A Truly Concurrent Semantics for a Simple Parallel Programming Language

This paper gives the first truly concurrent denotational, i.e., compositional semantics for a simple, deterministic parallel language. By truly concurrent we mean the denotational model does not rely on an interleaving of concurrent actions for its definition. Thus, our semantics does not reduce parallelism to nondeterminism, as is done in the more established approaches to concurrency. We also present a natural SOS-style operational semantics for our language, and we prove a congruence theorem relating the two semantics. This implies the operational model itself is compositional. The congruence theorem also implies the denotational model is adequate with respect to the operational semantics, and we characterize the relatively mild conditions under which the denotational semantics is fully abstract with respect to the operational semantics. Our simple language includes a (weak) sequential composition operator which takes advantage of the truly concurrent nature of the semantics, as well as a parallel composition operator which allows local events to execute asynchronously, while requiring synchronizing events to execute simultaneously. In addition, the language supports a restriction operator and includes recursion. The denotational semantics also is novel for its treatment of recursion. The meaning of a recursive process is defined using a least fixed point on a subdomain that is determined by the body of the recursion, and that varies from one process to another. Nonetheless, the recursion operators in the language have continuous interpretations in the denotational model. Our denotational model is based on a domain-theoretic generalization of Mazurkiewicz traces in which the concatenation operator, as well as the other operators from our language can be given continuous interpretations. 1 The support of the National Science Foundation and the US Office of Naval Research is gratefully acknowledged.