Translating between models of concurrency

Abstract

Hoare's Communicating Sequential Processes (CSP) (Hoare in Communicating Sequential Processes, Prentice-Hall Inc, Upper Saddle River, 1985) admits a rich universe of semantic models closely related to the van Glabbeek spectrum. In this paper we study finite observational models, of which at least six have been studied for CSP, namely traces, stable failures, revivals, acceptances, refusal testing and finite linear observations (Roscoe in Understanding concurrent systems. Texts in computer science, Springer, Berlin, 2010). (Others are known.) We show how to use the relatively recently-introduced priority operator (Roscoe in Understanding concurrent systems. Texts in Computer Science, Springer, Berlin, 2010) to transform refinement questions in these models into trace refinement (language inclusion) tests. Furthermore, we are able to generalise this to any (rational) finite observational model. As well as being of theoretical interest, this is of practical significance since the state-of-the-art refinement checking tool FDR4 (Gibson-Robinson et al. in Int J Softw Tools Technol Transf 18(2):149-167, 2016) currently only supports two such models. In particular we study how it is possible to check refinement in a discrete version of the Timed Failures model that supports Timed CSP.

Sound reasoning in tock-CSP

“…The refusals after the initial tock events are intersected for events other than tock, since we offer the non-tock events of both P and Q. This is specified by ( 23) and (24). They require that if p or q is a trace containing a single refusal then both must be such a trace, since lack of a refusal in P or Q indicates instability and so makes the external choice unstable.…”

“…To encode the refinement for -tock we tailor Mestel and Roscoe [23,24]'s model shifting technique to encoding refusals using traces, so that -tock refinement is reduced to traces refinement. We begin this section by explaining how refusals are encoded, followed by the encoding of -tock traces, and termination.…”

“…Because in C 1 there is an interleaving, termination of Skip does not lead to termination of C 1 [Skip], and instead refusal information is added exactly as for Stop U . To cater for termination, we define a third context C 3 and extend Σ tock with a fresh event tick that encodes , similarly to the approach in [24]. Unlike [24, p. 411], however, we do not hide tick, as this allows Skip to be erroneously refined by Stop U .…”

“…Because refusals before events other than tock need to be ignored to yield the right refinement relation, and not that of refusal testing, the construction is not trivial. A different encoding has also been considered by Roscoe [39] using the concept of slow-abstraction, and more recently Mestel [24] employed model-shifting, an approach that is similar to our encoding of -tock in FDR (described in Sect. 7), but whose treatment of termination is not adequate.…”