Resource traces: a domain for processes sharing exclusive resources

AbstractThe domain of partially terminated finite and infinite words is commonly used to give denotational semantics for process algebras such as CSP. In this well-known framework the denotational semantics of concurrency is derived via power-domains from that of non-deterministic choice and interleaving to the effect that the denotational semantics of a concurrent process is equal to the set of all its possible finite and infinite sequential behaviours. In this paper, we define a more versatile domain of the so-called finite and infinite resource traces which allows to capture the concurrent behaviour of a process and encode the static concurrency of a system directly into the domains definition. The approach we present refines the previous work of Diekert and Gastin (Lecture Notes in Computer Science, vol. 944, Springer, Berlin, pp. 15–26) on α- and δ-traces. We start with an alphabet of atomic actions, a set of resources, and a resource map assigning to each action the non-empty subset of resources it uses. Actions that do not share common resources are called independent and considered to be able to execute concurrently. A partially terminated concurrent process is specified by a resource trace which consists of two components: an already observed part represented as an action-labeled partial order (Mazurkiewicz trace), and a guard set containing the resources granted to the process for its further development. A process concatenation is then defined, which allows independent actions to execute concurrently. Specification refinement leads to a natural approximation ordering between processes. It confers to the set of all processes the structure of a coherently complete prime algebraic Scott domain, whereby, process concatenation is Scott-continuous in both arguments. Furthermore, we define a natural ultrametric on processes based on prefix information. The induced topology is shown to be equivalent to the compact Lawson topology induced by the approximation ordering. Process concatenation is moreover shown to be uniformly continuous with respect to the defined ultrametric. The mathematical theory we develop thus extends the central order and metric properties of the domain of partially terminated finite and infinite words which are needed in order to devise truly concurrent semantics for process algebras much in the style of classical CSP semantics