Effective operators in a topological setting (original) (raw)
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On effective topological spaces
The Journal of Symbolic Logic, 1998
Starting with D. Scott's work on the mathematical foundations of programming language semantics, interest in topology has grown up in theoretical computer science, under the slogan ‘open sets are semidecidable properties’. But whereas on effectively given Scott domains all such properties are also open, this is no longer true in general. In this paper a characterization of effectively given topological spaces is presented that says which semidecidable sets are open. This result has important consequences. Not only follows the classical Rice-Shapiro Theorem and its generalization to effectively given Scott domains, but also a recursion theoretic characterization of the canonical topology of effectively given metric spaces. Moreover, it implies some well known theorems on the effective continuity of effective operators such as P. Young and the author's general result which in its turn entails the theorems by Myhill-Shepherdson, Kreisel-Lacombe-Shoenfield and Ceĭtin-Moschovakis...
Computability on Topological Spaces via Domain Representations
New Computational Paradigms, 2008
Domains are ordered structures designed to model computation with approximations. We give an introduction to the theory of computability for topological spaces based upon representing topological spaces and algebras using domains. Among the topics covered are: different approaches to computability on topological spaces; orderings, approximations and domains; making domain representations; effective domains; classifying representations; type two effectivity and domains; special representations for inverse limits, regular spaces and metric spaces. Lastly, we sketch a variety of applications of the theory in algebra, calculus, graphics and hardware.
Relations between certain classes of effectively topological spaces
Mathematical Notes of the Academy of Sciences of the USSR, 1969
UDC 51. 01.16 Relations between properties of effectively topological spaces such as the existence of a countable regular base, a computable intersection, effective regularity, normality etc., are studied; an arithmetic (constructive) analog of Uryson's Theorem on the metrization of topological spaces with denumerable bases is established; it is shown that the conditions in the analog are independent.
Recursion and topology on 2⩽ω for possibly infinite computations
Theoretical Computer Science, 2004
In the context of possibly infinite computations yielding finite or infinite (binary) outputs, the space 2 ≤ω = 2 * ∪ 2 ω appears to be one of the most fundamental spaces in Computer Science. Though underconsidered, next to 2 ω , this space can be viewed (§3.5.2) as the simplest compact space native to computer science. In this paper we study some of its properties involving topology and computability. Though 2 ≤ω can be considered as a computable metric space in the sense of computable analysis, a direct and self-contained study, based on its peculiar properties related to words, is much illuminating. It is well known that computability for maps 2 ω → 2 ω reduces to continuity with recursive modulus of continuity. With 2 ≤ω , things get less simple. Maps 2 ω → 2 ≤ω or 2 ≤ω → 2 ≤ω induced by input/output behaviors of Turing machines on finite or infinite words-which we call semicomputable maps-are not necessarily continuous but merely lower semicontinuous with respect to the prefix partial ordering on 2 ≤ω. Continuity asks for a stronger notion of computability. We prove for (semi)continuous and (semi)computable maps F : I → O with I, O ∈ {2 ω , 2 ≤ω } a detailed representation theorem (Thm.82) via functions f : 2 * → 2 * following two approaches: bottom-up from f to F and top-down from F to f .
2015
The date of receipt and acceptance will be inserted by the editor Abstract First we introduce the notion of super-coherent topology which does not depend on any ordering. Then we show that a topology is super-coherent if and only if it is the Scott topology over a suitable algebraic dcpo. The main ideas of the paper are a by-product of the constructive ap-proach to domain theory through information bases which we have pro-posed in a previous work, but the presentation here does not rely on that foundational framework. Key words Scott Domain – Algebraic dcpo – Formal Topology 1
On some problems in computable topology
Logic Colloquium 2005, 2007
Computations in spaces like the real numbers are not done on the points of the space itself but on some representation. If one considers only computable points, i.e., points that can be approximated in a computable way, finite objects as the natural numbers can be used for this. In the case of the real numbers such an indexing can e.g. be obtained by taking the Gödel numbers of those total computable functions that enumerate a fast Cauchy sequence of rational numbers. Obviously, the numbering is only a partial map. It will be seen that this is not a consequence of a bad choice, but is so by necessity. The paper will discuss some consequences. All is done in a rather general topological framework.
A note on domain representability and formal topology
Two common approaches to constructive and effective topology are con- nected by showing that formal topologies have canonical representation in terms of Scott domains. Moreover a map lifting theorem for the representa- tion is proved. MSC 2000 classification: 54A05, 03D45, 03F65.
Bi-topological spaces and the Continuity Problem
2021
The Continuity Problem is the question whether effective operators are continuous, where an effective operator F is a function on a space of constructively given objects x, defined by mapping construction instructions for x to instructions for F pxq in a computable way. In the present paper the problem is dealt with in a bi-topological setting. To this end the topological setting developed by the author [23] is extended to the bi-topological case. Under very natural conditions it is shown that an effective operator F between bi-topological spaces T “ pT, τ, σq and T 1 “ pT 1, τ 1, σ1q is (effectively) continuous, if τ 1 is (effectively) regular with respect to σ1. A central requirement on T 1 is that bases of the neighbourhood filters of the points in T 1 can computably be enumerated in a uniform way, not only with respect to topology τ 1, but also with respect to σ1. As follows from an example by Friedberg, the last condition is indispensable. Conversely, it is proved that (effecti...