Sufficient conditions for the marked graph realisability of labelled transition systems

Abstract

This paper describes two results within the context of Petri net synthesis from labelled transition systems. We consider a set of structural properties of transition systems, and we show that, given such properties, it is possible to re-engineer a Petri net realisation into one which lies inside the set of marked graphs, a well-understood and useful class of Petri nets.The first result originates from Petri net based workflow specifications, where it is desirable that k customers can share a system without mutual interferences.In a Petri net representation of a workflow, the presence of k customers can be modelled by an initial k-marking, in which the number of tokens on each place is a multiple of k. For any initial k-marking with k ≥ 2, we show that other desirable assumptions such as reversibility and persistence suffice to guarantee marked graph realisability. For the case that k = 1, we show that the existence of certain cycles, along with other properties such as reversibility and persistence, again suffices to guarantee marked graph realisability.

Synthesis of Pure and Impure Petri nets With Restricted Place-environments: Complexity Issues

“…From the perspective of practical applications, they are particularly useful in, for example, some applications like hardware design [4,16] or as a proper model for systems with bulk services and arrivals [11]. On the other hand, as already mentioned, these classes have also been the subject of research aiming at Petri net synthesis for many years [15,17,18,19,20,21]. It turned out that these net classes provide some very useful features like, for example, persistency of their reachability graphs [11] that -in the sense of complexity issues-allow improved synthesis procedures that -instead on regionsrather rely on some basic structural properties of the input transition system.…”

Synthesis of Weighted Marked Graphs from Constrained Labelled Transition Systems: A Geometric Approach

“…In previous studies on the analysis or synthesis of Petri nets, structural restrictions encompassed plain nets (each weight equals 1; also called ordinary nets) [25], homogeneous nets (meaning that for each place p, all the output weights of p are equal) [28,23], free-choice nets (the net is homogeneous, and any two transitions sharing an input place have the same set of input places) [11,28], choice-free nets (each place has at most one output transition) [27,20], marked graphs (each place has at most one output transition and one input transition) [7,26,6,14], join-free nets (each transition has at most one input place) [28,10,21,23], etc.…”

“…The projection of a word w ∈ A * on a set A ′ ⊆ A of labels is the maximum subword of w whose labels belong to A ′ , noted w A ′ . For example, the projection of the word w = ℓ1 ℓ2 ℓ3 ℓ2 on the set {ℓ1, ℓ2} is the word ℓ1 ℓ2 ℓ2 6. Also called sometimes the synchronisation on transitions.…”

Efficient Synthesis of Weighted Marked Graphs with Circular Reachability Graph, and Beyond

“…In this paper, we study the solvability of LTS with weighted marked graphs (WMGs; each place has at most one output and one input) and choice-free nets (CF; each place has at most one output). Both classes are important for realworld applications, and are widely studied in the literature [35,26,18,11,34,10,20,9]. We focus mainly on finite circular LTS, i.e.…”

“…In this context, we investigate the cyclic solvability of a word w, meaning the existence of a Petri net solution to the finite circular LTS induced by the infinite cyclic word w ∞ . These restrictions appear in practical situations, since various complex applications can be decomposed into subsystems satisfying such constraints [25,4,23,6,10,22,16,17].…”