Unfolding CSP

Abstract

This paper demonstrates that a wide range of the CSP operators, in particular parallel composition, hiding, general choice, interleaving, and the non-deterministic OR, can be represented by confluent unfolding to a normal form. The relevant normal form is an enrichment of the CSP choice construction by the inclusion of non-determinism and silent actions. It is demonstrated that the majority of the equational laws presented by Hoare for these operators are valid not only for the failures equivalence but even for strong bisimilarity. This work is a prelude to embedding CSP in the bigraph model, a recent generic model for ubiquitous computing in which other process calculi have been faithfully embedded. The authors hope by this means to elucidate the differences and similarities between CSP and other calculi. The present paper presents the main ideas of the embedding in the formalism of CSP itself, and makes no use of bigraphs in doing so.

Appreciation from the second author Tony Hoare and I have exchanged ideas on concurrent processes for three decades. To a sufficiently distant observer it has seemed that we were doing the same thing, and that therefore we should not have made it look different. A closer view shows this to be false. Complementary things, yes; and both have been enriched by the cross-fertilisation.

A little history shows something of these different approaches. Around 1979 we first discovered our complementary interests in concurrency. Tony at first expressed his ideas through the medium of a programming language [6], and I through a prototypical algebraic theory [8]. The difference became plainer as time went on. Tony was keen to find a single formalism in which specifications of concurrent systems could be refined into programs. I, on the other hand, was keen to find a mathematical concept of process that could stand in analogy with the familiar notion of (single valued) function, and I was happy that specification should be done in an associated logic. We enjoyed discussing these things. I recall one discussion at a blackboard (or was it a whiteboard?) where Tony hinted to me his first ideas about failures semantics.

The present short paper, in honour of Tony and continuing our long, friendly and sometimes rivalrous collaboration, is another step towards harmonising our approaches.