The complexity of propositional linear temporal logics in simple cases

(Extended abstract)
  • S. Demri
  • Ph. Schnoebelen
Logic I
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1373)


It is well-known that model-checking and satisfiability for PLTL are PSPACE-complete. By contrast, very little is known about whether there exist some interesting fragments of PLTL with a lower worst-case complexity. Such results would help understand why PLTL model-checkers are successfully used in practice.

In this paper we investigate this issue and consider model-checking and satisfiability for all fragments of PLTL one obtains when restrictions are put on (1) the temporal connectives allowed, (2) the number of atomic propositions, and (3) the temporal height.


Modal Logic Temporal Logic Linear Temporal Logic Propositional Variable Atomic Proposition 
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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  1. [CL93]
    C.-C. Chen and I. P. Lin. The computational complexity of satisfiability of temporal Horn formulas in propositional linear-time temporal logic. Information Processing Letters, 45(3):131–136, March 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [DFR97]
    C. Dixon, M. Fisher, and M. Reynolds. Execution and proof in a Horn-clause temporal logic. In Proc. 2nd Int. Conf. on Temporal Logic (ICTL'97), Manchester, UK, July 1997, 1997. to appear.Google Scholar
  3. [DS97]
    S. Demri and Ph. Schnoebelen. The complexity of propositional linear temporal logics in simple cases. Research Report LSV-97-11, Lab. Specification and Verification, ENS de Cachan, Cachan, France, December 1997. Available at 1Google Scholar
  4. [EES90]
    E. A. Emerson, M. Evangelist, and J. Srinivasan. On the limits of efficient temporal decidability. In Proc. 5th IEEE Symp. Logic in Computer Science (LICS'90), Philadelphia, PA, USA, June 1990, pages 464–475, 1990.Google Scholar
  5. [EL87]
    E. A. Emerson and C. Lei. Modalities for model checking: Branching time logic strikes back. Science of Computer Programming, 8:275–306, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [Eme90]
    E. A. Emerson. Temporal and modal logic. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, vol. B, chapter 16, pages 995–1072. Elsevier Science Publishers, 1990.Google Scholar
  7. [Hal95]
    J. Y. Halpern. The effect of bounding the number of primitive propositions and the depth of nesting on the complexity of modal logic. Artificial Intelligence, 75(2):361–372, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [Har85]
    D. Harel. Recurring dominos: Making the highly undecidable highly understandable. Annals of Discrete Mathematics, 24:51–72, 1985.zbMATHMathSciNetGoogle Scholar
  9. [HR83]
    J. Y. Halpern and J. H. Reif. The propositional dynamic logic of deterministic, well-structured programs. Theor. Comp. Sci., 27(1–2):127–165, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Joh90]
    D. S. Johnson. A catalog of complexity classes. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, vol. A, chapter 2, pages 67–161. Elsevier Science Publishers, 1990.Google Scholar
  11. [Lam83]
    L. Lamport. What good is temporal logic ? In R. E. A. Mason, editor, Information Processing'83. Proc. IFIP 9th World Computer Congress, Sep. 1983, Paris, France, pages 657–668. North-Holland, 1983.Google Scholar
  12. [MP92]
    Z. Manna and A. Pnueli. The Temporal Logic of Reactive and Concurrent Systems: Specification. Springer-Verlag, 1992.Google Scholar
  13. [NO80]
    A. Nakamura and H. Ono. On the size of refutation Kripke models for some linear modal and tense logics. Studia Logica, 39(4):325–333, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [SC85]
    A. P. Sistla and E. M. Clarke. The complexity of propositional linear temporal logics. Journal of the ACM, 32(3):733–749, July 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [Spa93]
    E. Spaan. Complexity Logics. PhD thesis, ILLC, Amsterdam University, NL, March 1993.Google Scholar
  16. [WVS83]
    P. Wolper, M. Y. Vardi, and A. P. Sistla. Reasoning about infinite computation paths (extended abstract). In Proc. 24th IEEE Symp. Found. of Computer Science (FOCS'83), Tucson, USA, Nov. 1983, pages 185–194, 1983.Google Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • S. Demri
    • 1
  • Ph. Schnoebelen
    • 2
  1. 1.Leibniz-IMAGUniv. Grenoble & CNRS UMR 5522Grenoble CedexFrance
  2. 2.Lab. Specification and VerificationENS de Cachait & CNRS URA 2236Cachan CedexFrance

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