Characterizing Behavioural Congruences for Petri Nets

We exploit a notion of interface for Petri nets in order to design a set of net combinators. For such a calculus of nets, we focus on the behavioural congruences arising from four simple notions of behaviour, viz., traces, maximal traces, step, and maximal step traces, and from the corresponding four notions of bisimulation, viz., weak and weak step bisimulation and their maximal versions. We characterize such congruences via universal contexts and via games, providing in such a way an understanding of their discerning powers. Introduction In the early days of Petri net theory some important classes of Petri nets (PN) were introduced, such as Marked Graphs (MG), State Machines (SM), Free Choice nets (FC), Simple nets (SN), and inhibitory Petri nets (iPN). The ‘expressive power’ hierarchy among these classes is folklore, namely MG,SM ≺ FC ≺ SN ≺ PN ≺ iPN, where, for instance, SN ≺ PN reads that simple nets do not possess the modelling power of arbitrary Petri nets. In proving this fact, Patil [18] shows that no simple net possesses the behaviour of the so-called ‘3-smoker-net’, i.e., that for a certain notion ≈ of behavioural equivalence, ‘3-smoker-net’ 6≈ SN , for all simple nets SN. However, ≈ was not defined formally; the clever and convincing argument remained at a rather informal level. The same consideration applies to various decomposition results of that time. For instance, another folklore result at the Project MAC of the MIT was that every Petri net can be obtained as ‘composition’ of few, very simple components. Also in this case, the semantic and the composition operations were not defined formally. The proofs simply gave a decomposition technique preserving firing sequences and some ‘concurrency properties’. Looking back to these and to related works (e.g. [6, 4, 24, 10, 19]), it is now clear that the intuition behind ≈ was the today-well-know idea of behavioural congruence, i.e., for a fixed notion of behaviour B,