Bisimulation from Open Maps

Abstract

An abstract definition of bisimulation is presented. It makes possible a uniform definition of bisimulation across a range of different models for parallel computation presented as categories. As examples, transition systems, synchronisation trees, transition systems with independence (an abstraction from Petri nets), and labelled event structures are considered. On transition systems the abstract definition readily specialises to Milner's strong bisimulation. On event structures it explains and leads to a strengthening of the history-preserving bisimulation of Rabinovitch and Traktenbrot and van Glabeek and Goltz. A tie-up with open maps in a (pre)topos, as they appear in the work of Joyal and Moerdijk, brings to light a new model, presheaves on categories of pomsets, into which the usual category of labelled event structures embeds fully and faithfully. As an indication of its promise, this new presheaf model has``refinement'' operators. The general approach yields a logic, generalising Hennessy Milner logic, which is characteristic for the generalised notion of bisimulation.

Petri nets and other models of concurrency

“…This section aims at providing an illustration of this point. It is an exposition of a the categorical approach to bisimulation obtained from spans of open maps as defined in [35] -to which we refer for the missing proofs -with an additional treatment of SWNets in the general picture. The open map approach presented here has also been applied successfully to capture other familiar behavioural equivalences on nets, e.g., Hoare's trace equivalence [30] and Milner's weak bisimulation [49,50], both of which may be obtained by slightly changing the notion of path extension from the one presented here [55].…”

“…As an illustration, following [65] There is a general way of introducing labels to models in such a way that one may carry over adjunctions between unlabelled models to their labelled counterparts, following [35]. Here we sketch the idea, applicable to the categories of nets and event structures.…”

“…Following [35], a computation path represents a particular run or history of a process. For transition systems, a computation path is traditionally taken to be a finite sequence of transitions.…”

“…We conclude this section presenting a few general facts from [35] about how open morphisms and bisimilarity are preserved and reflected by functors, especially when part of a coreflection. For notational simplicity we shall assume the left adjoints of the coreflections are inclusions.…”

“…Once adjunctions are established between models, one may start comparing and transferring behavioural concepts from one model to another, formally via the adjoints L and R. In the final section of Part 1, we shall present on such example based on [35,63], introducing a general way of understanding Milner's seminal notion of bisimulation [50] across a range of different models, including net systems. It must be noted that there is more to the categorical view of models than we present here.…”