 160(1-2), 88–108 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Gabbay, D.M., Hodkinson, I., Reynolds, M.: Temporal logic: Mathematical Foundations and Computational Aspects, vol. 1. Clarendon Press, New York (1994)zbMATHGoogle Scholar
  9. 9.
    Gerth, R., Peled, D., Vardi, M.Y., Wolper, P.: Simple on-the-fly automatic verification of linear temporal logic. In: Dembinski, P., Sredniawa, M. (eds.) Protocol Specification, Testing and Verification. IFIP Conference Proceedings, vol. 38, pp. 3–18. Chapman & Hall (1995)Google Scholar
  10. 10.
    Kamp, H.: Tense logic and the theory of linear order. PhD thesis, University of California, Los Angeles (1968)Google Scholar
  11. 11.
    Kim, S.M., McNaughton, R., McCloskey, R.: A polynomial time algorithm for the local testability problem of deterministic finite automata. IEEE Trans. Comput. 40, 1087–1093 (1991)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Etessami, K.: A note on a question of Peled and Wilke regarding stutter-invariant LTL. Inform. Process. Lett. 75(6), 261–263 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    McNaughton, R., Papert, S.A.: Counter-free automata. MIT Press, Boston (1971)zbMATHGoogle Scholar
  14. 14.
    Peled, D., Wilke, T.: Stutter-invariant temporal properties are expressible without the next-time operator. Inform. Process. Lett. 63(5), 243–246 (1997)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Peled, D., Wilke, T., Wolper, P.: An algorithmic approach for checking closure properties of temporal logic specifications and ω-regular languages. Theor. Comput. Sci. 195(2), 183–203 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Perrin, D.: Recent Results on Automata and Infinite Words. In: Chytil, M., Koubek, V. (eds.) MFCS 1984. LNCS, vol. 176, pp. 134–148. Springer, Heidelberg (1984)Google Scholar
  17. 17.
    Perrin, D., Pin, J.-É.: Infinite Words: Automata, Semigroups, Logic and Games. Pure and Applied Mathematics, vol. 141. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
  18. 18.
    Pnueli, A.: The temporal logic of programs. In: FOCS, pp. 46–57. IEEE (1977)Google Scholar
  19. 19.
    Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inform. and Control 8(2), 190–194 (1965)zbMATHCrossRefGoogle Scholar
  20. 20.
    Prasad Sistla, A., Clarke, E.M.: The complexity of propositional linear temporal logics. J. ACM 32, 733–749 (1985)CrossRefGoogle Scholar
  21. 21.
    Vardi, M.Y., Wolper, P.: Reasoning about infinite computations. Inform. and Comput. 115(1), 1–37 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Wilke, T.: Classifying discrete temporal properties. Post-doctoral thesis, Christian-Albrechts-Universität zu Kiel (1998)Google Scholar
  23. 23.
    Wolfgang, T.: Star-free regular sets of ω-sequences. Inform. and Control 42(2), 148–156 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Wolper, P., Vardi, M.Y., Prasad Sistla, A.: Reasoning about infinite computation paths (extended abstract). In: FOCS, pp. 185–194. IEEE (1983)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sebastian Preugschat
    • 1
  • Thomas Wilke
    • 1
  1. 1.Christian-Albrechts-Universität zu KielGermany

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