Interval Domains and Computable Sequences: A Case Study of Domain Reductions

The interval domain as a model of approximations of real numbers is not unique, in fact, there are many variations of the interval domain. We study these variations with respect to domain reductions. The effectivity theory induced by these variations is not stable, and this paper investigates some of the rich structure found. We follow Mostowski (On computable sequences. Fund. Math., 44, 37-51) and use computable sequences to exhibit this structure

Domain Representable Spaces Defined by Strictly Positive Induction

Recursive domain equations have natural solutions. In particular there are domains defined by strictly positive induction. The class of countably based domains gives a computability theory for possibly non-countably based topological spaces. A $qcb_{0}$ space is a topological space characterized by its strong representability over domains. In this paper, we study strictly positive inductive definitions for $qcb_{0}$ spaces by means of domain representations, i.e. we show that there exists a canonical fixed point of every strictly positive operation on $qcb_{0}$ spaces.Comment: 48 pages. Accepted for publication in Logical Methods in Computer Scienc

Reducibility of domain representations and Cantor–Weihrauch domain representations

The paper looks at the spectrum of available domain representations of topological spaces. The spectrum is analysed via the notion of domain reducibility. This concept is related to the notion of reductions from TTE, and in fact all TTE representations (here referred to as Cantor-Weihrauch domain representations) and their reductions form a sub-spectrum of all available domain representations

Reducibility of Domain Representations and Cantor-Weihrauch Domain Representations

We introduce a notion of reducibility of representations of topological spaces and study some basic properties of this notion for domain representations. A representation reduces to another if its representing map factors through the other representation. Reductions form a pre-order on representations. A spectrum is a class of representations divided by the equivalence relation induced by reductions. We establish some basic properties of spectra, such as, non-triviality. Equivalent representations represent the same set of functions on the represented space. Within a class of representations, a representation is universal if all representations in the class reduce to it. We show that notions of admissibility, considered both for domains and within Weihrauch’s TTE, are universality concepts in the appropriate spectra. Viewing TTE representations as domain representations, the reduction notion here is a natural generalisation of the one from TTE. To illustrate the framework, we consider some domain representations of real numbers and show that the usual interval domain representation, which is universal among dense representations, does not reduce to various Cantor domain representations. On the other hand, however, we show that a substructure of the interval domain more suitable for efficient computation of operations is equivalent to the usual interval domain with respect to reducibility. 1