Advertisement

Hardness Results for the Synthesis of b-bounded Petri Nets

  • Ronny TredupEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11522)

Abstract

Synthesis for a type \(\tau \) of Petri nets is the following search problem: For a transition system A, find a Petri net N of type \(\tau \) whose state graph is isomorphic to A, if there is one. To determine the computational complexity of synthesis for types of bounded Petri nets we investigate their corresponding decision version, called feasibility. We show that feasibility is NP-complete for (pure) b-bounded P/T-nets if \(b\in \mathbb {N}^+\). We extend (pure) b-bounded P/T-nets by the additive group \(\mathbb {Z}_{b+1}\) of integers modulo \((b+1)\) and show feasibility to be NP-complete for the resulting type. To decide if A has the event state separation property is shown to be NP-complete for (pure) b-bounded and group extended (pure) b-bounded P/T-nets. Deciding if A has the state separation property is proven to be NP-complete for (pure) b-bounded P/T-nets.

Notes

Acknowledgements

I would like to thank Christian Rosenke and Uli Schlachter for their precious remarks. Also, I’m thankful to the anonymous reviewers for their helpful comments.

References

  1. 1.
    Aalst, W.M.P.: Process Mining Discovery - Conformance and Enhancement of Business Processes. Springer, Berlin (2011).  https://doi.org/10.1007/978-3-642-19345-3CrossRefzbMATHGoogle Scholar
  2. 2.
    Badouel, E., Bernardinello, L., Darondeau, P.: Polynomial algorithms for the synthesis of bounded nets. In: Mosses, P.D., Nielsen, M., Schwartzbach, M.I. (eds.) CAAP 1995. LNCS, vol. 915, pp. 364–378. Springer, Heidelberg (1995).  https://doi.org/10.1007/3-540-59293-8_207CrossRefGoogle Scholar
  3. 3.
    Badouel, E., Bernardinello, L., Darondeau, P.: The synthesis problem for elementary net systems is NP-complete. Theor. Comput. Sci. 186(1–2), 107–134 (1997).  https://doi.org/10.1016/S0304-3975(96)00219-8MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Badouel, E., Bernardinello, L., Darondeau, P.: Petri Net Synthesis. TTCSAES. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-47967-4CrossRefzbMATHGoogle Scholar
  5. 5.
    Badouel, E., Caillaud, B., Darondeau, P.: Distributing finite automata through petri net synthesis. Formal Asp. Comput. 13(6), 447–470 (2002).  https://doi.org/10.1007/s001650200022CrossRefzbMATHGoogle Scholar
  6. 6.
    Cortadella, J., Kishinevsky, M., Kondratyev, A., Lavagno, L., Yakovlev, A.: A region-based theory for state assignment in speed-independent circuits. IEEE Trans. CAD Integr. Circ. Syst. 16(8), 793–812 (1997).  https://doi.org/10.1109/43.644602CrossRefzbMATHGoogle Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)Google Scholar
  8. 8.
    Hiraishi, K.: Some complexity results on transition systems and elementary net systems. Theor. Comput. Sci. 135(2), 361–376 (1994).  https://doi.org/10.1016/0304-3975(94)90112-0MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Holloway, L.E., Krogh, B.H., Giua, A.: A survey of petri net methods for controlled discrete event systems. Discrete Event Dyn. Syst. 7(2), 151–190 (1997).  https://doi.org/10.1023/A:1008271916548CrossRefzbMATHGoogle Scholar
  10. 10.
    Moore, C., Robson, J.M.: Hard tiling problems with simple tiles. Discrete Comput. Geom. 26(4), 573–590 (2001).  https://doi.org/10.1007/s00454-001-0047-6MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Schlachter, U., Wimmel, H.: k-bounded petri net synthesis from modal transition systems. In: CONCUR. LIPIcs, vol. 85, pp. 6:1–6:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017).  https://doi.org/10.4230/LIPIcs.CONCUR.2017.6
  12. 12.
    Schmitt, V.: Flip-flop nets. In: Puech, C., Reischuk, R. (eds.) STACS 1996. LNCS, vol. 1046, pp. 515–528. Springer, Heidelberg (1996).  https://doi.org/10.1007/3-540-60922-9_42CrossRefGoogle Scholar
  13. 13.
    Tredup, R.: Fixed parameter tractability and polynomial time results for the synthesis of \(b\)-bounded petri nets. In: Donatelli, S., Haar, S. (eds.) PETRI NETS 2019. LNCS, vol. 11522, pp. 148–168. Springer, Cham (2019)Google Scholar
  14. 14.
    Tredup, R.: Hardness results for the synthesis of \(b\)-bounded petri nets (technical report). CoRR abs/1904.01094 (2019)Google Scholar
  15. 15.
    Tredup, R., Rosenke, C.: Narrowing down the hardness barrier of synthesizing elementary net systems. In: CONCUR. LIPIcs, vol. 118, pp. 16:1–16:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018).  https://doi.org/10.4230/LIPIcs.CONCUR.2018.16
  16. 16.
    Tredup, R., Rosenke, C.: The Complexity of synthesis for 43 boolean petri net types. In: Gopal, T.V., Watada, J. (eds.) TAMC 2019. LNCS, vol. 11436, pp. 615–634. Springer, Cham (2019).  https://doi.org/10.1007/978-3-030-14812-6_38CrossRefGoogle Scholar
  17. 17.
    Tredup, R., Rosenke, C., Wolf, K.: Elementary net synthesis remains NP-complete even for extremely simple inputs. In: Khomenko, V., Roux, O.H. (eds.) PETRI NETS 2018. LNCS, vol. 10877, pp. 40–59. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-91268-4_3CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Universität Rostock, Institut für Informatik, Theoretische InformatikRostockGermany

Personalised recommendations