Abstract
Recent studies investigated the problems of analysing Petri nets and synthesising them from labelled transition systems (LTS) with two labels (transitions) only. In this paper, we extend these works by providing new conditions for the synthesis of Weighted Marked Graphs (WMGs), a well-known and useful class of weighted Petri nets in which each place has at most one input and one output.
Some of these new conditions do not restrict the number of labels; the other ones consider up to 3 labels. Additional constraints are investigated: when the LTS is either finite or infinite, and either cyclic or acyclic. We show that one of these conditions, developed for 3 labels, does not extend to 4 nor to 5 labels. Also, we tackle geometrically the WMG-solvability of finite, acyclic LTS with any number of labels.
E. Erofeev—Supported by DFG through grant Be 1267/16-1 ASYST.
T. Hujsa—Supported by the STAE foundation/project DAEDALUS, Toulouse, France.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A set A of k arcs in a LTS G defines a cycle of G if the elements of A can be ordered as a sequence \(a_1 \ldots a_k\) such that, for each \(i \in \{1, \ldots , k\}\), \(a_i = (n_i,\ell _i,n_{i+1})\) and \(n_{k+1} = n_1\), i.e. the i-th arc \(a_i\) goes from node \(n_i\) to node \(n_{i+1}\) until the first node \(n_1\) is reached, closing the path. Cycles are also sometimes called circuits, circles and oriented cycles.
- 2.
The projection of a word \(w\in A^*\) on a set \(A' \subseteq A\) of labels is the maximum subword of w whose labels belong to \(A'\), noted
. For example, the projection of the word \(w = \ell _1 \, \ell _2 \, \ell _3 \, \ell _2\) on the set \(\{\ell _1 ,\, \ell _2\}\) is the word \(\ell _1 \, \ell _2 \, \ell _2\). - 3.
Also called sometimes the synchronisation on transitions.
. For example, the projection of the word References
Murata, T.: Petri nets: properties, analysis and applications. Proc. IEEE 77(4), 541–580 (1989)
Teruel, E., Silva, M.: Structure theory of equal conflict systems. Theoret. Comput. Sci. 153(1&2), 271–300 (1996)
Hujsa, T., Devillers, R.: On liveness and deadlockability in subclasses of weighted Petri nets. In: van der Aalst, W., Best, E. (eds.) PETRI NETS 2017. LNCS, vol. 10258, pp. 267–287. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-57861-3_16
Desel, J., Esparza, J.: Free Choice Petri Nets. Cambridge Tracts in Theoretical Computer Science, vol. 40. Cambridge University Press, New York (1995)
Teruel, E., Colom, J.M., Silva, M.: Choice-free Petri nets: a model for deterministic concurrent systems with bulk services and arrivals. IEEE Trans. Syst. Man Cybern. Part A 27(1), 73–83 (1997)
Hujsa, T., Delosme, J.-M., Munier-Kordon, A.: On the reversibility of well-behaved weighted choice-free systems. In: Ciardo, G., Kindler, E. (eds.) PETRI NETS 2014. LNCS, vol. 8489, pp. 334–353. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-07734-5_18
Commoner, F., Holt, A., Even, S., Pnueli, A.: Marked directed graphs. J. Comput. Syst. Sci. 5(5), 511–523 (1971)
Teruel, E., Chrzastowski-Wachtel, P., Colom, J.M., Silva, M.: On weighted T-systems. In: Jensen, K. (ed.) ICATPN 1992. LNCS, vol. 616, pp. 348–367. Springer, Heidelberg (1992). https://doi.org/10.1007/3-540-55676-1_20
Best, E., Hujsa, T., Wimmel, H.: Sufficient conditions for the marked graph realisability of labelled transition systems. Theoret. Comput. Sci. 750, 101–116 (2017)
Devillers, R., Hujsa, T.: Analysis and synthesis of weighted marked graph Petri nets. In: Khomenko, V., Roux, O.H. (eds.) PETRI NETS 2018. LNCS, vol. 10877, pp. 19–39. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-91268-4_2
Delosme, J.M., Hujsa, T., Munier-Kordon, A.: Polynomial sufficient conditions of well-behavedness for weighted join-free and choice-free systems. In: 13th International Conference on Application of Concurrency to System Design, pp. 90–99, July 2013
