Simultaneous Petri Net Synthesis

Abstract

Petri net synthesis deals with the problem whether, given a labelled transition system TS , one can find a Petri net N with an initial marking M 0 such that the reachability graph of (N, M 0 ) is isomorphic to TS . This may be preceded by a pre-synthesis phase that will quickly reject ill-formed transition systems (and give structural reasons for the failure) and otherwise build data structures needed by the proper synthesis. The last phase proceeds by solving systems of linear inequalities, and may still fail but for less transparent reasons. In this paper, we consider an extended problem. A finite set of transition systems {TS 1 , . . . , TS m } shall be called simultaneously Petri net solvable if there is a single Petri net N with several initial markings {M 01 , . . . , M 0m }, such that for every i = 1, . . . , m, the reachability graph of (N, M 0i ) is isomorphic to TS i . The focus will be on choice-free nets, that is, nets without structural choices, and we explore how previously published efficient algorithms for the pre-synthesis and proper synthesis of bounded and choice-free Petri nets can be generalised for the simultaneous pre-synthesis and synthesis of such multi-marked nets. At the same time, the choice-free pre-synthesis of a single transition system shall be strengthened by introducing new structural checks.

Articulation of Transition Systems and Its Application to Petri Net Synthesis

“…Other possible issues are to examine how this may be specialised for some subclasses of Petri nets, like Choice-Free ones, where each place has at most one outgoing transition: this is exactly the class of Petri nets allowing fully distributed implementations [6], and they present interesting behavioural properties [10,5] which could be exploited.…”

A New Property of Choice-Free Petri Net Systems

“…A generalisation of the synthesis problem, called the simultaneous synthesis problem, has been introduced in [9]. It consists in considering several transition systems {TS 1 , .…”

“…We then have: Theorem 1. [9] A simultaneous choice-free synthesis problem is solvable if each of its augmented individual synthesis problems is, and the solution may be obtained easily by aggregating all those individual solutions.…”

“…For this first phase, a series of checks arising from the analysis of choice-free systems, has been elaborated in [9], mainly based on the following observation [16]:…”

“…Those last years, several works have been dedicated to the (simultaneous) synthesis of choice-free Petri nets 1 , among others [1,3,8,9]. Since this means a loss in the expressive power, we may thus wonder why this class of systems could be especially interesting, besides the known structural properties of their behaviours.…”