Models for concurrency: Towards a classification

Abstract

Models for concurrency can be classified with respect to three relevant parameters: behaviour/ system, interleaving/noninterleaving, linear/branching time. When modelling a process, a choice concerning such parameters corresponds to choosing the level of abstraction of the resulting semantics.In this paper, we move a step towards a classification of models for concurrency based on the parameters above. Formally, we choose a representative of any of the eight classes of models obtained by varying the three parameters, and we study the formal relationships between them using the language of category theory. O. IntroductionMuch effort in the development of the theory of concurrency has been devoted to the study of suitable models for concurrent and distributed processes, and to the formal understanding of their semantics.As a result, in addition to standard models like languages, automata and transition systems [6,13], models like Petri nets [1211 process algebras [9,4], Hoare traces [5], Mazurkiewicz traces [8], synchronisation trees [20] and event structures [10,21] have been introduced.The idea common to the models above is that they are based on atomic units of change -transitions, actions, events or symbols from an alphabet -which are indivisible and constitute the steps out of which computations are built.The difference between the models may be expressed in terms of the parameters according to which models are often classified. For instance, a distinction made explicitly in the theory of Petri nets, but sensible in a wider context, is that between so-called

Weak Bisimulation and Open Maps

“…The starting point is the work on categorical models for concurrency, such as [24,19,21,22], which has concentrated on understanding the structure and relationships between classes of models, ranging from synchronisation trees to transition systems, event structures and Petri nets. Morphisms between models account for the possibility of a model to simulate the behaviour of another.…”

Petri nets and other models of concurrency

“…The results on categorical relationships between models is a small part of a general picture, as illustrated in [80], in which a number of important constructions from concurrency theory emerges as parts of (coreflective) 'unfoldings' and 'sequentialisations', and (reflective) 'determinizations'. Also, several results in the literature [74,79,62] concern trace structures and other models for concurrency -including the pomsets of Pratt [69] and the partial words of Grabowsky [28].…”

“…As we discussed in the introduction, we tend to classify models can for concurrency according to their 'level of abstraction' (see, e.g., [80,97,79]), that is, according to those aspects of the behaviour of distributed systems they focus on and those they deliberately abstract from. In stating and proving relationships between models viewed under this perspective, the language of category theory as proven in many contexts to be very useful, as it is capable of abstracting away from unwanted details of the individual models and, therefore, of expressing the more essential aspects succinctly and in great generality.…”

“…The modified adjoints are essentially the adjoints presented in the previous sections, simply carrying the label parts across from one model to the other. Furthermore, this coreflection is part of greater collection [97,80,79] of which here we are interested in the small portion shown in the diagram below.…”

“…These result are part of a greater picture of relationships between models for concurrency, see e.g. [97,80,79]. We have chosen to present a few results in some detail, at the expense of the range of models covered.…”