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Decidability Border for Petri Nets with Data: WQO Dichotomy Conjecture

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Application and Theory of Petri Nets and Concurrency (PETRI NETS 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,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.

The work is partially supported by the Polish National Center of Science 2012/07/B/ST6/01497.

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Notes

  1. 1.

    Restriction to only relational structures is for the sake of simplicity.

  2. 2.

    We deliberately do not distinguish a structure \(\mathcal{A}\) from its domain set.

  3. 3.

    A class has amalgamation property if for every two embeddings \(h_1 : \mathcal{A}\rightarrow \mathcal{B}_1\) and \(h_2 : \mathcal{A} \rightarrow \mathcal{B}_2\) there is a structure \(\mathcal{C}\) and two embeddings \(g_1 : \mathcal{B}_1 \rightarrow \mathcal{C}\) and \(g_2 : \mathcal{B}_2 \rightarrow \mathcal{C}\) such that \(g_1 \circ h_1 = g_2 \circ h_2\) (see [12, 25] for details).

  4. 4.

    We could further restrict the codomain to nonempty finite multisets over P.

References

  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)

    Chapter  MATH  Google Scholar 

  2. Bojańczyk, M., Braud, L., Klin, B., Lasota, S.: Towards nominal computation. In: Proceedings of the POPL, pp. 401–412 (2012)

    Google Scholar 

  3. Bojańczyk, M., Klin, B., Lasota, S.: Automata theory in nominal sets. Logical Methods Comput. Sci. 10(3), 1–44 (2014). Paper 4

    MathSciNet  MATH  Google Scholar 

  4. Bojańczyk, M., Klin, B., Lasota, S., Toruńczyk, S.: Turing machines with atoms. In: LICS, pp. 183–192 (2013)

    Google Scholar 

  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. 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. 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. 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)

    Article  Google Scholar 

  9. Delzanno, G.: Constraint multiset rewriting. Technical report DISI-TR-05-08, DISI, Universitá di Genova (2005)

    Google Scholar 

  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. Finkel, A., Schnoebelen, P.: Well-structured transition systems everywhere! Theor. Comput. Sci. 256(1–2), 63–92 (2001)

    Google Scholar 

  12. Fraïssé, R.: Theory of Relations. North-Holland (1953)

    Google Scholar 

  13. Genrich, H.J., Lautenbach, K.: System modelling with high-level petri nets. Theor. Comput. Sci. 13, 109–136 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  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. Henson, W.: Countable homogeneous relational structures and \(\aleph _0\)-categorical theories. J. Symb. Logic 37, 494–500 (1972)

    Article  MATH  Google Scholar 

  16. Higman, G.: Ordering by divisibility in abstract algebras. Proc. London Math. Soc. (3) 2(7), 326–336 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  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_26

    Chapter  Google Scholar 

  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)

    Chapter  MATH  Google Scholar 

  19. Jensen, K.: Coloured petri nets and the invariant-method. Theor. Comput. Sci. 14, 317–336 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  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. Kopczyński, E., Toruńczyk, S.: LOIS: an application of SMT solvers (submitted, 2016). http://www.mimuw.edu.pl/~erykk/lois/lois-sat.pdf

  22. Kruskal, J.B.: Well-quasi-ordering, the tree theorem, and Vazsonyi’s conjecture. Trans. Am. Math. Soc. 95(2), 210–225 (1960)

    MathSciNet  MATH  Google Scholar 

  23. Latka, B.J.: Finitely constrained classes of homogeneous directed graphs. J. Symbolic Logic 59(1), 124–139 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  MATH  Google Scholar 

  25. Macpherson, D.: A survey of homogeneous structures. Discrete Math. 311(15), 1599–1634 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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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.

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  1. Institute of Informatics, University of Warsaw, Warsaw, Poland

    Sławomir Lasota

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  1. Sławomir Lasota
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Correspondence to Sławomir Lasota .

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  1. Université Pierre and Marie Curie, Paris, France

    Fabrice Kordon

  2. Department Informatik, Universität Hamburg Department Informatik, Hamburg, Germany

    Daniel Moldt

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Lasota, S. (2016). Decidability Border for Petri Nets with Data: WQO Dichotomy Conjecture. In: Kordon, F., Moldt, D. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2016. Lecture Notes in Computer Science(), vol 9698. Springer, Cham. https://doi.org/10.1007/978-3-319-39086-4_3

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  • DOI: https://doi.org/10.1007/978-3-319-39086-4_3

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