On CSP and the Algebraic Theory of Effects

TL;DR

This paper applies the algebraic theory of effects to CSP, classifying its operations as constructors or deconstructors, and explores their algebraic properties and connections to functional programming and monads.

Contribution

It introduces an algebraic framework for CSP operations, characterizes free and initial algebras, and discusses integrating CSP with Moggi's computational lambda calculus.

Findings

Algebraic characterization of CSP constructors and deconstructors.

Representation of CSP's action and choice as constructors.

Proposal of a polymorphic CSP-Moggi's lambda calculus combination.

Abstract

We consider CSP from the point of view of the algebraic theory of effects, which classifies operations as effect constructors or effect deconstructors; it also provides a link with functional programming, being a refinement of Moggi's seminal monadic point of view. There is a natural algebraic theory of the constructors whose free algebra functor is Moggi's monad; we illustrate this by characterising free and initial algebras in terms of two versions of the stable failures model of CSP, one more general than the other. Deconstructors are dealt with as homomorphisms to (possibly non-free) algebras. One can view CSP's action and choice operators as constructors and the rest, such as concealment and concurrency, as deconstructors. Carrying this programme out results in taking deterministic external choice as constructor rather than general external choice. However, binary deconstructors, such as the CSP concurrency operator, provide unresolved difficulties. We conclude by presenting a combination of CSP with Moggi's computational {\lambda}-calculus, in which the operators, including concurrency, are polymorphic. While the paper mainly concerns CSP, it ought to be possible to carry over similar ideas to other process calculi.