On Functions Weakly Computable by Petri Nets and Vector Addition Systems

  • J. Leroux
  • Ph. Schnoebelen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8762)


We show that any unbounded function weakly computable by a Petri net or a VASS cannot be sublinear. This answers a long-standing folklore conjecture about weakly computing the inverses of some fast-growing functions. The proof relies on a pumping lemma for sets of runs in Petri nets or VASSes.


Computable Function Reachability Problem Containment Problem Primitive Recursive Hardness Proof 
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  1. 1.
    Abdulla, P.A., Čerāns, K., Jonsson, B., Tsay, Y.-K.: Algorithmic analysis of programs with well quasi-ordered domains. Information & Computation 160(1-2), 109–127 (2000)CrossRefzbMATHGoogle Scholar
  2. 2.
    Araki, T., Kasami, T.: Some decision problems related to the reachability problem for Petri nets. Theoretical Computer Science 3(1), 85–104 (1976)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Baker Jr., H.G.: Rabin’s proof of the undecidability of the reachability set inclusion problem of vector addition systems. Memo 79, Computation Structures Group, Project MAC, M.I.T. (July 1973)Google Scholar
  4. 4.
    Bouajjani, A., Emmi, M.: Analysis of recursively parallel programs. In: POPL 2012, pp. 203–214. ACM (2012)Google Scholar
  5. 5.
    Demri, S., Figueira, D., Praveen, M.: Reasoning about data repetitions with counter systems. In: LICS 2013, pp. 33–42. IEEE (2013)Google Scholar
  6. 6.
    Demri, S., Jurdziński, M., Lachish, O., Lazić, R.: The covering and boundedness problems for branching vector addition systems. Journal of Computer and System Sciences 79(1), 23–38 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dickson, L.E.: Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. Amer. Journal Math. 35, 413–422 (1913)CrossRefzbMATHGoogle Scholar
  8. 8.
    Dufourd, C., Finkel, A., Schnoebelen, Ph.: Reset nets between decidability and undecidability. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 103–115. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  9. 9.
    Figueira, D., Figueira, S., Schmitz, S., Schnoebelen, Ph.: Ackermannian and primitive-recursive bounds with Dickson’s Lemma. In: LICS 2011, pp. 269–278. IEEE (2011)Google Scholar
  10. 10.
    Finkel, A., Schnoebelen, Ph.: Well-structured transition systems everywhere! Theoretical Computer Science 256(1–2), 63–92 (2001)Google Scholar
  11. 11.
    Haase, C., Schmitz, S., Schnoebelen, Ph.: The power of priority channel systems. In: D’Argenio, P.R., Melgratti, H. (eds.) CONCUR 2013 – Concurrency Theory. LNCS, vol. 8052, pp. 319–333. Springer, Heidelberg (2013)Google Scholar
  12. 12.
    Hack, M.: Decidability Questions for Petri Nets. PhD thesis, Massachusetts Institute of Technology, Available as report MIT/LCS/TR-161 (June 1976)Google Scholar
  13. 13.
    Hack, M.: The equality problem for vector addition systems is undecidable. Theoretical Computer Science 2(1), 77–95 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Haddad, S., Schmitz, S., Schnoebelen, Ph.: The ordinal-recursive complexity of timed-arc Petri nets, data nets, and other enriched nets. In: LICS 2012, pp. 355–364. IEEE (2012)Google Scholar
  15. 15.
    Hopcroft, J., Pansiot, J.-J.: On the reachability problem for 5-dimensional vector addition systems. Theoretical Computer Science 8(2), 135–159 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jančar, P.: Decidability of a temporal logic problem for Petri nets. Theoretical Computer Science 74(1), 71–93 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jančar, P.: Nonprimitive recursive complexity and undecidability for Petri net equivalences. Theoretical Computer Science 256(1-2), 23–30 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jančar, P.: Undecidability of bisimilarity for Petri nets and some related problems. Theoretical Computer Science 148(2), 281–301 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Karp, R.M., Miller, R.E.: Parallel program schemata. Journal of Computer and System Sciences 3(2), 147–195 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kruskal, J.B.: The theory of well-quasi-ordering: A frequently discovered concept. Journal of Combinatorial Theory, Series A 13(3), 297–305 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lazić, R.: The reachability problem for vector addition systems with a stack is not elementary. CoRR, abs/1310.1767 (2013)Google Scholar
  22. 22.
    Leroux, J.: Vector addition systems reachability problem (a simpler solution). In: The Alan Turing Centenary Conference (Turing-100). EasyChair Proceedings in Computing, vol. 10, pp. 214–228. EasyChair (2012)Google Scholar
  23. 23.
    Leroux, J., Praveen, M., Sutre, G.: Hyper-Ackermannian bounds for pushdown vector addition systems. In: CSL-LICS 2014. ACM (2014)Google Scholar
  24. 24.
    Mayr, E.W.: The complexity of the finite containment problem for Petri nets. Master’s thesis, Massachusetts Institute of Technology, Available as report MIT/LCS/TR-181 (June 1977)Google Scholar
  25. 25.
    Mayr, E.W., Meyer, A.R.: The complexity of the finite containment problem for Petri nets. Journal of the ACM 28(3), 561–576 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Müller, H.: Weak Petri net computers for Ackermann functions. Elektronische Informationsverarbeitung und Kybernetik 21(4-5), 236–246 (1985)MathSciNetGoogle Scholar
  27. 27.
    Reinhardt, K.: Reachability in Petri nets with inhibitor arcs. Electr. Notes Theor. Comput. Sci. 223, 239–264 (2008)CrossRefGoogle Scholar
  28. 28.
    Schmitz, S.: Complexity hierarchies beyond elementary. Research Report 1312.5686 [cs.CC], Computing Research Repository (December 2013)Google Scholar
  29. 29.
    Schmitz, S., Schnoebelen, Ph.: Algorithmic aspects of WQO theory. Lecture notes (2012)Google Scholar
  30. 30.
    Schnoebelen, Ph.: Verifying lossy channel systems has nonprimitive recursive complexity. Information Processing Letters 83(5), 251–261 (2002)Google Scholar
  31. 31.
    Schnoebelen, Ph.: Revisiting Ackermann-hardness for lossy counter machines and reset Petri nets. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 616–628. Springer, Heidelberg (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • J. Leroux
    • 1
  • Ph. Schnoebelen
    • 2
  1. 1.LaBRIUniv. Bordeaux & CNRSFrance
  2. 2.LSVENS Cachan & CNRSFrance

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