Categorical Models for Fairness: Completion vs Delay

Abstract

We study two different approaches to semantics for fairness within thecategorical framework of presheaf models for concurrency [1, 2]. The first approach, used in e.g. dataflow models, is based on representing finite or infinite completed observations. A completed observation is the result of an infinite computation. A finite completed observation is then the result of an infinite computation that only produces finitely many actions, i.e. at some point it stops producing output but stays active (as opposed to being deadlocked). The second approach is that of e.g. Milner's SCCS with finite delay [6], having an observable delay action and representing also finite incomplete observations, but only infinite completed observations.More precisely, we consider (separated) presheaves over respectively the category Comp of incomplete finite, and completed infinite and finite observations, and the category Inf of incompleted finite and completed infinite observations, possibly containing delays. The latter model was applied previously by the author [5, 4] to give a denotational semantics of SCCS with finite delay shown to be fully abstract with respect to Hennessy and Stirlings extended bisimulation [3]. One of the motivations for studying finite delay is the ability to encode a fair asynchronous parallel operator. However, the model contains a lot more processes than just those obtained from the encoding. Moreover, the introduction of delay actions makes the modelling of synchronous communication less obvious. This raises the questions of identifying the behaviours of fair processes and giving a “delay free” representation of the processes, both answered by the present paper. We apply the first model to give a denotational semantics for a CSP-like process language, FCSP, with a fair asynchronous parallel operator with synchronous communication, an idle process (the identity for the fair parallel operator) and a deadlocked process (represented by the incomplete observation with no actions). The canonical notion of bisimulation obtained from span of open maps is sensitive to both fairness and deadlock.We give a syntactic encoding of FCSP into SCCS with finite delay. Benefiting from the categorical presentation we show that, semantically, the encoding corresponds to the inverse image of the geometric morphism induced by the functor gamma from Inf to Comp which removes delay actions and adds the idle completion, if the sequence contained infinitely many delay actions but only finitely many non-delay actions.