Advertisement

Decidability Border for Petri Nets with Data: WQO Dichotomy Conjecture

  • Sławomir LasotaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9698)

Abstract

In Petri nets with data, every token carries a data value, and executability of a transition is conditioned by a relation between data values involved. Decidability status of various decision problems for Petri nets with data may depend on the structure of data domain. For instance, if data values are only tested for equality, decidability status of the reachability problem is unknown (but decidability is conjectured). On the other hand, the reachability problem is undecidable if data values are additionally equipped with a total ordering.

We investigate the frontiers of decidability for Petri nets with various data, and formulate the WQO Dichotomy Conjecture: under a mild assumption, either a data domain exhibits a well quasi-order (in which case one can apply the general setting of well-structured transition systems to solve problems like coverability or boundedness), or essentially all the decision problems are undecidable for Petri nets over that data domain.

Notes

Acknowledgments

In first place, I am very grateful Wojtek Czerwiński and Paweł Parys, with whom I currently work on the WQO Dichotomy Conjecture, for many fruitful discussions and for reading a draft of this paper. Furthermore, I thank Sylvain Schmitz for our discussion and for his interesting ideas towards resolving the conjecture. Finally, I would like to thank my colleagues: Mikołaj Bojańczyk, Lorenzo Clemente, Bartek Klin, Asia Ochremiak and Szymek Toruńczyk for the joint research effort on sets with atoms, a long-term research project, of which the present note constitutes a small part.

References

  1. 1.
    Abdulla, P.A., Nylén, A.: Timed petri nets and BQOs. In: Colom, J.-M., Koutny, M. (eds.) ICATPN 2001. LNCS, vol. 2075, pp. 53–70. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. 2.
    Bojańczyk, M., Braud, L., Klin, B., Lasota, S.: Towards nominal computation. In: Proceedings of the POPL, pp. 401–412 (2012)Google Scholar
  3. 3.
    Bojańczyk, M., Klin, B., Lasota, S.: Automata theory in nominal sets. Logical Methods Comput. Sci. 10(3), 1–44 (2014). Paper 4MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bojańczyk, M., Klin, B., Lasota, S., Toruńczyk, S.: Turing machines with atoms. In: LICS, pp. 183–192 (2013)Google Scholar
  5. 5.
    Cervesato, I., Durgin, N.A., Lincoln, P., Mitchell, J.C., Scedrov, A.: A meta-notation for protocol analysis. In: Proceedings of the CSFW 1999, pp. 55–69 (1999)Google Scholar
  6. 6.
    Cherlin, G.: The classification of countable homogeneous directed graphs and countable homogeneous n-tournaments. Memoirs of the American Mathematical Society, vol. 621. American Mathematical Society (1998)Google Scholar
  7. 7.
    Clemente, L., Lasota, S.: Reachability analysis of first-order definable pushdown systems. In: Proceedings of the CSL 2015, pp. 244–259 (2015)Google Scholar
  8. 8.
    Delzanno, G.: An overview of MSR(C): a clp-based framework for the symbolic verification of parameterized concurrent systems. Electr. Notes Theor. Comput. Sci. 76, 65–82 (2002)CrossRefGoogle Scholar
  9. 9.
    Delzanno, G.: Constraint multiset rewriting. Technical report DISI-TR-05-08, DISI, Universitá di Genova (2005)Google Scholar
  10. 10.
    Finkel, A., Goubault-Larrecq, J.: Forward analysis for wsts, part I: completions. In: Proceedings of the STACS 2009, pp. 433–444 (2009)Google Scholar
  11. 11.
    Finkel, A., Schnoebelen, P.: Well-structured transition systems everywhere! Theor. Comput. Sci. 256(1–2), 63–92 (2001)Google Scholar
  12. 12.
    Fraïssé, R.: Theory of Relations. North-Holland (1953)Google Scholar
  13. 13.
    Genrich, H.J., Lautenbach, K.: System modelling with high-level petri nets. Theor. Comput. Sci. 13, 109–136 (1981)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Haddad, S., Schmitz, S., Schnoebelen, P.: The ordinal-recursive complexity of timed-arc Petri nets, data nets, and other enriched nets. In: Proceedings of the LICS 2012, pp. 355–364 (2012)Google Scholar
  15. 15.
    Henson, W.: Countable homogeneous relational structures and \(\aleph _0\)-categorical theories. J. Symb. Logic 37, 494–500 (1972)CrossRefGoogle Scholar
  16. 16.
    Higman, G.: Ordering by divisibility in abstract algebras. Proc. London Math. Soc. (3) 2(7), 326–336 (1952)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hofman, P., Lasota, S., Lazic, R., Leroux, J., Schmitz, S., Totzke, P.: Coverability trees for petri nets with unordered data. In: Jacobs, B., Löding, C. (eds.) FOSSACS 2016. LNCS, vol. 9634, pp. 445–461. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-49630-5_26CrossRefGoogle Scholar
  18. 18.
    Jacobsen, L., Jacobsen, M., Møller, M.H., Srba, J.: Verification of timed-arc petri nets. In: Černá, I., Gyimóthy, T., Hromkovič, J., Jefferey, K., Králović, R., Vukolić, M., Wolf, S. (eds.) SOFSEM 2011. LNCS, vol. 6543, pp. 46–72. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  19. 19.
    Jensen, K.: Coloured petri nets and the invariant-method. Theor. Comput. Sci. 14, 317–336 (1981)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Klin, B., Szynwelski, M.: SMT solving for functional programming over infinite structures. In: Mathematically Structured Functional Programming (accepted for publication, 2016)Google Scholar
  21. 21.
    Kopczyński, E., Toruńczyk, S.: LOIS: an application of SMT solvers (submitted, 2016). http://www.mimuw.edu.pl/~erykk/lois/lois-sat.pdf
  22. 22.
    Kruskal, J.B.: Well-quasi-ordering, the tree theorem, and Vazsonyi’s conjecture. Trans. Am. Math. Soc. 95(2), 210–225 (1960)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Latka, B.J.: Finitely constrained classes of homogeneous directed graphs. J. Symbolic Logic 59(1), 124–139 (1994)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lazić, R.S., Newcomb, T., Ouaknine, J., Roscoe, A.W., Worrell, J.B.: Nets with tokens which carry data. In: Kleijn, J., Yakovlev, A. (eds.) ICATPN 2007. LNCS, vol. 4546, pp. 301–320. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  25. 25.
    Macpherson, D.: A survey of homogeneous structures. Discrete Math. 311(15), 1599–1634 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rosa-Velardo, F., de Frutos-Escrig, D.: Decidability and complexity of petri nets with unordered data. Theor. Comput. Sci. 412(34), 4439–4451 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (http://creativecommons.org/licenses/by-nc/2.5/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Authors and Affiliations

  1. 1.Institute of InformaticsUniversity of WarsawWarsawPoland

Personalised recommendations