Using Transition Invariants for Reachability Analysis of Petri Nets

Petri nets are an important formal paradigm for modeling and analysis of discrete event systems. The related areas of application of Petri nets include deadlock avoidance and prevention, supervisory control, forbidden state detection, different aspects of flexible manufacturing systems, and many others (Zhou & DiCesare, 1993; Holloway et al., 1997; Boel et al., 1995). Quite often, given a discrete-event system, the designer is interested in determining whether the system can transit from an initial state to another, target state as a result of some operations from a predefined set. In terms of Petri nets, the answer to this question is obtained as a solution of a reachability problem. The reachability problem in Petri nets is formulated as follows: for any Petri net PN, with an initial marking M0, and for some other marking M, determine whether the relation M ∈ R(PN, M0) is true, where R(PN, M0) is the reachability set of PN for its initial marking M0 (Murata, 1989). The decidability of the reachability problem has been proved for a number of restricted classes of Petri nets, and there are efficient algorithms for such classes as acyclic Petri nets, marked graphs, and others (Kodama & Murata, 1988; Caprotti et al., 1995; Kostin, 1997). It has been shown that the reachability problem is decidable for generalized Petri nets as well (Mayr, 1984). The fundamental contribution of the paper (Mayr, 1984) is in proving that the reachability problem for generalized Petri nets is decidable. However, being highly important theoretically, the practical use of the algorithm described in that paper is limited. Actually, the algorithm creates a series of so called regular constrained refined graphs, each of which is a generalization of the basic coverability tree. As the author admits, the first refined graph would enumerate the whole reachability set of the given Petri net. In practice, two different approaches are used most often to determine the reachability of a marking in Petri nets. The first approach is based on the creation and investigation of a complete or reduced reachability graph. The main drawback of this approach is a state explosion problem. A closely related technique is the use of stubborn sets. The main purpose of the stubborn sets technique is to choose, for each marking of the net, a set of transitions to fire that is large enough to preserve some desired information about the Petri net, but is as small as possible to get a significant reduction of the resulting reachability graph (Varpaaniemi, 1998). Unfortunately, generation of minimal or reduced reachability graphs in finite state systems is known to be an NP-hard problem (Peled, 1993). If Petri net has no