On the Relationship between Scott Domains, Synchronization Trees, and Metric Spaces

Scott's theory of information systems (Scott, 1982) is intended to provide an easy way to define partial order structures (domains) for denotational semantics. This paper illustrates the new method by considering a simple modal logic, due to Hennessy and Milner (1980), as an example of an information system. The models of formulas in this logic are the rigid synchronization trees of Milner (1980). We characterize the domain defined by the Hennessy-Milner information system as the complete partial order of synchronization forests: nonempty closed sets of synchronization trees. "Closed" means closed with respect to a natural metric distance on synchronization trees, first defined by de Bakker and Zucker (1982) and characterized by Golson and Rounds (1983). After notational preliminaries and background results, Section 3 treats the Hennessy-Milner information system. The background results (Brookes and Rounds, 1983; Golson and Rounds, 1983) are used as lemmas in the characterization of the partial order. Section 4 then shows how to use metric space methods to extend certain natural tree operations to forests. These operations become continuous in the partial order sense when so extended, and therefore can be used to provide a denotational semantics for concurrency which allows the full power of least fixed point methods for recursion (Sect. 5).