A Framework for Classical Petri Net Problems: Conservative Petri Nets as an Application

  • Ernst W. Mayr
  • Jeremias Weihmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8489)


We present a framework based on permutations of firing sequences and on canonical firing sequences to approach computational problems involving classes of Petri nets with arbitrary arc multiplicities. As an example of application, we use these techniques to obtain PSPACE-completeness for the reachability and the covering problems of conservative Petri nets, generalizing known results for ordinary 1-conservative Petri nets. We also prove PSPACE-completeness for the RecLFS and the liveness problems of conservative Petri nets, for which, in case of ordinary 1-conservative Petri nets, PSPACE-membership but no matching lower bound has been known. Last, we show PSPACE-completeness for the containment and equivalence problems of conservative Petri nets. PSPACE-hardness of the problems mentioned above still holds if they are restricted to ordinary 1-conservative Petri nets.


Turing Machine Transition Sequence Reachability Problem Liveness Problem Polynomial Time Reduction 
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  1. 1.
    Cardoza, E., Lipton, R., Meyer, A.R.: Exponential space complete problems for Petri nets and commutative semigroups (preliminary report). In: Proceedings of the 8th ACM Symposium on Theory of Computing (STOC 1976), pp. 50–54. ACM (1976)Google Scholar
  2. 2.
    Esparza, J.: Petri nets, commutative context-free grammars, and basic parallel processes. Fundamenta Informaticae 31(1), 13–25 (1997)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Esparza, J., Nielsen, M.: Decibility issues for Petri nets - a survey. Journal of Information Processing and Cybernetics 30(3), 143–160 (1994)zbMATHGoogle Scholar
  4. 4.
    Hack, M.: The recursive equivalence of the reachability problem and the liveness problem for Petri nets and vector addition systems. In: IEEE Conference Record of the 15th Annual Symposium on Switching and Automata Theory, pp. 156–164 (1974)Google Scholar
  5. 5.
    Howell, R.R., Jancar, P., Rosier, L.E.: Completeness results for single-path Petri nets. Information and Computation 106(2), 253–265 (1993)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Howell, R.R., Rosier, L.E.: Completeness results for conflict-free vector replacement systems. Journal of Computer and System Sciences 37(3), 349–366 (1988)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Howell, R.R., Rosier, L.E., Yen, H.C.: Normal and sinkless Petri nets. In: Csirik, J., Demetrovics, J., Gécseg, F. (eds.) Proceedings of the 1989 International Conference on Fundamentals of Computation Theory (FCT 1989). LNCS, vol. 380, pp. 234–243. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  8. 8.
    Huynh, D.T.: The complexity of semilinear sets. In: de Bakker, J., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 324–337. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  9. 9.
    Huynh, D.T.: A simple proof for the \(\Sigma_2^p\) upper bound of the inequivalence problem for semilinear sets. Elektronische Informationsverarbeitung und Kybernetik 22, 147–156 (1986)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Jones, N.D., Landweber, L.H., Lien, Y.E.: Complexity of some problems in Petri nets. Theoretical Computer Science 4(3), 277–299 (1977)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lien, Y.E.: A note on transition systems. Information Sciences 10(4), 347–362 (1976)CrossRefGoogle Scholar
  13. 13.
    Lien, Y.E.: Termination properties of generalized Petri nets. SIAM Journal on Computing 5(2), 251–265 (1976)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mayr, E.W.: An algorithm for the general Petri net reachability problem. SIAM Journal on Computing 13(3), 441–460 (1984)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mayr, E.W., Meyer, A.R.: The complexity of the word problems for commutative semigroups and polynomial ideals. Advances in Mathematics 46(3), 305–329 (1982)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mayr, E.W., Weihmann, J.: Completeness results for generalized communication-free Petri nets with arbitrary edge multiplicities. In: Abdulla, P.A., Potapov, I. (eds.) RP 2013. LNCS, vol. 8169, pp. 209–221. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  17. 17.
    Teruel, E., Silva, M.: Well-formedness of equal conflict systems. In: Valette, R. (ed.) ICATPN 1994. LNCS, vol. 815, pp. 491–510. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  18. 18.
    Watanabe, T., Mizobata, Y., Onaga, K.: Legal firing sequence and related problems of Petri nets. In: Proceedings of the 3rd International Workshop on Petri Nets and Performance Models (PNPM 1989), pp. 277–286 (1989)Google Scholar
  19. 19.
    Yen, H.C.: On reachability equivalence for BPP-nets. Theoretical Computer Science 179, 301–317 (1997)MathSciNetCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Ernst W. Mayr
    • 1
  • Jeremias Weihmann
    • 1
  1. 1.Technische Universität MünchenGarchingGermany

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