An Efficient Characterization of Petri Net Solvable Binary Words

  • David de Frutos Escrig
  • Maciej Koutny
  • Łukasz Mikulski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10877)


We present a simple characterization of the set of Petri net solvable binary words. It states that they are exactly the extensions of the prefixes of Petri net cyclic solvable words, by some prefix \(x^k\), where x is any letter of the binary alphabet being considered, and k is any natural number. We derive several consequences of this characterization which, in a way, shows that the set of solvable words is ‘smaller than expected’. Therefore, the existing conjecture that all of them can be generated by quite simple net is not only confirmed, but indeed reinforced. As a byproduct of the characterization, we also present a linear time algorithm for deciding whether a binary word is solvable. The key idea is that the connection with the cyclic solvable words induces certain structural regularity. Therefore, one just needs to look for possible irregularities, which can be done in a structural way, resulting in a rather surprising linearity of the decision algorithm. Finally, we employ the obtained results to provide a characterization of reversible binary transition systems.


Petri net Binary word Word solvability Reversibility Binary transition system 



This research was supported by Cost Action IC1405. The first author was partially supported by the Spanish projects TRACES (TIN2015-67522-C3-3) and N-GREENS (S2013/ICE-2731).


  1. 1.
    Badouel, E., Bernardinello, L., Darondeau, P.: Petri Net Synthesis. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2015). Scholar
  2. 2.
    Barylska, K., Best, E., Erofeev, E., Mikulski, Ł., Piątkowski, M.: Conditions for Petri net solvable binary words. In: Koutny, M., Desel, J., Kleijn, J. (eds.) Transactions on Petri Nets and Other Models of Concurrency XI. LNCS, vol. 9930, pp. 137–159. Springer, Heidelberg (2016). Scholar
  3. 3.
    Barylska, K., Erofeev, E., Koutny, M., Mikulski, Ł., Piątkowski, M.: Reversing transitions in bounded Petri nets. Fundam. Inform. 157(1–4), 341–357 (2018)CrossRefGoogle Scholar
  4. 4.
    Barylska, K., Koutny, M., Mikulski, Ł., Piątkowski, M.: Reversible computation vs. reversibility in Petri nets. Sci. Comput. Program. 151, 48–60 (2018)CrossRefGoogle Scholar
  5. 5.
    Best, E., Erofeev, E., Schlachter, U., Wimmel, H.: Characterising Petri net solvable binary words. In: Kordon, F., Moldt, D. (eds.) PETRI NETS 2016. LNCS, vol. 9698, pp. 39–58. Springer, Cham (2016). Scholar
  6. 6.
    Erofeev, E., Barylska, K., Mikulski, Ł., Piątkowski, M.: Generating all minimal Petri net unsolvable binary words. In: Proceedings of PSC 2016, pp. 33–46 (2016)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • David de Frutos Escrig
    • 1
  • Maciej Koutny
    • 2
  • Łukasz Mikulski
    • 3
  1. 1.Dpto. Sistemas Informáticos y Computación, Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain
  2. 2.School of ComputingNewcastle UniversityNewcastle upon TyneUK
  3. 3.Faculty of Mathematics and Computer ScienceNicolaus Copernicus University in ToruńToruńPoland

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