Characterisation of the state spaces of marked graph Petri nets

Abstract

The structure of the reachability graph of a marked graph Petri net is fully characterised. Exact structural conditions are given for a given labelled transition system to be generated by a marked graph. Two cases are distinguished, corresponding to the boundedness or the unboundedness of the net, and, respectively, to the finiteness or the infiniteness of the transition system. Dedicated synthesis procedures are presented for both cases, and it is shown that there is always a unique minimal solution. The synthesis procedures allow this minimal net, its initial marking, and the marking bounds of its places to be computed from the labelled transition system.

Sufficient conditions for the marked graph realisability of labelled transition systems

“…These results have direct consequences for Petri net synthesis, since they provide conditions for the existence of a marked graph realisation of the specification. Moreover, the proofs of both results are constructive, the second one directly, and the first one by means of Theorem 7 and the algorithm described in [13]. They provide synthesis algorithms that re-engineer a Petri net realisation into a marked graph solution.…”

“…Proposition 1. Classic theory [7,13] Let Σ be a Petri net system. Then RG(Σ) is totally reachable and deterministic.…”

“…Theorem 7. Marked graph synthesis [13] Assume that a finite transition system T S is totally reachable, deterministic, persistent, backward persistent, reversible, and satisfies P1. Then there is a connected, live and bounded marked graph Σ with RG(Σ ) ∼ = T S. ♦ Figure 4.…”

“…In fact, according to the results of[13], it is the only place-minimal such graph (up to isomorphism).…”

Incremental Process Discovery using Petri Net Synthesis

“…Algorithmic solutions for the synthesis of Petri nets using linear algebra techniques 1 [4,5,6] and combinatorial techniques [7,8,9] have been used in the context of process discovery [10,11]. Indeed, the goal of process discovery [12,13] is to construct a model of a system under observation from a set of its execution traces, a so-called log of the system.…”