Language Emptiness of Continuous-Time Parametric Timed Automata

  • Nikola Beneš
  • Peter Bezděk
  • Kim G. Larsen
  • Jiří Srba
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9135)


Parametric timed automata extend the standard timed automata with the possibility to use parameters in the clock guards. In general, if the parameters are real-valued, the problem of language emptiness of such automata is undecidable even for various restricted subclasses. We thus focus on the case where parameters are assumed to be integer-valued, while the time still remains continuous. On the one hand, we show that the problem remains undecidable for parametric timed automata with three clocks and one parameter. On the other hand, for the case with arbitrary many clocks where only one of these clocks is compared with (an arbitrary number of) parameters, we show that the parametric language emptiness is decidable. The undecidability result tightens the bounds of a previous result which assumed six parameters, while the decidability result extends the existing approaches that deal with discrete-time semantics only. To the best of our knowledge, this is the first positive result in the case of continuous-time and unbounded integer parameters, except for the rather simple case of single-clock automata.


Time Slot Corner Point Time Division Multiple Access Safety Problem Integer Parameter 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alur, R., Courcoubetis, C., Dill, D.: Model-checking for real-time systems. In: LICS 1990. pp. 414–425. IEEE (1990)Google Scholar
  2. 2.
    Alur, R., Dill, D.: A theory of timed automata. Theoretical Computer Science 126(2), 183–235 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Alur, R., Henzinger, T., Vardi, M.: Parametric real-time reasoning. In: Proceedings of 25th Annual Symposium on Theory of Computing (STOC 1993), pp. 592–601. ACM Press (1993)Google Scholar
  4. 4.
    André, É., Chatain, T., Fribourg, L., Encrenaz, E.: An inverse method for parametric timed automata. ENTCS 223, 29–46 (2008)Google Scholar
  5. 5.
    Behrmann, Gerd, Fehnker, Ansgar, Hune, Thomas, Larsen, Kim Guldstrand, Pettersson, Paul, Romijn, Judi M.T., Vaandrager, Frits W.: Minimum-cost reachability for priced timed automata. In: Di Benedetto, Maria Domenica, Sangiovanni-Vincentelli, Alberto L. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 147–161. Springer, Heidelberg (2001) CrossRefGoogle Scholar
  6. 6.
    Beneš, N., Bezděk, P., Larsen, K.G., Srba, J.: Language emptiness of continuous-time parametric timed automata (2015). CoRR abs/1504.07838Google Scholar
  7. 7.
    Bouyer, P., Brihaye, T., Markey, N.: Improved undecidability results on weighted timed automata. Inform. Proc. Letters 98(5), 188–194 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bozzelli, L., La Torre, S.: Decision problems for lower/upper bound parametric timed automata. Formal Methods in Syst. Design 35(2), 121–151 (2009)zbMATHCrossRefGoogle Scholar
  9. 9.
    Bruyère, Véronique, Raskin, Jean-François: Real-time model-checking: parameters everywhere. In: Pandya, Paritosh K., Radhakrishnan, Jaikumar (eds.) FSTTCS 2003. LNCS, vol. 2914, pp. 100–111. Springer, Heidelberg (2003) CrossRefGoogle Scholar
  10. 10.
    Bundala, Daniel, Ouaknine, Joël: Advances in parametric real-time reasoning. In: Csuhaj-Varjú, Erzsébet, Dietzfelbinger, Martin, Ésik, Zoltán (eds.) MFCS 2014, Part I. LNCS, vol. 8634, pp. 123–134. Springer, Heidelberg (2014) Google Scholar
  11. 11.
    Bérard, Beatrice, Haddad, Serge, Jovanović, Aleksandra, Lime, Didier: Parametric interrupt timed automata. In: Abdulla, Parosh Aziz, Potapov, Igor (eds.) RP 2013. LNCS, vol. 8169, pp. 59–69. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  12. 12.
    Doyen, L.: Robust parametric reachability for timed automata. Information Processing Letters 102(5), 208–213 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Feo-Arenis, Sergio, Westphal, Bernd, Dietsch, Daniel, Muñiz, Marco, Andisha, Ahmad Siyar: The wireless fire alarm system: ensuring conformance to industrial standards through formal verification. In: Jones, Cliff, Pihlajasaari, Pekka, Sun, Jun (eds.) FM 2014. LNCS, vol. 8442, pp. 658–672. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  14. 14.
    Hune, Thomas, Romijn, Judi M.T., Stoelinga, Mariëlle, Vaandrager, Frits W.: Linear parametric model checking of timed automata. In: Margaria, Tiziana, Yi, W. (eds.) TACAS 2001. LNCS, vol. 2031, pp. 189–203. Springer, Heidelberg (2001) CrossRefGoogle Scholar
  15. 15.
    Jovanović, Aleksandra, Lime, Didier, Roux, Olivier H.: Integer parameter synthesis for timed automata. In: Piterman, Nir, Smolka, Scott A. (eds.) TACAS 2013 (ETAPS 2013). LNCS, vol. 7795, pp. 401–415. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  16. 16.
    Kuzmin, E., Chalyy, D.: Decidability of boundedness problems for Minsky counter machines. Automatic Control and Computer Sciences 44(7), 387–397 (2010)CrossRefGoogle Scholar
  17. 17.
    Miller, Joseph S.: Decidability and complexity results for timed automata and semi-linear hybrid automata. In: Lynch, Nancy A., Krogh, Bruce H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 296–310. Springer, Heidelberg (2000) CrossRefGoogle Scholar
  18. 18.
    Minsky, M.: Computation: Finite and Infinite Machines. Prentice (1967)Google Scholar
  19. 19.
    Wang, F.: Parametric timing analysis for real-time systems. Information and Computation 130(2), 131–150 (1996)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Nikola Beneš
    • 1
  • Peter Bezděk
    • 1
  • Kim G. Larsen
    • 2
  • Jiří Srba
    • 2
  1. 1.Faculty of InformaticsMasaryk University BrnoBrnoCzech Republic
  2. 2.Department of Computer ScienceAalborg UniversityAalborgDenmark

Personalised recommendations