Satisfiability and Model Checking for MSO-Definable Temporal Logics Are in PSPACE

  • Paul Gastin
  • Dietrich Kuske
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2761)


Temporal logics over Mazurkiewicz traces have been extensively studied over the past fifteen years. In order to be usable for the verification of concurrent systems they need to have reasonable complexity for the satisfiability and the model checking problems. Whenever a new temporal logic was introduced, a new proof (usually non trivial) was needed to establish the complexity of these problems. In this paper, we introduce a unified framework to define local temporal logics over traces. We prove that the satisfiability problem and the model checking problem for asynchronous Kripke structures for local temporal logics over traces are decidable in PSPACE. This subsumes and sometimes improves all complexity results previously obtained on local temporal logics for traces.


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  1. 1.
    Alur, R., Peled, R., Penczek, W.: Model checking of causality properties. In: LICS 1995, pp. 90–100. IEEE Computer Society Press, Los Alamitos (1995)Google Scholar
  2. 2.
    Büchi, J.R.: On a decision method in restricted second order arithmetics. In: Nagel, E., et al. (eds.) Proc. Intern. Congress on Logic, Methodology and Philosophy of Science, pp. 1–11. Stanford University Press, Stanford (1960)Google Scholar
  3. 3.
    Diekert, V.: A pure future local temporal logic beyond cograph-monoids. In: Ito, M. (ed.) Proc. of the RIMS Symposium on Algebraic Systems, Formal Languages and Conventional and Unconventional Computation Theory, Kyoto, Japan (2002)Google Scholar
  4. 4.
    Diekert, V., Gastin, P.: Local temporal logic is expressively complete for cograph dependence alphabets. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 55–69. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    Diekert, V., Gastin, P.: Local temporal logic is expressively complete for cograph dependence alphabets. Tech. Rep. LIAFA, Universit´e Paris 7, France (2003),
  6. 6.
    Diekert, V., Rozenberg, G.: The Book of Traces. World Scientific Publ. Co., Singapore (1995)CrossRefGoogle Scholar
  7. 7.
    Gastin, P., Mukund, M.: An elementary expressively complete temporal logic for Mazurkiewicz traces. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 938–949. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    Kamp, H.W.: Tense logic and the theory of linear order. PhD thesis, University of California, Los Angeles, USA (1968)Google Scholar
  9. 9.
    Kupferman, O., Piterman, N., Vardi, M.Y.: Extended temporal logic revisited. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 519–535. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Mukund, M., Sohoni, M.: Keeping trace of the latest gossip: bounded timestamps suffice. In: Shyamasundar, R.K. (ed.) FSTTCS 1993. LNCS, vol. 761, pp. 388–399. Springer, Heidelberg (1993)Google Scholar
  11. 11.
    Mukund, M., Thiagarajan, P.S.: Linear time temporal logics over Mazurkiewicz traces. In: Penczek, W., Szałas, A. (eds.) MFCS 1996. LNCS, vol. 1113, pp. 62–92. Springer, Heidelberg (1996)Google Scholar
  12. 12.
    Rabinovich, A., Maoz, S.: An infinite hierarchy of temporal logics over branching time. Information and Computation 171(2), 306–332 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Thiagarajan, P.S.: A trace based extension of linear time temporal logic. In: Proc. of LICS 1994, pp. 438–447. IEEE Computer Society Press, Los Alamitos (1994)Google Scholar
  14. 14.
    Thiagarajan, P.S.: A trace consistent subset of PTL. In: Lee, I., Smolka, S.A. (eds.) CONCUR 1995. LNCS, vol. 962, pp. 438–452. Springer, Heidelberg (1995)Google Scholar
  15. 15.
    Thomas, W.: Automata on infinite objects. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, pp. 133–191. Elsevier Science Publ. B.V., Amsterdam (1990)Google Scholar
  16. 16.
    Vardi, M.Y., Wolper, P.: Reasonning about infinite computations. Information and Computation 115, 1–37 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Wolper, P.: Temporal logic can be more expressive. Inf. and Control 56, 72–99 (1983)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Paul Gastin
    • 1
  • Dietrich Kuske
    • 2
  1. 1.LIAFAUniversité Paris 7Paris
  2. 2.TU DresdenInstitut für AlgebraDresden

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