Abstract

Abstract. This paper surveys some classic and recent results on the finite axiomatizability of bisimilarity over CCS-like languages. It focuses, in particular, on non-finite axiomatizability results stemming from the semantic interplay between parallel composition and nondeterministic choice. The paper also highlights the role that auxiliary operators, such as Bergstra and Klop’s left and communication merge and Hennessy’s merge operator, play in the search for a finite, equational axiomatization of parallel composition both for classic process algebras and for their real-time extensions. 1 The Problem and its History Process algebras are prototype description languages for reactive systems that arose from the pioneering work of figures like Bergstra, Hoare, Klop and Milner. Well-known examples of such languages are ACP [18], CCS [44], CSP [40] and Meije [13]. These algebraic description languages for processes differ in the basic collection of operators that they offer for building new process descriptions from existing ones. However, since they are designed to allow for the description and analysis of systems of interacting processes, all these languages contain some form of parallel composition (also known as merge) operator allowing one to put two process terms in parallel with one another. These operators usually interleave the behaviours of their arguments, and support some form of synchronization between them. For example, Milner’s CCS offers the binary operator ||, whose intended semantics is described by the following classic rules in the style of Plotkin [49]. x µ → x ′ x | | y µ → x ′ | | y y µ → y ′ x | | y µ → x | | y ′ x α → x ′ , y ¯α → y ′ x | | y τ → x ′ | | y ′ (In the above rules, the symbol µ stands for an action that a process may perform, α and ¯α are two observable actions that may synchronize, and τ is a symbol denoting the result of their synchronization.)

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