Topological Domain Theory

Topological domain theory is a generalization of domain theory that includes a wider collection of topological spaces than traditional domain theory. The generalization overcomes certain known limitations of domain theory, which is unable to model various awkward combinations of computational features.

The development of topological domain theory was supported by an EPSRC research grant "Topological Models for Computational Metalanguages" 2003-6, which employed Matthias Schröder and Ingo Battenfeld. Here is the final report for the project (postscript, pdf).

Overviews of topological domain theory

A manifesto for topological domain theory was presented in talks at Dagstuhl and Kyoto, 2002.

An up-to-date overview of research in topological domain theory will appear in Gordon Plotkin's forthcoming Festschrift:

Mathematical development of topological domain theory

Topological domain theory is based on the remarkable closure properties of qcb spaces (topological quotients of countably based spaces). These study of these spaces was initiated independently by Schröder and by Menni and Simpson in the late 1990's. Qcb spaces provide a link between general topology in mathematics, realizability semantics in computer science, and computability theory (especially, Weihrauch's type two theory of effectivity). Here is a selection of background material developing these aspects of qcb spaces. Topological domain theory involves restricting qcb spaces to ones enjoying a form of chain completeness sufficient for allowing domain-theoretic constructions. The technical development of this is carried out in the references below.

Computational effects in topological domain theory

The various non-functional aspects of computation (e.g. nondeterminism, imperative features, control facilities) are collectively known as computational effects. Plotkin and Power have argued that many computational effects can be modelled using free algebras for equational theories expressing natural equalities between efferct-triggering operations. The papers below adapt this to topological domain theory. The last paper above also makes use of an alternative "observational" approach to computational effects, and depends mathematically upon the following motivating example for the approach.

Related contributions to computability on continuous structures

The following papers address related issues in finding appropriate topological structure for representing diverse aspects of continuous computation.