Parallel Product of Event Structures

Event structures were introduced by Nielsen, Plotkin and Winskel [5] as a model for computational processes. These were essentially sets of events with relations expressing causal dependence and conflicts between them. These structures, also called prime event structures, were shown to be connected both with a class of acyclic Petri nets (called by the authors occurrence nets) and with a class of domains (finitary prime algebraic coherent domains). Though a very pleasant model of computation, prime event structures can lead to tedious definitions for more sophisticated constructions such as parallel product or exponentiation. In subsequent papers [8,9], Winskel introduced a more general class of event structures called stable event structures, for which such constructions are defined categorically. Although stable event structures themselves are slightly more complicated than prime event structures, constructions on them can be defined in an elegant way. In general, category theory has proven very useful for computer science. When constructions are formulated in categorical terms, not only can the required universal properties provide some guideline in the definitions, the associated morphisms may contain suggestions of computational interest as well. Moving to more general structures, however, can sometimes shield the intuitions about the model. In the case of stable event structures, the causality relation on events is not explicitly given, but is derived from an enabling relation on events. This makes such event structures a little hard to visualize, and one may wonder whether a more manageable notion could be devised. Moreover, unlike prime event structures, it is not clear if stable event structures correspond to a class of Petri nets, in the sense of [5]. Flow event structures were proposed in [2] as an intermediate between prime event structures and stable event structures, suitable both for graphical representation and for an easy definition of process operators. Among these operators the critical one is the