Parametric Metric Interval Temporal Logic

  • Barbara Di Giampaolo
  • Salvatore La Torre
  • Margherita Napoli
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6031)


We study an extension of the logic MITL with parametric constants. In particular, we define a logic, denoted PMITL (parametric MITL), where the subscripts of the temporal operators are intervals with possibly a parametric endpoint. We consider typical decision problems, such as emptiness and universality of the set of parameter valuations under which a given parametric formula is satisfiable, or whether a given parametric timed automaton is a model of a given parametric formula. We show that when each parameter is used with a fixed polarity and only parameter valuations which evaluate parametric intervals to non-singular time intervals are taken into consideration, then the considered problems are decidable and Expspace-complete. We also investigate the computational complexity of these problems for natural fragments of PMITL, and show that in meaningful fragments of the logic they are Pspace-complete. Finally, we discuss other natural parameterizations of MITL, which indeed lead to undecidability.


Model Check Temporal Logic Linear Temporal Logic Parametric Interval Parametric Expression 
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  1. 1.
    Alur, R., Courcoubetis, C., Dill, D.L.: Model-checking in dense real-time. Inf. Comput. 104(1), 2–34 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alur, R., Etessami, K., La Torre, S., Peled, D.: Parametric temporal logic for “model measuring”. ACM Trans. Comput. Log. 2(3), 388–407 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alur, R., Feder, T., Henzinger, T.A.: The benefits of relaxing punctuality. J. ACM 43(1), 116–146 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Alur, R., Henzinger, T.A., Vardi, M.Y.: Parametric real-time reasoning. In: STOC, pp. 592–601 (1993)Google Scholar
  5. 5.
    Bozzelli, L., La Torre, S.: Decision problems for lower/upper bound parametric timed automata. Formal Methods in System Design 35(2), 121–151 (2009)zbMATHCrossRefGoogle Scholar
  6. 6.
    Bruyère, V., Dall’Olio, E., Raskin, J.F.: Durations and parametric model-checking in timed automata. ACM Trans. Comput. Log. 9(2) (2008)Google Scholar
  7. 7.
    Bruyère, V., Raskin, J.F.: Real-time model-checking: Parameters everywhere. Logical Methods in Computer Science 3(1) (2007)Google Scholar
  8. 8.
    Campos, S.V.A., Clarke, E.M., Grumberg, O.: Selective quantitative analysis and interval model checking: Verifying different facets of a system. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 257–268. Springer, Heidelberg (1996)Google Scholar
  9. 9.
    Chevallier, R., Encrenaz-Tiphène, E., Fribourg, L., Xu, W.: Timed verification of the generic architecture of a memory circuit using parametric timed automata. Formal Methods in System Design 34(1), 59–81 (2009)zbMATHCrossRefGoogle Scholar
  10. 10.
    Courcoubetis, C., Yannakakis, M.: Minimum and maximum delay problems in real-time systems. Formal Methods in System Design 1(4), 385–415 (1992)zbMATHCrossRefGoogle Scholar
  11. 11.
    Emerson, E.A.: Temporal and modal logic. In: Handbook of Theoretical Computer Science. Formal Models and Sematics (B), vol. B, pp. 995–1072 (1990)Google Scholar
  12. 12.
    Emerson, E.A., Trefler, R.J.: Parametric quantitative temporal reasoning. LICS, 336–343 (1999)Google Scholar
  13. 13.
    Hune, T., Romijn, J., Stoelinga, M., Vaandrager, F.W.: Linear parametric model checking of timed automata. J. Log. Algebr. Program. 52-53, 183–220 (2002)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Kupferman, O., Piterman, N., Vardi, M.Y.: From liveness to promptness. Formal Methods in System Design 34(2), 83–103 (2009)zbMATHCrossRefGoogle Scholar
  15. 15.
    Pnueli, A.: The temporal logic of programs. In: FOCS, pp. 46–57. IEEE, Los Alamitos (1977)Google Scholar
  16. 16.
    Wang, F.: Parametric timing analysis for real-time systems. Inf. Comput. 130(2), 131–150 (1996)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Barbara Di Giampaolo
    • 1
  • Salvatore La Torre
    • 1
  • Margherita Napoli
    • 1
  1. 1.Università degli Studi di SalernoFiscianoItaly

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