The Equational Theory of Prebisimilarity over Basic CCS with Divergence ⋆ Luca Aceto a,∗ , Silvio Capobianco a , Anna Ingolfsdottir a , Bas Luttik b a School of Computer Science, Reykjav´ ık University, Iceland b Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, The Netherlands Abstract This paper studies the equational theory of prebisimilarity, a bisimulation-based preorder introduced by Hennessy and Milner in the early 1980s, over basic CCS with the divergent process Ω. It is well known that prebisimilarity affords a finite ground-complete axiomatization over this language; this study proves that this ground-complete axiomatization is also complete in the presence of an infinite set of actions. Moreover, in sharp contrast to this positive result, it is shown that prebisimilarity is not finitely based over basic CCS with the divergent process Ω when the set of actions is finite and non-empty. Key words: PACS: 1. Introduction The notion of prebisimilarity [5,9,13,17], or bisim- ulation preorder, has played a key role in the study of domain-theoretic and term models for Milner’s CCS [9] and related languages [1,3,7]. In particu- lar, as shown in [4], Abramsky’s classic domain- theoretic, synchronization-tree-based model for bisimilarity (presented in [1]) is fully abstract with respect to (the finitary part of) prebisimilarity for a natural class of GSOS languages [6]. This means that two processes specified in any GSOS language satisfying the restrictions considered in [4] are re- lated by (the finitary part of) prebisimilarity if, and only if, their denotations are similarly related within ⋆ The first three authors were partly supported by the project “The Equational Logic of Parallel Processes” (nr. 060013021) of The Icelandic Research Fund. * Corresponding author. Email addresses: luca@ru.is (Luca Aceto), silvio@ru.is (Silvio Capobianco), annai@ru.is (Anna Ingolfsdottir), s.p.luttik@tue.nl (Bas Luttik). Abramsky’s denotational model. This behavioural semantics for processes is therefore in complete agreement with the denotational semantics, thus achieving results for concurrent processes modulo a bisimulation-like behavioural semantics that are akin to those presented in the classic papers [12,15]. The simplest process language over which pre- bisimilarity can be naturally defined is basic CCS extended with the divergent process Ω. We refer to this language as BCCS Ω in what follows. This is a language that contains only the basic process al- gebraic operators from CCS [14], but is sufficiently powerful to express all finite synchronization trees in Abramsky’s domain. Intuitively, Ω is the pro- cess about whose behaviour we have no information whatsoever, and is essentially a syntactic represen- tation of the least element in Abramsky’s domain. In this paper, we study the equational theory of prebisimilarity over the language BCCS Ω . It is well known that prebisimilarity affords a finite ground- complete axiomatization over this language (see, Preprint submitted to Elsevier 26 May 2008