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This is the second of two papers in which we present the x-calculus, a calculus of mobile processes. We provide a detailed presentation of some of the theory of the calculus developed to date, and in particular we establish most of the results stated in the companion paper. k?

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.

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We present the a-calculus, a calculus of communicating systems in which one can naturally express processes which have changing structure. Not only may the component agents of a system be arbitrarily linked, but a communication between neighbours may carry information which changes that linkage. The calculus is an extension of the process algebra CCS, following work by Engberg and Nielsen, who added mobility to CCS while preserving its algebraic properties. The rr-calculus gains simplicity by removing all distinction between variables and constants; communication links are identified by names, and computation is represented purely as the communication of names across links. After an illustrated description of how the n-calculus generalises conventional process algebras in treating mobility, several examples exploiting mobility are given in some detail. The important examples are the encoding into the n-calculus of higher-order functions (the I-calculus and combinatory algebra), the transmission of processes as values, and the representation of data structures as processes. The paper continues by presenting the algebraic theory of strong bisimilarity and strong equivalence, including a new notion of equivalence indexed by distinctions-i.e., assumptions of inequality among names. These theories are based upon a semantics in terms of a labeled transition system and a notion of strong bisimulation, both of which are expounded in detail in a companion paper. We also report briefly on work-in-progress based upon the corresponding notion of weak bisimulation, in which internal actions cannot be observed.

The -calculus is a model of concurrent computation based upon the notion of naming. It is rst presented in its simplest and original form, with the help of several illustrative applications. Then it is generalized from monadic to polyadic form. Semantics is done in terms of both a reduction system and a version of labelled transitions called commitment; the known algebraic axiomatization of strong bisimilarity i s g i v en in the new setting, and so also is a characterization in modal logic. Some theorems about the replication operator are proved.Justi cation for the polyadic form is provided by the concepts of sort and sorting which it supports. Several illustrations of di erent sortings are given. One example is the presentation of data structures as processes which respect a particular sorting; another is the sorting for a known translation of the -calculus into -calculus. For this translation, the equational validity o f -conversion is proved with the help of replication theorems. The paper ends with an extension of the -calculus to !-order processes, and a brief account of the demonstration by D a vide Sangiorgi that higher-order processes may b e faithfully encoded at rst-order. This extends and strengthens the original result of this kind given by Bent Thomsen for second-order processes.