Linear time analysis of properties of conflict-free and general Petri nets

En:

Theor Comput Sci 2011;412(4-5):320-338

Fecha:

2011

Formato:

application/pdf

Tipo de documento:

info:eu-repo/semantics/article
info:ar-repo/semantics/artículo
info:eu-repo/semantics/publishedVersion

Descriptores:

Boundedness -  Conflict-free Petri nets -  Coverability -  Directed hypergraphs -  Incremental algorithms -  Liveness -  Marked graphs -  Petri nets -  Boundedness

Descripción:

We introduce the notion of a T-path within Petri nets, and propose to adopt the model of directed hypergraphs in order to determine properties of nets; in particular, we study the relationships between T-paths and firable sequences of transitions. Let us consider a Petri net P=〈P,T,A,M0〉 and the set of places with a positive marking in M0, i.e., P0=p|M0(p)>0. If we regard the net as a directed graph, the existence of a simple path from any place in P0 to a transition t is, of course, a necessary condition for the potential firability of t. This is sufficient only if the net is a state machine, where |•t|=|t•|=1 for all t∈T. In this paper we show that the existence of a T-path from any subset of P0 to a transition t is a more restrictive condition and is, again, a necessary condition for the potential firability of t. But, in this case: (a) if P is a conflict-free Petri net, this is also a sufficient condition, (b) if P is a general Petri net, t is potentially firable by increasing the number of tokens in P0. For conflict-free nets (CFPN) we consider the following problems: (a) determining the set of firable transitions, (b) determining the set of coverable places, (c) determining the set of live transitions, (d) deciding the boundedness of the net. For all these problems we provide algorithms requiring linear space and time, i.e., O(|P|+|T|+|A|), for a net P=〈P,T,A,M0〉. Previous results for this class of networks are given by Howell et al. (1987) [20], providing algorithms for solving problems in conflict-free nets in O(|P|×|T|) time and space. Given a Petri net and a marking M, the well-known coverability problem consists in finding a reachable marking M′ such that M′