Are Two Binary Operators Necessary to Obtain a Finite Axiomatisation of Parallel Composition?

TL;DR

This paper investigates whether a single auxiliary binary operator can enable a finite axiomatization of bisimilarity in a simplified CCS setting, concluding it cannot satisfy all natural assumptions simultaneously.

Contribution

It proves that no single auxiliary binary operator can satisfy all three natural assumptions needed for finite axiomatization in the simplified CCS fragment.

Findings

No auxiliary binary operator satisfies all three assumptions.

Finite axiomatisation of bisimilarity requires at least two operators.

Results are specific to the simplified, recursion-, relabelling-, and restriction-free CCS fragment.

Abstract

Bergstra and Klop have shown that bisimilarity has a finite equational axiomatisation over ACP/CCS extended with the binary left and communication merge operators. Moller proved that auxiliary operators are necessary to obtain a finite axiomatisation of bisimilarity over CCS, and Aceto et al. showed that this remains true when Hennessy's merge is added to that language. These results raise the question of whether there is one auxiliary binary operator whose addition to CCS leads to a finite axiomatisation of bisimilarity. We contribute to answering this question in the simplified setting of the recursion-, relabelling-, and restriction-free fragment of CCS. We formulate three natural assumptions pertaining to the operational semantics of auxiliary operators and their relationship to parallel composition, and prove that an auxiliary binary operator facilitating a finite axiomatisation of bisimilarity in the simplified setting cannot satisfy all three assumptions.