Polynomial-Time Many-One reductions for Petri nets

  • Catherine Dufourd
  • Alain Finkel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1346)


We apply to Petri net theory the technique of polynomialtime many-one reductions. We study boundedness, reachability, deadlock, liveness problems and some of their variations. We derive three main results. Firstly, we highlight the power of expression of reachability which can polynomially give evidence of unboundedness. Secondly, we prove that reachability and deadlock are polynomially-time equivalent; this improves the known recursive reduction and it complements the result of Cheng and al. [4]. Moreover, we show the polynomial equivalence of liveness and t-liveness. Hence, we regroup the problems in three main classes: boundedness, reachability and liveness. Finally, we give an upper bound on the boundedness for post self-modified nets: \(2^{O(size(N)^2 *\log size(N))} \). This improves a decidability result of Valk [18].

Key words

Petri net theory Complexity Theory Program Verification Equivalences 


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  1. 1.
    T. Araki and T. Kasami. Some decision problems related to the reachability problem for Petri nets. TCS, 3(1):85–104, 1977.MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Z. Bouziane. Algorithmes primitifs récursifs et problèmes Expspace-complets pour les réseaux de Petri cycliques. PhD thesis, LSV, école Normale Supérieure de Cachan, France, November 1996.Google Scholar
  3. 3.
    E. Cardoza, R. Lipton, and A. Meyer. Exponential space complete problems for Petri nets and commutative semigroups. In Proc. of the 8th annual ACM Symposium on theory of computing, pages 50–54, May 1976.Google Scholar
  4. 4.
    A. Cheng, J. Esparza, and J. Palsberg. Complexity result for 1-safe nets. TCS, 147:117–136, 1995.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Desel and J. Esparza. Free Choice Petri Nets. Cambridge University Press, 1995.Google Scholar
  6. 6.
    C. Dufourd and A. Finkel. A polynomial λ-bisimilar normalization for Petri nets. Technical report, LIFAC, ENS de Cachan, July 1996. Presented at AFL'96, Salgótarján, Hungary, 1996.Google Scholar
  7. 7.
    J. Esparza and M. Nielsen. Decidability issues on Petri nets — a survey. Bulletin of the EATCS, 52:254–262, 1994.Google Scholar
  8. 8.
    M. Hack. Decidability questions for Petri Nets. PhD thesis, M.I.T., 1976.Google Scholar
  9. 9.
    J.E. Hopcroft and J.D. Ullman. Introduction to automata theory, languages, and computation. Addison-Wesley, 1979.Google Scholar
  10. 10.
    M. Jantzen. Complexity of Place/Transition nets. In Petri nets: central models and their properties, volume 254 of LNCS, pages 413–434. Springer-Verlag, 1986.Google Scholar
  11. 11.
    R.M. Karp and R.E. Miller. Parallel program schemata. Journal of Computer and System Sciences, 3:146–195, 1969.MathSciNetGoogle Scholar
  12. 12.
    R. Kosaraju. Decidability of reachability in vector addition systems. In Proc. of the 14th Annual ACM Symposium on Theory of Computing, San Francisco, pages 267–281, May 1982.Google Scholar
  13. 13.
    R.J. Lipton. The reachability problem requires exponential space. Technical Report 62, Yale University, Department of computer science, January 1976.Google Scholar
  14. 14.
    E.W. Mayr. An algorithm for the general Petri net reachability problem. SIAM Journal on Computing, 13(3):441–460, 1984.MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    E.W. Mayr and R. Meyer. The complexity of the word problem for commutative semigroups and polynomial ideals. Advances in Mathematics, 46:305–329, 1982.MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    J.L. Peterson. Petri Net Theory and the Modeling of Systems. Prentice Hall, 1981.Google Scholar
  17. 17.
    C. Rackoff. The covering and boundedness problems for vector addition systems. TCS, 6(2), 1978.Google Scholar
  18. 18.
    R. Valk. Self-modifying nets, a natural extension of Petri nets. In Proc. of ICALP'78, volume 62 of LNCS, pages 464–476. Springer-Verlag, September 1978.Google Scholar
  19. 19.
    R. Valk. Generalizations of Petri nets. In Proc. of the 10th Symposium on Mathematical Fondations of Computer Science, volume 118 of LNCS, pages 140–155. Springer-Verlag, 1981.Google Scholar
  20. 20.
    R. Valk and G. Vidal-Naquet. Petri nets and regular languages. Journal of Computer and System Sciences, 23(3):299–325, 1981.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Catherine Dufourd
    • 1
  • Alain Finkel
    • 1
  1. 1.LSV, CNRS URA 2236; ENS de CachanCachan CedexFrance

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