A framework allowing a unified and rigorous definition of the semantics of concurrency is proposed. The mathematical model introduces processes as elements of process domains which are obtained as solutions of domain equations in the sense of Scott and Plotkin. Techniques of metric topology as proposed, e.g., by Nivat are used to solve such equations. Processes are then used as meanings of statements in languages with concurrency. Three main concepts are treated, viz. parallellism (arbitrary interleaving of sequences of elementary actions), synchronization, and communication. These notions are embedded in languages which also feature classical sequential concepts such as assignment, tests, iteration or recursion, and guarded commands. In the definitions, a sequence of process domains of increasing complexity is used. The languages discussed include Milner's calculus for communicating systems and Hoare's communicating sequential processes. The paper concludes with a section with brief remarks on miscellaneous notions in concurrency. and two appendices with mathematical details.

A general framework for the denotational treatment of concurrency is introduced. The key idea is the notion of process which is element of a domain obtained as solution of a domain equation in the style as considered previously by Plotkin. We use tools from metric topology as advocated by Nivat to solve this equation, show how operations upon processes can be defined conveniently, and illustrate the approach with the definition of a variety of concepts as encountered in the study of concurrency. Only few proofs of the supporting mathematical theory are given; full proofs will appear in the final version of the paper.

A survey is given of work performed by the authors in recent years concerning the semantics of imperative concurrency. Four sample languages are presented for which a number of operational and denotational semantic models are developed. All languages have parallel execution through interleaving, and the last three have as well a form of synchronization. Three languages are uniform, i.e., they have uninterpreted elementary actions; the fourth is nonuniform and has assignment, tests and value-passing communication. The operational models build on Hennessy-Plotkin transition systems; as denotational structures both metric spaces and cpo domains are employed. Two forms of nondeterminacy are distinguished, viz. the local and global variety. As associated model-theoretic distinction that of linear time versus branching time is investigated. In the former we use streams, i.e. finite or infinite sequences of actions; in the latter the (metrically based) notion of process is introduced. We furthermore study a model with only finite observations. Ready sets also appear, used as technical tool to compare various semantics. Altogether, ten models tor the four languages are described, and precise statements on (the majority of) their interrelationships are made . The paper supplies no proofs; for these references to technical papers by the authors are provided.Contents:

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Abstract.An overview is given of work we have done in recent years on the semantics of concurrency, concentrating on semantic models built on metric structures. Three contrasting themes are discussed, viz. (i) uniform or schematic versus nonuniform or interpreted languages; (ii) operational versus denotational semantics, and (iii) linear time versus branching time models. The operational models are based on Plotkin's transition systems. Language constructs which receive particular attention are recursion and merge, synchronization and global nondeterminacy, process creation, and communication with value passing. Various semantic equivalence results are established. Both in the definitions and in the derivation of these equivalences, essential use is made of Banach's theorem for contracting functions.1985 Mathematics Subject Classification: 68Q55, 68Ql0.