Title: Topological Models in Comparative Semantics

During the last three decades several diierent styles of semantics for programming languages have been developed. This thesis compares two of them: the operational and the denotational approach. We show how to give operational and denotational semantics to programming languages, and how to compare diierent semantic models for a given language. Both in the deenition of the denotational semantics and in the comparison of semantic models we make use of metric topology. The work reported in this thesis started with the search for comparative semantics for a fragment of ACPr, a real time language introduced by Baeten and Bergstra (1991). The principal construction of this language is the integration statement. In its most general form the integration statement gives rise to unbounded nondeterminism. We rst considered a nite version of integration. An operational semantics was given for the language by means of a labelled transition system. Furthermore, a denotational semantics based on a complete metric space was constructed for the language. We proved that the two semantic models are equivalent by means of standard tools. Inspired by a theorem of Michael (1951), a result from general topology which roughly tells us that a compact union of compact sets is again compact, we next considered the language with a compact version of integration. The operational semantics could be extended without any problems. For the extension of the denotational semantics we needed some new ingredients. In the deenition of the denotational semantics we exploited the already mentioned theorem of Michael and the fact that a continuous image of a compact set is compact, a result from general topology due to Alexandroo (1927). The equivalence proof of the two semantic models was the most diicult part to generalize. To prove the equivalence by uniqueness of xed point, a proof principle based on Banach's unique xed point theorem (1922) and introduced in semantics by Kok and Rutten (1990), the operational semantics should be compact. The compactness of an operational semantics is usually derived from the fact that the labelled transition system inducing the semantics satisses the niteness condition nitely branching. However, the labelled transition system at hand is not nitely branching. It does not even satisfy the weaker niteness condition image nite. We provided the labelled transition system with some additional metric structure. The enriched labelled transition system was shown to be compactly branching, a generalization of nitely branching. This generalization enabled us to …