Abstract
A set of necessary conditions for a Petri net to be plain, pure and safe is given. Some applications of these conditions both in practice (for Petri net synthesis), and in theory (e.g., as part of a characterisation of the reachability graphs of live and safe marked graphs) are described.
K. Barylska—Co-funded by project PO KL Information technologies: Research and their interdisciplinary applications, Agreement UDA-POKL.04.01.01-00-051/10-00 and by the Polish National Science Center (grant No.2013/09/D/ST6/03928).
U. Schlachter—Supported by DFG (German Research Foundation) through grant Be 1267/15-1 ARS (Algorithms for Reengineering and Synthesis).
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Notes
- 1.
Unless infinite nets are considered, which we shall exclude in this paper.
- 2.
Characterisations have been obtained for more restricted classes, e.g. in [4].
- 3.
S can also be considered as a set of vertices and \(\rightarrow \) as a set of edges of a directed graph, labelled by letters from T.
- 4.
In this paper, no arc weights greater than 1 will be considered.
- 5.
This can be verified easily by playing the “token game” in the latter.
- 6.
Note that p is the only unsafe place (with bound 2), and that \(L'\) is the only non-safe reachable marking. In this sense, the example demonstrates that the conjecture is sharp. We also believe that it is the smallest example with this property.
- 7.
- 8.
Note that there is also a short path \(s_a[a\rangle 1[c\rangle 2[d\rangle 4[a\rangle 5[c\rangle s_b\) containing a two times. However, this path is not indicative of an unsafe place from a to b, since \(s_a\notin Seq(b)\).
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The authors are grateful to the reviewers for their helpful comments.
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Barylska, K., Best, E., Schlachter, U., Spreckels, V. (2017). Properties of Plain, Pure, and Safe Petri Nets. In: Koutny, M., Kleijn, J., Penczek, W. (eds) Transactions on Petri Nets and Other Models of Concurrency XII. Lecture Notes in Computer Science(), vol 10470. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55862-1_1
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