Unifying Petri Net Semantics with Token Flows

In this paper we advocate a unifying technique for description of Petri net semantics. Semantics, i.e. a possible behaviour, is basically a set of node-labelled and arc-labelled directed acyclic graphs, called token flows, where the graphs are distinguished up to isomorphism. The nodes of a token flow represent occurrences of transitions of the underlying net, so they are labelled by transitions. Arcs are labelled by multisets of places. Namelly, an arc between an occurrence x of a transition a and an occurrence y of a transition b is labelled by a multiset of places, saying how many tokens produced by the occurrence x of the transition a is consumed by the occurrence y of the transition b . The variants of Petri net behaviour are given by different interpretation of arcs and different structure of token flows, resulting in different sets of labelled directed acyclic graphs accepted by the net. We show that the most prominent semantics of Petri nets, namely processes of Goltz and Reisig, partial languages of Petri nets introduced by Grabowski, rewriting terms of Meseguer and Montanari, step sequences as well as classical occurrence (firing) sequences correspond to different subsets of token flows. Finally, we discuss several results achieved using token flows during the last four years, including polynomial test for the acceptance of a partial word by a Petri net, synthesis of Petri nets from partial languages and token flow unfolding.