In order to provide both a sound semantics for programming languages that come with a data type for the real numbers and to study computability on topological spaces like the reals, the elements of such spaces are represented by “total” elements of some Scott-Ershov domain, where what is meant by a “total” domain element depends on the concrete representation. Baire space, e.g., that is the space of all infinite sequences over a fixed infinite set, is represented by the domain of all finite and infinite sequences with the prefix ordering, called Baire domain; similarly for Cantor space, i.e. the space of all infinite sequences over a fixed finite set. In both cases the infinite sequences are the total elements.
In previous work, the author has studied various kinds of such domain representations. In the present paper he compares representations of a given space via an appropriate adaption of the usual reducibility notion used in computability theory. Depending on which reduction functions are admitted, different reducibility notions are obtained. As usual, each reducibility notion defines a pre-order among representations. A spectrum over a represented space is a collection of equivalence classes of representations of the given space with respect to a fixed reducibility notion ordered by the induced partial order.
The author establishes basic properties of spectra, such as non-triviality. Moreover, he proves that equivalent representations represent the same set of functions on the represented space.
Representations that are largest within a class of representations and with respect to a fixed reducibility notion are called universal. With respect to continuous reduction, every \(T_0\) space has a universal domain representation. In the case of partial reduction functions, a domain representation is universal if, and only if, the domain representation is a retract representation. Several admissibility notions that have been considered both for domains as well as Weihrauch’s Type-Two Theory of Effectivity (TTE) turn out to be universality concepts in the appropriate spectra.
To illustrate the framework, the author finally considers domain representations of the real numbers. It is shown that the usual interval domain representation, which is universal among dense representations, does not reduce to various Cantor domain representations. A substructure of the interval domain is exhibited, which is more suitable for efficient computation of real number operations and which is equivalent to the usual interval domain with respect to computable reducibility. With respect to continuous reducibility the spectrum of all domain representations of the real numbers is infinite.