Hujsa, T., Delosme, J.M., Munier-Kordon, A.: Polynomial sufficient conditions of well-behavedness and home markings in subclasses of weighted Petri nets. ACM Trans. Embed. Comput. Syst. 13(4s), 141:1–141:25 (2014)
Barylska, K., Best, E., Erofeev, E., Mikulski, L., Piatkowski, M.: On binary words being Petri net solvable. In: Proceedings of the International Workshop on Algorithms & Theories for the Analysis of Event Data, ATAED 2015, Brussels, Belgium, pp. 1–15 (2015)
Barylska, K., Best, E., Erofeev, E., Mikulski, L., Piatkowski, M.: Conditions for Petri net solvable binary words. Trans. Petri Nets Other Models Concurr. 11, 137–159 (2016)
Erofeev, E., Barylska, K., Mikulski, L., Piatkowski, M.: Generating all minimal Petri net unsolvable binary words. In: Proceedings of the Prague Stringology Conference 2016, Prague, Czech Republic, pp. 33–46 (2016)
Erofeev, E., Wimmel, H.: Reachability graphs of two-transition Petri nets. In: Proceedings of the International Workshop on Algorithms & Theories for the Analysis of Event Data 2017, Zaragoza, Spain, pp. 39–54 (2017)
Best, E., Devillers, R.: Synthesis and reengineering of persistent systems. Acta Inf. 52(1), 35–60 (2015)
Hujsa, T., Delosme, J.M., Munier-Kordon, A.: On liveness and reversibility of equal-conflict Petri nets. Fundamenta Informaticae 146(1), 83–119 (2016)
Hujsa, T.: Contribution to the study of weighted Petri nets. Ph.D. thesis, Pierre and Marie Curie University, Paris, France (2014)
Devillers, R., Erofeev, E., Hujsa, T.: Synthesis of weighted marked graphs from constrained labelled transition systems. In: Proceedings of the International Workshop on Algorithms & Theories for the Analysis of Event Data, Bratislava, Slovakia, pp. 75–90 (2018)
Crespi-Reghizzi, S., Mandrioli, D.: A decidability theorem for a class of vector-addition systems. Inf. Process. Lett. 3(3), 78–80 (1975)
Devillers, R.: Products of transition systems and additions of Petri nets. In: Desel, J., Yakovlev, A. (eds.) Proceedings of 16th International Conference on Application of Concurrency to System Design (ACSD 2016), pp. 65–73 (2016)
Devillers, R.: Factorisation of transition systems. Acta Informatica 55, 339–362 (2017)
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). https://doi.org/10.1007/978-3-319-39086-4_4
Doignon, J.P.: Convexity in cristallographical lattices. J. Geom. 3(1), 71–85 (1973)
Keller, R.M.: A fundamental theorem of asynchronous parallel computation. In: Feng, T. (ed.) Parallel Processing. LNCS, vol. 24, pp. 102–112. Springer, Heidelberg (1975). https://doi.org/10.1007/3-540-07135-0_113
Badouel, E., Bernardinello, L., Darondeau, P.: Petri Net Synthesis. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47967-4
David, R., Alla, H.: Discrete, Continuous, and Hybrid Petri Nets, 2nd edn. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-10669-9
Acknowledgements
We would like to thank the anonymous referees for their involvement and useful suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer-Verlag GmbH Germany, part of Springer Nature
About this chapter
Cite this chapter
Devillers, R., Erofeev, E., Hujsa, T. (2019). Synthesis of Weighted Marked Graphs from Constrained Labelled Transition Systems: A Geometric Approach. In: Koutny, M., Pomello, L., Kristensen, L. (eds) Transactions on Petri Nets and Other Models of Concurrency XIV. Lecture Notes in Computer Science(), vol 11790. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-60651-3_7
Download citation
DOI: https://doi.org/10.1007/978-3-662-60651-3_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-60650-6
Online ISBN: 978-3-662-60651-3
eBook Packages: Computer ScienceComputer Science (R0)