Model Checking the Quantitative μ-Calculus on Linear Hybrid Systems

  • Diana Fischer
  • Łukasz Kaiser
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6756)


In this work, we consider the model-checking problem for a quantitative extension of the modal μ-calculus on a class of hybrid systems. Qualitative model checking has been proved decidable and implemented for several classes of systems, but this is not the case for quantitative questions, which arise naturally in this context. Recently, quantitative formalisms that subsume classical temporal logics and additionally allow to measure interesting quantitative phenomena were introduced. We show how a powerful quantitative logic, the quantitative μ-calculus, can be model-checked with arbitrary precision on initialised linear hybrid systems. To this end, we develop new techniques for the discretisation of continuous state spaces based on a special class of strategies in model-checking games and show decidability of a class of counter-reset games that may be of independent interest.


Model Check Hybrid System Discrete Strategy Hybrid Automaton Arbitrary Precision 
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., Halbwachs, N., Henzinger, T.A., Ho, P.-H., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: The algorithmic analysis of hybrid systems. Theoretical Computer Science 138(1), 3–34 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Colcombet, T., Löding, C.: Regular cost functions over finite trees. In: LICS, pp. 70–79. IEEE Computer Society, Los Alamitos (2010)Google Scholar
  3. 3.
    de Alfaro, L.: Quantitative verification and control via the mu-calculus. In: Amadio, R.M., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, pp. 103–127. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    de Alfaro, L., Faella, M., Stoelinga, M.: Linear and Branching Metrics for Quantitative Transition Systems. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 97–109. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Figueira, D., Figueira, S., Schmitz, S., Schnoebelen, P.: Ackermann and primitive-recursive bounds with dickson’s lemma. CoRR, abs/1007.2989 (2010)Google Scholar
  6. 6.
    Fischer, D., Grädel, E., Kaiser, L.: Model checking games for the quantitative μ-calculus. Theory Comput. Syst. 47(3), 696–719 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Gawlitza, T., Seidl, H.: Computing game values for crash games. In: Namjoshi, K.S., Yoneda, T., Higashino, T., Okamura, Y. (eds.) ATVA 2007. LNCS, vol. 4762, pp. 177–191. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Henzinger, T.A., Horowitz, B., Majumdar, R.: Rectangular hybrid games. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 320–335. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  9. 9.
    Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s decidable about hybrid automata? In: Proceedings of STOC 1995, pp. 373–382. ACM, New York (1995)Google Scholar
  10. 10.
    McAloon, K.: Petri nets and large finite sets. Theoretical Computer Science 32, 173–183 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    McIver, A., Morgan, C.: Results on the quantitative μ-calculus qMμ. ACM Trans. Comput. Log. 8(1) (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Diana Fischer
    • 1
  • Łukasz Kaiser
    • 2
  1. 1.Mathematische Grundlagen der InformatikRWTH Aachen UniversityGermany
  2. 2.CNRS & LIAFAUniversité Paris DiderotParis 7France

Personalised recommendations