In search of lost time: Axiomatising parallel composition in process algebras

TL;DR

This survey examines the axiomatisation of parallel composition in process algebras, addressing longstanding open problems and establishing new boundaries for finite axiomatisability in various bisimilarity contexts.

Contribution

It provides new insights into the limitations of finite axiomatisations in CCS, especially regarding auxiliary operators and rooted weak bisimilarity.

Findings

Adding a single auxiliary operator does not yield finite axiomatisation of bisimilarity.

Finite axiomatisation is impossible for rooted weak bisimilarity over CCS.

Clarifies the boundary between finite and non-finite axiomatisability in the linear time-branching time spectrum.

Abstract

This survey reviews some of the most recent achievements in the saga of the axiomatisation of parallel composition, along with some classic results. We focus on the recursion, relabelling and restriction free fragment of CCS and we discuss the solutions to three problems that were open for many years. The first problem concerns the status of Bergstra and Klop's auxiliary operators left merge and communication merge in the finite axiomatisation of parallel composition modulo bisimiliarity: We argue that, under some natural assumptions, the addition of a single auxiliary binary operator to CCS does not yield a finite axiomatisation of bisimilarity. Then we delineate the boundary between finite and non-finite axiomatisability of the congruences in van Glabbeek's linear time-branching time spectrum over CCS. Finally, we present a novel result to the effect that rooted weak bisimilarity has no finite complete axiomatisation over CCS.