Compositional Semantics for Open Petri Nets based on Deterministic Processes

In order to model the behaviour of open concurrent systems by means of Petri nets, we introduce open Petri nets, a generalisation of the ordinary model where some places, designated as open, represent an interface between the system and the environment. Besides generalising the token game to reflect this extension, we define a truly concurrent semantics for open nets by extending the Goltz-Reisig process semantics of Petri nets. We introduce a composition operation over open nets, characterised as a pushout in the corresponding category, suitable for modelling both interaction through open places and synchronisation of transitions. The deterministic process semantics is shown to be compositional with respect to such a composition operation. If a net Z3 results as the composition of two nets Z1 and Z2, having a common subnet Z0, then any two deterministic processes of Z1 and Z2 that 'agree' on the common part, can be 'amalgamated' to produce a deterministic process of Z3. Conversely, any deterministic process of Z3 can be decomposed into processes of the component nets. The amalgamation and decomposition operations are shown to be inverse to each other, leading to a bijective correspondence between the deterministic processes of Z3 and the pair of deterministic processes of Z1 and Z2 that agree on the common subnet Z0. Technically, our result is similar to the amalgamation theorem for data-types in the framework of algebraic specification. A possible application field of the proposed constructions and results is the modelling of interorganisational workflows, recently studied in the literature. This is illustrated by a running example