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Reducibility of Domain Representations and CantorWeihrauch Domain Representations
, 2006
"... 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 preorder on representatio ..."
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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 preorder 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, nontriviality. 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.
A rich hierarchy of functionals of finite types (manuscript
, 2008
"... Abstract. We are considering typed hierarchies of total, continuous functionals using complete, separable metric spaces at the base types. We pay special attention to the socalled Urysohn space constructed by P. Urysohn. One of the properties of the Urysohn space is that every other separable metric ..."
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Abstract. We are considering typed hierarchies of total, continuous functionals using complete, separable metric spaces at the base types. We pay special attention to the socalled Urysohn space constructed by P. Urysohn. One of the properties of the Urysohn space is that every other separable metric space can be isometrically embedded into it. We discuss why the Urysohn space may be considered as the universal model of possibly infinitary outputs of algorithms. The main result is that all our typed hierarchies may be topologically embedded, type by type, into the corresponding hierarchy over the Urysohn space. As a preparation for this, we prove an effective density theorem that is also of independent interest. 1.
Additional services for Mathematical Structures in
, 2009
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Domain representations of spaces of compact subsets
, 2010
"... We present a method for constructing from a given domain representation of a space X with underlying domain D, a domain representation of a subspace of compact subsets of X where the underlying domain is the Plotkin powerdomain of D. We show that this operation is functorial over a category of domai ..."
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We present a method for constructing from a given domain representation of a space X with underlying domain D, a domain representation of a subspace of compact subsets of X where the underlying domain is the Plotkin powerdomain of D. We show that this operation is functorial over a category of domain representations with a natural choice of morphisms. We study the topological properties of the space of representable compact sets and isolate conditions under which all compact subsets of X are representable. Special attention is paid to admissible representations and representations of metric spaces.
Cantor–Weihrauch domain representations
, 2007
"... We hope you have now downloaded a proof of your paper in pdf format, and also the Transfer of ..."
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We hope you have now downloaded a proof of your paper in pdf format, and also the Transfer of
doi:10.1093/comjnl/bxs121 Interval Domains and Computable Sequences: A Case Study of Domain Reductions
, 2012
"... 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. ..."
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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.
12345efghi UNIVERSITY OF WALES SWANSEA REPORT SERIES
"... Computability on topological spaces via domain representations by V StoltenbergHansen and J V Tucker Report # CSR 22007Computability on topological spaces via domain representations ..."
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Computability on topological spaces via domain representations by V StoltenbergHansen and J V Tucker Report # CSR 22007Computability on topological spaces via domain representations
Stability of representations of effective partial algebras
, 2011
"... Key words Numberings, recursive equivalence, computable stability, effective partial algebras, computable ..."
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Key words Numberings, recursive equivalence, computable stability, effective partial algebras, computable