Classifying models of concurrency

Classifying models of concurrency with respect to the kind of mathematical objects that are used to represent processes, I find it convenient to distinguish 5 types of models.

1. GRAPH oriented models. Here a concurrent system is represented by a process graph, or state-transition diagram, also called automaton. Or by a richer graph-like object. A variant are labelled transition systems, in which a concurrent system is not denoted by a whole graph, but by a vertex in a graph; the entire domain is then one large graph.
2. NET oriented models, in which a concurrent system is represented by a Petri net, or a net-like object.
3. EVENT orient models, in which a concurrent system is represented by a set of events (action occurrences) together with some structure on this set, determining the relations between the events. This class of models includes the various brands of event structures.
4. EXPLICIT models. In the models mentioned above, a concurrent system is not really modeled as a single graph/net/event structure, but actually by an equivalence class of such objects. Thus different graphs/nets/etc. represent the same concurrent system. Which equivalence relation is divided out is partly determined by which properties of concurrent systems are considered to be relevant. The choice of equivalence is important in proving the correctness of specifications w.r.t. implementations.
   The explicit models on the other hand are not quotient models, but offer one `explicit' object to represent a system. This object is usually a mathematically coded set of the relevant properties, or of the possible executions, of the represented system.
5. TERM models. Here a concurrent system is represented as a term in a system description language. Well known system description languages are CCS and CSP. Instead of speaking of term models as type of model, one could say that systems can be denoted either syntactically (by means of an expression in a language) or semantically (in a model of type 1-4). The notion of a term model is the result of unifying these views. Usually, the meaning of an expression is given by means of a mapping from the used set of terms into another model. This mapping constitutes the semantics of the used language.

• the equivalence relation that is divided out (explicitly in term, graph, net and event oriented models; implicitly in explicit models),
• the scope of the model (which systems can be represented and which cannot)
• and the chosen structure (the selection of operators, relations and morphisms defined on the model).