Structure Theory of Petri Nets

Abstract

Abstract. The aim of this tutorial is to give a concise, but nonetheless not too narrow, overview of definitions and results pertaining centrally to Petri net structure theory. The Petri net model considered in these notes are the classical place/transition nets, as they have been defined in the First Advanced Course held in 1979 and originate back to the late Sixties. Structure theory asks what behavioural properties of a Petri net can be derived from its structural properties. Other aspects of Petri nets are neglected to a large extent in the present notes, such as various extensions and generalisations of central notions and results, as well as almost all algorithmic and complexity-theoretic consequences that accompany the structure-theoretic results. Because full proofs can easily be retrieved from the literature, they are not given, unless they are small and perhaps somewhat characteristic for Petri net oriented reasoning. Proof ideas are often sketched, however, and the sharpness of various results is accentuated by means of examples and counterexamples. A list for further reading is also provided.

Markings in Perpetual Free-Choice Nets Are Fully Characterized by Their Enabled Transitions

“…For particular subclasses of Petri net there is a relationship between structural properties and behavioral properties like liveness and boundedness [12]. In this paper, we focus on free-choice nets [13].…”

“…Moreover, one can reach this marking without firing transitions from the cluster that needs to become enabled. A simplified proof was given in [26] and another proof sketch was provided in [12]. The free-choice net in Figure 6 is live and bounded.…”

“…Structure theory is a branch in Petri nets [8,20,21,22,23] that asks what behavioral properties can be derived from it structural properties [10,12,13]. Many different subclasses have been studied.…”

“…Examples include state machines, marked graphs, free-choice nets, asymmetric choice nets, and nets without TP and PT handles. Structure theory also studies local structures such as traps and siphons that may reveal information about the behavior of the Petri net and includes linear algebraic characterizations of behavior involving the matrix equation or invariants [12,13,20].…”

“…The definite book on the structure theory of free-choice nets is [13]. Also, see [12] for pointers to literature. Therefore, it is surprising that the question whether markings are uniquely identified by the set of enabled transitions (i.e., lucency) has not been explored in literature.…”