Characterizing Stable Inequalities of Petri Nets

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9115)

Abstract

One way to express correctness of a Petri net \(N\) is to specify a linear inequality \(U\), requiring each reachable marking of \(N\) to satisfy \(U\). A linear inequality \(U\) is stable if it is preserved along steps. If \(U\) is stable, then verifying correctness reduces to checking \(U\) in the initial marking of \(N\). In this paper, we characterize classes of stable linear inequalities of a given Petri net by means of structural properties. Thereby, we generalize classical results on traps, co-traps, and invariants. We show how to decide stability of a given inequality. For a certain class of inequalities, we present a polynomial time decision procedure.

Keywords

Petri net analysis Inductive invariants Linear inequalities Stable properties Traps Co-traps Invariants 

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References

  1. 1.
    Abdallah, I., ElMaraghy, H.: Deadlock prevention and avoidance in fms: A petri net based approach. The International Journal of Advanced Manufacturing Technology 14(10), 704–715 (1998)CrossRefGoogle Scholar
  2. 2.
    Cardoza, E., Lipton, R., Meyer, A.R.: Exponential space complete problems for petri nets and commutative semigroups. In: Proceedings of the 8th Annual ACM Symposium on Theory of Computing, pp. 50–54 (1976)Google Scholar
  3. 3.
    Colom, J.M., Silva, M.: Convex geometry and semiflows in P/T nets. A comparative study of algorithms for computation of minimal p-semiflows. In: Proceedings of 10th International Conference on Applications and Theory of Petri Nets, Bonn, Germany, June 1989. Advances in Petri Nets 1990, pp. 79–112 (1989)Google Scholar
  4. 4.
    Desel, J., Neuendorf, K.P., Radola, M.D.: Proving nonreachability by modulo-invariants. Theoretical Computer Science 153(1–2), 49–64 (1996)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Desel, J.: Struktur und Analyse von Free-Choice-Petrinetzen. Deutscher Universitätsverlag, DUV Informatik (1992)Google Scholar
  6. 6.
    Emerson, E.A., Halpern, J.Y.: “sometimes” and “not never” revisited: On branching versus linear time temporal logic. J. ACM 33(1), 151–178 (1986)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ezpeleta, J., Colom, J., Martinez, J.: A petri net based deadlock prevention policy for flexible manufacturing systems. IEEE Transactions on Robotics and Automation 11(2), 173–184 (1995)CrossRefGoogle Scholar
  8. 8.
    Ezpeleta, J., Couvreur, J., Silva, M.: A new technique for finding a generating family of siphons, traps and st-components. application to colored petri nets. In: Advances in Petri Nets 1993, Papers from the 12th International Conference on Applications and Theory of Petri Nets, Gjern, Denmark, pp. 126–147 (1991)Google Scholar
  9. 9.
    Fischer, M.J., Rabin, M.O.: Super-exponential complexity of presburger arithmetic, pp. 27–41 (1974)Google Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W.H. Freeman & Co., New York, NY, USA (1990)Google Scholar
  11. 11.
    Ginsburg, S., Spanier, E.H.: Semigroups, presburger formulas, and languages. Pacific Journal of Mathematics 16(2), 285–296 (1966)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hack, M.: Analysis production schemata by Petri nets. Master’s thesis, Massachusetts Institute of Technology, Cambridge, Mass (1972)Google Scholar
  13. 13.
    Heiner, M., Gilbert, D., Donaldson, R.: Petri nets for systems and synthetic biology. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 215–264. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  14. 14.
    Lautenbach, K.: Linear algebraic techniques for place/transition nets. In: Brauer, W., Reisig, W., Rozenberg, G. (eds.) Petri Nets: Central Models and Their Properties. Lecture Notes in Computer Science, vol. 254, pp. 142–167. Springer, Berlin Heidelberg (1987)CrossRefGoogle Scholar
  15. 15.
    Leroux, J.: The general vector addition system reachability problem by presburger inductive invariants. Logical Methods in Computer Science 6(3) (2010)Google Scholar
  16. 16.
    Leroux, J.: Vector addition systems reachability problem (a simpler solution). In: Voronkov, A. (ed.) The Alan Turing Centenary Conference, Turing-100, Manchester UK June 22–25, 2012, Proceedings. EPiC Series, vol. 10, pp. 214–228 (2012)Google Scholar
  17. 17.
    Mitchell, J.E.: Branch-and-cut algorithms for combinatorial optimization problems. Handbook of applied optimization, pp. 65–77 (2002)Google Scholar
  18. 18.
    Murata, T.: Petri nets: properties, analysis and applications. In: Proceedings of the IEEE, pp. 541–580 (Apr 1989)Google Scholar
  19. 19.
    Papadimitriou, C.H.: On the complexity of integer programming. J. ACM 28(4), 765–768 (1981)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Pascoletti, K.H.: Diophantische Systeme und Lösungsmethoden zur Bestimmung aller Invarianten in Petri-Netzen. GMD-Bericht Nr. 160, R. Oldenbourg Verlag (1986)Google Scholar
  21. 21.
    Reisig, W.: Understanding Petri Nets: Modeling Techniques, Analysis Methods. Springer, Case Studies (2013)Google Scholar
  22. 22.
    Silva, M., Teruel, E., Colom, J.M.: Linear algebraic and linear programming techniques for the analysis of place or transition net systems. In: Lectures on Petri Nets I: Basic Models, Advances in Petri Nets, pp. 309–373 (1996)Google Scholar
  23. 23.
    Starke, P.H.: Analyse von Petri-Netz-Modellen, pp. 1–253 (1990)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany

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