Abstract. Since a nondeterministic and concurrent program may, in general, communicate repeatedly with its environment, its meaning cannot be presented naturally as an input/output function (as is often done in the denotational approach to semantics). In this paper, an alternative is put forth. First, a definition is given of what it is for two programs or program parts to be equivalent for all observers; then two program parts are said to be observation congruent iff they are, in all program contexts, equivalent. The behavior of a program part, that is, its meaning, is defined to be its observation congruence class.The paper demonstrates, for a sequence of simple languages expressing finite (terminating) behaviors, that in each case observation congruence can be axiomatized algebraically. Moreover, with the addition of recursion and another simple extension, the algebraic language described here becomes a calculus for writing and specifying concurrent programs and for proving their properties.

Given a set of processes and a set of tests on these processes we show how to define in a natural way three different eyuitalences on processes. ThesP equivalences are applied to a particular language CCS. We give associated complete proof systems and fully abstract models. These models have a simple representation in terms of trees. 86 R. de Nicr)la, M.C.B. Hennessy a unary postfixed predicate on computations, which we denote by fi: CQ if (ij c is unsuccessful, or (ii) I" contains a state S, such that ST and is not preceded by a successful state. By convention a state precedes itself. We also use 4 to denote the negation of 9. The usual notion on input-output can be viewed as a simple instance of the general setting, as can be seen from the following example. Example. Let 9 denote a set of nondeterministic programs for computing over the natural numbers, with the property that they either compute forever or they halt with some natural number as output. For each pair of natural numbers (n, m) we have an observer 0((n, m)',. This observer, when applied to a program p, will attempt to discover if p will give output m on input n. Thus Comp(O(n, m), p) will consist of all computations of the form (i) a computation generated by p on input n, followed by (ii) if this computation halts examine the output. If it is IPI, then go to a successful state ss. If it is not m, go to a deadlocked state sD.

Abstract. Given a set of processes and a set of tests on these processes we show how to define in a natural way three different eyuitalences on processes. ThesP equivalences are applied to a particular language CCS. We give associated complete proof systems and fully abstract models. These models have a simple representation in terms of trees.

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