Dualities between nets and automata induced by schizophrenic objects

Abstract

The so-called synthesis problem for nets, which consists in deciding whether a given graph is isomorphic to the case graph of some net, and then constructing the net, has been solved in the litterature for various types of nets, ranging from elementary nets to Petri nets. The common principle for the synthesis is the idea of regions in graphs, representing possible extensions of places in nets. When the synthesis problem has a solution, the set of regions viewed as properties of states provides a set-theoretic representation of the transition system. We show that such correspondences between nets and transition systems can be described as dualities induced by schizophrenic objects, leading further the analogy with classical representation theorems and giving us a means to describe uniformly previously known translations between nets and automata.