Analysing Mutual Exclusion using Process Algebra with Signals

Abstract

In contrast to common belief, the Calculus of Communicating Systems (CCS) and similar process algebras lack the expressive power to accurately capture mutual exclusion protocols without enriching the language with fairness assumptions. Adding a fairness assumption to implement a mutual exclusion protocol seems counter-intuitive. We employ a signalling operator, which can be combined with CCS, or other process calculi, and show that this minimal extension is expressive enough to model mutual exclusion: we confirm the correctness of Peterson's mutual exclusion algorithm for two processes, as well as Lamport's bakery algorithm, under reasonable assumptions on the underlying memory model. The correctness of Peterson's algorithm for more than two processes requires stronger, less realistic assumptions on the underlying memory model.

Progress, Justness, and Fairness

“…In this setting, one may expect that necessary participants in a synchronisation are always affected (npc(t) ⊆ afc(t)). However, in a process algebra with signals [7,18] we find transitions t with npc(t) ⊈ afc(t). Let t model the action of a driver seeing a red traffic light.…”

“…In applications where npc afc, two variants of fairness of components and fairness of groups of components can be distinguished, namely by taking either npc or afc to be the function comp : Tr → P(C ) assumed in Section 5. Since the specification formalisms dealing with fairness found in the literature, with the exception of our own work [18,22,28], did not give rise to a distinction between necessary and affected participants in a synchronisation, in our treatment of fairness we assume that npc = afc (= comp), leaving a treatment of the general case for future work.…”

“…In [18,27] we argue that a justness assumption is essential for the validity of any mutual exclusion protocol. It appears that in the correctness arguments of famous mutual exclusion protocols appearing in the literature [35,49] the relevant justness assumption is taken for granted, without explicit acknowledgement that the correctness of the protocol rests on the validity of this assumption.…”

Justness

“…This example is not a contrived corner case, but a rather typical illustration of an issue that is central to the study of distributed systems. Other illustrations of this phenomena occur in [10, Section 9.1], [14,Section 10], [11,Section 1.4], [12] and [6,Section 4]. The criterion of justness aims to ensure the liveness property occurring in these examples.…”

“…In the same vain, [15] offered a formalisation for the transition systems generated by CCS [20], and its extension ABC, the Algebra of Broadcast Communication [15], a variant of CBS, the Calculus of Broadcasting Systems [25]. The same was done for CCS extended with signals in [6]. These formalisations coinductively define B-justness, where B ranges over sets of transitions deemed to be blocking, as a family of predicates on paths, and proceed by a case distinction on the operators in the language.…”