Generalizing Domain Theory

Domain theory began in an attempt to provide mathematical models for high-level programming languages, an area where it has proved to be particularly useful. It is perhaps the most widely-used method for devising semantic models for such languages. This paper is a survey of some generalizations of domain theory that have arisen in efforts to solve related problems. In each case, a description is given of the problem and of the solution generalizing domain theory it inspired. The problems range from the relation of domain theory to other approaches for providing semantic models, particularly in process algebra, to issues surrounding the notion of a computational model, an approach inspired by the recent work of Abbas Edalat. 1 The Basics { How Domain Theory Began This section is a brief outline of some of the \basic ingredients" of domain theory and the applications that inspired them. Domain theory began in an attempt by Dana Scott to nd mathematical models for high-level programming languages. Upon his arrival in Oxford in the mid 1960s, Scott found Christopher Strachey and his colleagues at the Programming Research Group using the untyped lambda calculus of Church and Curry as a model for programming, something Scott found disturbing because he regarded it as a \formal and unmotivated" notation (cf. 11]). He thus set out to nd alternative models for Strachey and his colleagues to use. Because programs can call other programs, and indeed, can even call themselves, Scott was led to consider objects X in some category or other which satisfy the property that they contain a copy of their space of selfmaps in the category. Of course, Cantor's Lemma implies the only such objects in the category of sets and functions are degenerate (i.e., they consist of a single point), and so no such objects can be found there. But the reals have only as many continuous selfmaps as there are real numbers (because of their having a dense, countable subset on which all continuous selfmaps are completely determined), so it is potentially II possible to nd such objects X among topological spaces. While attempting to nd appropriate models of partially deened maps, Scott realized there had to be T 0 spaces isomorphic to their space of continuous selfmaps, and he then constructed such an object in the category of algebraic lattices and so-called Scott continuous maps. In the years since Scott constructed the rst model …