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Discrete Event Dynamic Systems

, Volume 26, Issue 3, pp 413–437 | Cite as

Determinization of timed Petri nets behaviors

  • Jan KomendaEmail author
  • Sébastien Lahaye
  • Jean-Louis Boimond
Article

Abstract

In this paper we are interested in sequentialization of formal power series with coefficients in the semiring \((\mathbb {R}\cup \{- \infty \},\max ,+)\) which represent the behavior of timed Petri nets. Several approaches make it possible to derive nondeterministic (max, + ) automata modeling safe timed Petri nets. Their nondeterminism is a serious drawback since determinism is a crucial property for numerous results on (max, + ) automata (in particular, for applications to performance evaluation and control) and existing procedures for determinization succeed only for restrictive classes of (max, + ) automata. We present a natural semi-algorithm for determinization of behaviors based on the semantics of bounded timed Petri nets. The resulting deterministic (max, + ) automata can be infinite, but a sufficient condition called strong liveness is proposed to ensure the termination of the semi-algorithm. It is shown that strong liveness is closely related to bounded fairness, which has been widely studied for Petri nets and other models for concurrency. Moreover, if the net cannot be sequentialized we propose a restriction of its logical behavior so that the sufficient condition becomes satisfied for the restricted net. The restriction is based on the synchronous product with non injectively labeled scheduler nets that are built in an incremental hierarchical way from simple scheduler nets.

Keywords

Timed Petri nets (max + ) automata Determinization 

Notes

Acknowledgments

This work has been supported by RVO 67985840, by GAČR project 15-02532S, and by Czech Ministry of Education project LH13012.

References

  1. Alur R, Dill D (1994) A theory of timed automata. Theor Comput Sci 126:183–235MathSciNetCrossRefzbMATHGoogle Scholar
  2. Baccelli F, Cohen G, Olsder GJ, Quadrat JP (1992) Synchronization and linearity. WileyGoogle Scholar
  3. Badouel E, Bouillard A, Darondeau P, Komenda J (2011) Residuation of tropical series: rationality issues. In: Joint 50th IEEE conference on decision and control and european control conference (CDC-ECC’11)Google Scholar
  4. David R, Alla H (2010) Discrete, continuous and hybrid Petri nets, 2nd edn. Springer, ParisCrossRefzbMATHGoogle Scholar
  5. Gaubert S (1995) Performance evaluation of (max, + ) automata. IEEE TAC 40 (12):2014–2025MathSciNetzbMATHGoogle Scholar
  6. Gaubert S, Mairesse J (1999a) Asymptotic analysis of heaps of pieces and application to timed Petri nets. In: Petri nets and performance models (PNPM’99), pp 158–169Google Scholar
  7. Gaubert S, Mairesse J (1999b) Modeling and analysis of timed Petri nets using heaps of pieces. IEEE TAC 44(4):683–698MathSciNetzbMATHGoogle Scholar
  8. Giua A (2013) Supervisory control of Petri nets with language specifications. In: Seatzu C, Silva M, van Schuppen J H (eds) Control of discrete-event systems – Automata and Petri net perspectives, no. 433 in Lecture Notes in Control and Information Sciences. doi: 10.1007/978-1-4471-4276-8. Springer Verlag London Ltd., London, pp 235–255
  9. Gohari P, Wonham WM (2005) Efficient implementation of fairness in discrete-event systems using queues. IEEE TAC 50(11):1845–1849MathSciNetGoogle Scholar
  10. Kirsten D (2008) A burnside approach to the termination of Mohri’s algorithm for polynomially ambiguous min-plus-automata. RAIRO - Theor Informatics Appl 42 (3):553–581MathSciNetCrossRefzbMATHGoogle Scholar
  11. Klimann I, Lombardy S, Mairesse J, Prieur C (2004) Deciding unambiguity and sequentiality from a finitely ambiguous max-plus automaton, TCSGoogle Scholar
  12. Komenda J, Lahaye S, Boimond J L (2009) Supervisory control of (max, + ) automata: A behavioral approach. Discret Event Dyn Syst 19(4):525–549MathSciNetCrossRefzbMATHGoogle Scholar
  13. Komenda J, Lahaye S, Boimond J L (2013) Séquentialisation des réseaux de Petri temporisés, JESA, special issue including the proceedings of MSR 2013 43(1-3)Google Scholar
  14. Lahaye S, Komenda J, Boimond JL (2014a) Compositions of (max, + ) automata, Discrete Event Dynamic SystemsGoogle Scholar
  15. Lahaye S, Komenda J, Boimond J L (2014b) Modeling of timed Petri nets using deterministic (max, + ) automata. In: Proceedings of WODES, Cachan, France, May 22-24, 2014Google Scholar
  16. Lime D, Roux OH, Jard C (2012) Clock transition systems. In: CS & P, pp 227–238Google Scholar
  17. Lombardy S, Sakarovitch J (2006) Sequential? Theor Comput Sci 359 (1-2):224–244MathSciNetCrossRefzbMATHGoogle Scholar
  18. Merlin P M (1974) A study of the recoverability of computing systems, PhD thesis. University of California, IrvineGoogle Scholar
  19. Mohri M (1997) Finite-state transducers in language and speech processing. Comput Linguis 23(2):269–311MathSciNetGoogle Scholar
  20. Ramchandani C (1973) Analysis of asynchronous concurrent systems by timed Petri nets, Ph.d. thesis, M.I.TGoogle Scholar
  21. Rutten J J M M (2003) Behavioural differential equations: a coinductive calculus of streams, automata, and power series. Theor Comput Sci 308(1-3):1–53MathSciNetCrossRefzbMATHGoogle Scholar
  22. Silva M, Murata T (1992) B-fairness and structural b-fairness in Petri net models of concurrent systems. J Comput Syst Sci 44(3):447–477MathSciNetCrossRefzbMATHGoogle Scholar
  23. Sreenivas R S (1997) On supervisory policies that enforce global fairness and bounded fairness in partially controlled Petri nets. Discret Event Dyn Syst 7(2):191–208CrossRefzbMATHGoogle Scholar
  24. Thistle J G, Wonham W M (1991) Control of omega-automata, church’s problem, and the emptiness problem for tree omega-automata. In: Computer Science Logic, 5th Workshop, CSL ’91, Berne, Switzerland, October 7-11, Proceedings, Springer, Lecture Notes in Computer Science, vol 626, pp 367–382Google Scholar
  25. Yamalidou K, Moody J O, Lemmon M D, Antsaklis P J (1996) Feedback control of Petri nets based on place invariants. Automatica 32(1):15–28MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Jan Komenda
    • 2
    Email author
  • Sébastien Lahaye
    • 1
  • Jean-Louis Boimond
    • 1
  1. 1.LARISLUNAM UniversitéAngersFrance
  2. 2.Institute of Mathematics - Brno BranchCzech Academy of SciencesPragueCzech Republic

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