Abstract not found

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.

Abstract not found

on topological spaces via domain representations

Given a continuous functional f: X → Y and y ∈ Y, we wish to compute x ∈ X such that f(x) = y, if such an x exists. We show that if x is unique and X and Y are subspaces of Kleene–Kreisel spaces of continuous functionals with X exhaustible, then x is computable uniformly in f, y and the exhaustion functional ∀X: 2 X → 2. We also establish a version of the above for computational metric spaces X and Y, where is X computationally complete and has an exhaustible set of Kleene–Kreisel representatives. Examples of interest include functionals defined on compact spaces X of analytic functions. Our development includes a discussion of the generality of our constructions, bringing QCB spaces into the picture, in addition to general topological considerations. Keywords and phrases. Higher-type computability, Kleene–Kreisel spaces of continuous functionals, exhaustible set, searchable set, QCB space, admissible representation, topology in the theory of computation with infinite objects. 1

We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a start on our programme by proposing a general logical framework for describing physical theories, and analysing several possible formulations of a simple Newtonian theory from the point of view of determinism. Our emphasis throughout is on clarifying the precise physical and metaphysical assumptions that typically underlie a claim that some physical theory is ‘deterministic’. A sequel paper is planned, in which we shall apply similar methods to the analysis of other physical theories. Along the way, we discuss some possible repercussions of this kind of investigation for both physics and logic. 1

In recent work we developed the notion of exhaustible set as a higher-type computational counter-part of the topological notion of compact set. In this paper we give applications to the computation of solutions of higher-type equations. Given a continuous functional f: X → Y and y ∈ Y, we wish to compute x ∈ X such that f(x) = y, if such an x exists. We show that if x is unique and X and Y are subspaces of Kleene– Kreisel spaces of continuous functionals with X exhaustible, then x is computable uniformly in f, y and the exhaustibility condition. We also establish a version of this for computational metric spaces X and Y, where is X computationally complete and has an exhaustible set of Kleene–Kreisel representatives. Examples of interest include evaluation functionals defined on compact spaces X of bounded sequences of Taylor coefficients with values on spaces Y of real analytic functions defined on a compact set. A corollary is that it is semi-decidable whether a function defined on such a compact set fails to be analytic, and that the Taylor coefficients of an analytic function can be computed extensionally from the function. Keywords and phrases. Higher-type computability, Kleene–Kreisel spaces of continuous functionals, exhaustible set, searchable set, computationally compact set, QCB space, admissible representation, topology in the theory of computation. 1