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

We introduce the notion of T -path within Petri nets, and propose a simple approach, based on previous work developed for directed hypergraphs, in order to determine structural properties of nets; in particular, we study the relationships between T -paths in a Petri net and firable sequences of transitions. Let us consider a 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 reachable 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(|A| + |P | + |T |), for a net P = 〈P, T,A,M0〉. Previous results for this class of networks are given by Howell, Rosier and Yen [18], 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 ′ ≥ M ; this problem is known to be EXPSPACE-hard [30]. For general Petri nets we provide a partial answer to this problem. M is coverable by augmentation if it is coverable from an augmented marking M ′ 0 of the initial marking M0: M ′ 0 ≥ M0 and, for all p ∈ P , M ′ 0(p) = 0 if M0(p) = 0. We solve this problem in linear time. The algorithms to compute T -paths are incremental: it is possible to modify the network (adding new places, transitions, arcs, tokens), and update the set of potentially firable transitions and reachable places without recomputing them from scratch. This feature results meaningful when used during the interactive design of a system.