Concrete models of computation for topological algebras

Abstract

A concrete model of computation for a topological algebra is based on a representation of the algebra made from functions on the natural numbers. The functions computable in a concrete model are computable in the representation in the classical sense of the Chruch-Turing Thesis. Moreover, the functions turn out to be continuous in the topology of the algebra. In this paper we consider different concrete models for computing in topological algebras and prove their mutual equivalence in certain commonly occurring circumstances. For topological algebras, the concrete models we use are: effective representation by algebraic domains (Stoltenberg-Hansen and Tucker); effective representation by continuous domains (Edelat); effective representation by type two recursion on Baire space (Weihrauch). And for metric and normed algebras we use: effective metric spaces (Moschovakis) and computability structures (Pour-El and Richards). The result are evidence that computability theory for topological algebras is a stable theory independent of the specific models of computation, just as classical computability theory for discrete algebras is stable.