A polynomial-time graph algorithm to decide liveness of some basic classes of bounded Petri nets

  • Kamel Barkaoui
  • Michel Minoux
Submitted Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 616)


This paper is related to structural analysis of Petri nets where liveness and boundedness issues are addressed through the analysis of the combinatorial properties of the underlying graph. We first recall a number of basic results about liveness and boundedness involving combinatorial substructures (deadlocks and traps). It is then shown that testing whether a bounded Extended Free Choice net or a Non Self-Controlling net is structurally live can be reduced to the search for a strongly connected deadlock which is not a trap. This problem, in turn, is shown to be solvable in polynomial time through a purely combinatorial algorithm making combined use of Tarjan's strong connectivity algorithm and Minoux's LTUR algorithm for solving Horn satisfiability problems. Once structural liveness has been proved, testing liveness for a given initial marking is already known to be polynomially solvable.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    K.Barkaoui: Contribution aux méthodes d'analyse des réseaux de Petri par la théorie des graphes. Thèse Doct, Université Paris 6, 1988.Google Scholar
  2. 2.
    K.Barkaoui, M.Minoux: On the Petri nets analysis based on graph theory. 6th SIAM Conference on Discrete Mathematics, June1992.Google Scholar
  3. 3.
    K.Barkaoui, B.Lemaire: An effective characterization of minimal deadlocks and traps based on graph theory. Proceedings of 10th Inter.conf. on appl and theory of Petri nets Bonn, June 1989.Google Scholar
  4. 4.
    C.Berge: Graphes et hypergraphes. Dunod, Paris, 1970.Google Scholar
  5. 5.
    G.Berthelot: Tansformations and decompositions of nets. Petti nets: Central models and their properties. Advances in Petri nets, Part 1, LNCS 254, Ed by W.Brauer, W.Reisig, G.Rozenberg, Springer-Verlag, 1987Google Scholar
  6. 6.
    E.Best, P.S.Thiagarajan: Some classes of live and safe Petri nets. Concurrency and net, Advances in Petri nets, K.Voss, HJ.Genrich, G.Rozenberg, Springer-Verlag Ed, 1987.Google Scholar
  7. 7.
    G.W.Brams: Réseaux de Petri-Théorie. Tomel Masson, Paris, 1982.Google Scholar
  8. 8.
    F.Commoner. Deadlocks in Petri nets. Applied Data Res Inc Wakefield, MA1972.Google Scholar
  9. 9.
    J.Esparza, M.Silva: A polynomial-time algorithm to decide liveness of bounded free choice nets. Hildesheimer Informatikberichte 12/90 Institut fur Informatik, Univ Hildesheim.Google Scholar
  10. 10.
    W.Griese: Lebendigkeit in NSK Petri-netzen. Tech.Univ.Munchen Info N∘6, 1979.Google Scholar
  11. 11.
    M. Hack: Analysis of production schemata by Petri nets. TR-94, MIT, Project MAC, Boston 1972, Corrected 1974.Google Scholar
  12. 12.
    M.Jantzen, R.Valk: Formal properties of place transition nets. Net thory and Applications LNCS N∘ 84, W.Brauer Springer-Verlag Ed, 1980.Google Scholar
  13. 13.
    N.Jones, L.Landweber, Y.Lien: Complexity of some problems in Petri nets. TCS Vol 4, 1977.Google Scholar
  14. 14.
    N.Karmarkar. A new polynomial-time algorithm in linear programming. Proceedings of the 16th Anual ACM STOC, 1984.Google Scholar
  15. 15.
    K.Lautenbach: Linear algebraic techniques for Place/Transition nets. Petri nets: Central models and their properties. Advances in Petri nets, Part 1, LNCS 254, Ed by W.Brauer, W.Reisig, G.Rozenberg, Springer-Verlag, 1987.Google Scholar
  16. 16.
    K.Lautenbach: Linear algebraic calculation of deadlocks and traps. Concurrency and net, Advances in Petri nets, K.Voss, H.J.Genrich, G.Rozenberg. Springer-Verlag Ed, 1987.Google Scholar
  17. 17.
    R.Lipton: The reachability problem and the boundedness problem for Petri nets are exponential-space hard. Tech.Report.N∘ 62, Yale University, New Haven 1976.Google Scholar
  18. 18.
    J.Martinez, M.Silva: A simple and fast algorithm to obtain all invariants of a generalized Petri net. Informatik-Fachbrichte 52, C. Ed by C.Girault, W.Reizig, Springer-Verlag, 1982.Google Scholar
  19. 19.
    E.W Mayr: An algorithm for the general Perti net reachability problem. SIAM. J. of Computing 13, 1984.Google Scholar
  20. 20.
    G.Memmi: Méthodes d'analyse des réseaux de Petri, réseaux à files, application aux systèmes en temps réel. Thése Doct. Etat, Université Paris 6, 1983.Google Scholar
  21. 21.
    MMinoux: LTUR: A simplified linear-time unit resolution algorithm for Horn formulae with computer implementation. Info.Process-Lett. N∘29, 1988.Google Scholar
  22. 22.
    M.Minoux, K.Barkaoui: Polynomial time algorithms for proving or disproving Commoner's structural property in Petri nets. Proc of 9th Inter conf on application and theory of Petri nets, Venice, 1988.Google Scholar
  23. 23.
    M.Minoux, K.Barkaoui: Deadlocks and traps in Petri nets as Horn-satisfiality solutions and some related polynomially solvable problems. Discrete Applied Mathematics N∘29, 1990.Google Scholar
  24. 24.
    W.Reisig: Petri nets, An Introduction. EATCS, Monographs on theoretical Computer Science. Springer Verlag, 1985.Google Scholar
  25. 25.
    W.Reisig: Place-Transition Systems. Petri nets: Central models and their properties. Advances in Petri nets, Part 1, LNCS 254, Ed by W.Brauer, W.Reisig, G.Rozenberg, Springer-Verlag, 1987.Google Scholar
  26. 26.
    R.Tarjan: Depth-first search and linear graph algorithms. SIAM. Jour. Comput. Vol N∘2, 1972.Google Scholar
  27. 27.
    J.M.Toudic: Algorithmes d'analyse structurelle des réseaux de Petri. Thèse 3ème cycle, Université Paris 6, 1985.Google Scholar
  28. 28.
    A.Valmari: Stubborn sets. Proc of 10th Inter.conf. on appl and theory of Petri nets, Bonn, 1989Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Kamel Barkaoui
    • 1
  • Michel Minoux
    • 2
  1. 1.Laboratoire CEDRICConservatoire National des Arts et MétiersParisFrance
  2. 2.Laboratoire MASIUniversité Paris 6ParisFrance

Personalised recommendations