Symbolic State Space of Stopwatch Petri Nets with Discrete-Time Semantics (Theory Paper)

  • Morgan Magnin
  • Didier Lime
  • Olivier (H. ) Roux
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5062)


In this paper, we address the class of bounded Petri nets with stopwatches (SwPNs), which is an extension of T-time Petri nets (TPNs) where time is associated with transitions. Contrary to TPNs, SwPNs encompass the notion of actions that can be reset, stopped and started. Models can be defined either with discrete-time or dense-time semantics. Unlike dense-time, discrete-time leads to combinatorial explosion (state space is computed by an exhaustive enumeration of states). We can however take advantage from discrete-time, especially when it comes to SwPNs: state and marking reachability problems, undecidable even for bounded nets, become decidable once discrete-time is considered. Thus, to mitigate the issue of combinatorial explosion, we now aim to extend the well-known symbolic handling of time (using convex polyhedra) to the discrete-time setting. This is basically done by computing the state space of discrete-time nets as the discretization of the state space of the corresponding dense-time model. First, we prove that this technique is correct for TPNs but not for SwPNs in general: in fact, for the latter, it may add behaviors that do not really belong to the evolution of the discrete-time net. To overcome this problem, we propose a splitting of the general polyhedron that encompasses the temporal information of the net into an union of simpler polyhedra which are safe with respect to the symbolic successor computation. We then give an algorithm that computes symbolically the state space of discrete-time SwPNs and finally exhibit a way to perform TCTL model-checking on this model.


Verification using nets Time Petri nets symbolic state space stopwatches dense-time discrete-time 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Merlin, P.: A study of the recoverability of computing systems. PhD thesis, Department of Information and Computer Science, University of California, Irvine, CA (1974)Google Scholar
  2. 2.
    Ramchandani, C.: Analysis of asynchronous concurrent systems by timed Petri nets. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA Project MAC Report MAC-TR-120 (1974)Google Scholar
  3. 3.
    Roux, O., Déplanche, A.M.: A t-time Petri net extension for real time-task scheduling modeling. European Journal of Automation (JESA) 36, 973–987 (2002)Google Scholar
  4. 4.
    Bucci, G., Fedeli, A., Sassoli, L., Vicario, E.: Time state space analysis of real-time preemptive systems. IEEE transactions on software engineering 30, 97–111 (2004)CrossRefGoogle Scholar
  5. 5.
    Roux, O.H., Lime, D.: Time Petri nets with inhibitor hyperarcs. Formal semantics and state space computation. In: Cortadella, J., Reisig, W. (eds.) ICATPN 2004. LNCS, vol. 3099, pp. 371–390. Springer, Heidelberg (2004)Google Scholar
  6. 6.
    Berthomieu, B., Lime, D., Roux, O.H., Vernadat, F.: Reachability problems and abstract state spaces for time petri nets with stopwatches. Journal of Discrete Event Dynamic Systems (DEDS) (to appear, 2007)Google Scholar
  7. 7.
    Henzinger, T.A., Manna, Z., Pnueli, A.: What good are digital clocks? In: Automata, Languages and Programming, pp. 545–558 (1992)Google Scholar
  8. 8.
    Magnin, M., Molinaro, P., Roux, O.H.: Decidability, expressivity and state-space computation of stopwatch petri nets with discrete-time semantics. In: 8th International Workshop on Discrete Event Systems (WODES 2006), Ann Arbor, USA (2006)Google Scholar
  9. 9.
    Popova, L.: On time petri nets. Journal Inform. Process. Cybern, EIK (formerly: Elektron. Inform. verarb. Kybern) 27, 227–244 (1991)zbMATHGoogle Scholar
  10. 10.
    Popova-Zeugmann, L.: Essential states in time petri nets (1998)Google Scholar
  11. 11.
    Popova-Zeugmann, L., Schlatter, D.: Analyzing paths in time petri nets. Fundamenta Informaticae 37, 311–327 (1999)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Molinaro, P., Delfieu, D., Roux, O.H.: Improving the calculus of the marking graph of Petri net with bdd like structure. In: 2002 IEEE international conference on systems, man and cybernetics (SMC 2002), Hammamet, Tunisia (2002)Google Scholar
  13. 13.
    Berthomieu, B., Diaz, M.: Modeling and verification of time dependent systems using time Petri nets. IEEE transactions on software engineering 17, 259–273 (1991)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Gardey, G., Roux, O.H., Roux, O.F.: State space computation and analysis of time Petri nets. Theory and Practice of Logic Programming (TPLP) (to appear, 2006); Special Issue on Specification Analysis and Verification of Reactive SystemsGoogle Scholar
  15. 15.
    Berthomieu, B., Vernadat, F.: State class constructions for branching analysis of time petri nets. In: Garavel, H., Hatcliff, J. (eds.) TACAS 2003. LNCS, vol. 2619, pp. 442–457. Springer, Heidelberg (2003)Google Scholar
  16. 16.
    Boucheneb, H., Gardey, G., Roux, O.H.: TCTL model checking of time Petri nets. Technical Report IRCCyN number RI2006-14 (2006)Google Scholar
  17. 17.
    Lime, D., Roux, O.: Expressiveness and analysis of scheduling extended time Petri nets. In: 5th IFAC International Conference on Fieldbus Systems and their Applications (FET 2003), Aveiro, Portugal. Elsevier Science, Amsterdam (2003)Google Scholar
  18. 18.
    Berthomieu, B., Menasche, M.: An enumerative approach for analyzing time Petri nets. IFIP Congress Series 9, 41–46 (1983)Google Scholar
  19. 19.
    Dill, D.: Timing assumptions and verification of finite-state concurrent systems. In: Sifakis, J. (ed.) CAV 1989. LNCS, vol. 407, pp. 197–212. Springer, Heidelberg (1990)Google Scholar
  20. 20.
    Magnin, M., Lime, D., Roux, O.: An efficient method for computing exact state space of Petri nets with stopwatches. In: Third International Workshop on Software Model-Checking (SoftMC 2005), Edinburgh, Scotland, UK. Electronic Notes in Theoretical Computer Science, Elsevier, Amsterdam (2005)Google Scholar
  21. 21.
    Gardey, G., Lime, D., Magnin, M., Roux, O.H.: Roméo: A tool for analyzing time Petri nets. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 418–423. Springer, Heidelberg (2005)Google Scholar
  22. 22.
    Hadjidj, R., Boucheneb, H.: On-the-fly tctl model checking for time petri nets using state class graphs. In: ACSD, pp. 111–122. IEEE Computer Society, Los Alamitos (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Morgan Magnin
    • 1
  • Didier Lime
    • 1
  • Olivier (H. ) Roux
    • 1
  1. 1.IRCCyN, CNRS UMR 6597NantesFrance

Personalised recommendations