An Extended Expansion Theorem

Closed CCS (CCCS) is a CCS-like algebra of processes with a generalized form of prefixing based on a full-fledged algebra of transitions rather than on basic actions only. The basic idea is that the generalized prefixing operator takes a transition t, or rather its observation w, a process E and yields the process t.E. From an operational standpoint, the process t.E may evolve to E by performing a transition labelled by w. By exploiting the algebra of transitions, we define a general form of expansion theorem which is the heart of a finite axiomatizatiou of a strong observational equivalence for finite CCCS agents. By adding the axioms concerning the interpretation of the operations of the algebra of observations, we still obtain a sound and complete axiomatization of the corresponding bisimulation equivalence. For instance, it is possible to define the classical expansion theorem, or versions of it which handle partial ordering based observations. 1 I n t r o d u c t i o n Many different models have been proposed to specify the behaviour of concurrent and distributed systems. We single out two approaches: the interleaving approach ([Mi180,Mi189], [BHR84,Ho85], [AB84], [BK84], [tIeSS]) and the true concurrency approach ([Re85], [NPW81], [Pr86], [BC88], [DM87,DDM88a, DDM90a], [0187], [RT88]). The debate and the arguments between the supporters of either these antagonist approaches have not yet led to a clear measure of superiority of one over the other. In our view, the main merit of the former is its well-established theory, whilst the latter gives an intuitively more convincing representation of concurrent system behaviour. This paper aims at giving a contribution in filling the gap between the two approaches by providing also the latter with an axiomatic theory which can be naturally specialized to the interleaving case. The development of the interleaving approach to concurrency is well illustrated by Milner's work on CCS [MilS0,Mi189]. Considering concurrent systems as structured entities, which interact by some synchronization mechanisms, naturally leads to the definition of operations for building new systems from existing ones: every system can be seen as a term of the free algebra over this set of operations. The resulting process description language (the one proposed by Milner is called Calculus of Communicating Systems (CCS)) comes equipped with an operational semantics which takes the form of a labelled transition system [PloS1]. However, these operational descriptions are too intensional. More abstract semantics of the language are obtained by introducing behaviourai equivalences which identify process terms which exhibit the same behaviour in accordance with certain observational scenarios. Many of these behavioural equivalences are based on the notion of bisimulation [Par81]. Several bisimuiation based equivalences are completely characterized by equational laws between process terms; in other words, the equivalences are actually congruences obtained by making the quotient of the free a~ebra "Work partially supported by ESPRIT Basic Research Action 3011, CEDISYS, and by Progetto Finalizzato Information e Galcolo Pa~allelo, obiettivo LAMBRUSCO