Topological Models for Higher Order Control Flow

Semantic models are presented for two simple imperative languages with higher order constructs. In the rst language the interesting notion is that of second order assignment x := s, for x a procedure variable and s a statement. The second language extends this idea by a form of higher order communication, with statements c ! s and c ? x, for c a channel. We develop operational and denotational models for both languages, and study their relationships. Both in the deenitions and the comparisons of the semantic models, convenient use is made of some tools from (metric) topology. The operational models are based on (SOS-style) transition systems; the denotational deenitions use domains speciied as solutions of domain equations in a category of 1-bounded complete ultrametric spaces. In establishing the connection between the two kinds of models, fruitful use is made of Rutten's processes as terms technique. Another new tool consists in the use of metric transition systems, with a metric deened on the conngurations of the system. In addition to higher order programming notions, we use higher order deenitional techniques, e.g., in deening the semantic mappings as xed points of (contractive) higher order operators. By Banach's theorem, such xed points are unique, yielding another important proof principle for our paper.