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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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Cited by 9 (4 self)
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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.
Spectra of algebraic fields and subfields
 in Mathematical Theory and Computational Practice: Fifth Conference on Computability in Europe, CiE 2009, eds. K. AmbosSpies, B. Löwe, & W. Merkle, Lecture Notes in Computer Science 5635
"... Abstract. An algebraic field extension of Q or Z/(p) may be regarded either as a structure in its own right, or as a subfield of its algebraic closure F (either Q or Z/(p)). We consider the Turing degree spectrum of F in both cases, as a structure and as a relation on F, and characterize the sets o ..."
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Cited by 2 (2 self)
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Abstract. An algebraic field extension of Q or Z/(p) may be regarded either as a structure in its own right, or as a subfield of its algebraic closure F (either Q or Z/(p)). We consider the Turing degree spectrum of F in both cases, as a structure and as a relation on F, and characterize the sets of Turing degrees that are realized as such spectra. The results show a connection between enumerability in the structure F and computability when F is seen as a subfield of F.
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
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
The Real Number Structure is Effectively Categorical
, 1997
"... On countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show tha ..."
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On countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show that with respect to computable equivalence there is one and only one equivalence class of representations of the real numbers which make the basic operations computable. This characterizes the real numbers in terms of the theory of effective algebras or computable structures, and is reflected by observations made in real number computer arithmetic. We also give further evidence for the wellknown nonappropriateness of the representation to some base b by proving that strictly less functions are computable with respect to these representations than with respect to a standard representation of the real numbers. Furthermore we consider basic constructions of representations and the countab...
Computability on topological spaces . . .
, 1997
"... Our aim in this thesis is to study a uniform method to introduce computability on large, usually uncountable, mathematical structures. The method we choose is domain representations using ScottErshov domains. Domain theory is a theory of approximations and incorporates a natural computability theor ..."
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Our aim in this thesis is to study a uniform method to introduce computability on large, usually uncountable, mathematical structures. The method we choose is domain representations using ScottErshov domains. Domain theory is a theory of approximations and incorporates a natural computability theory. This provides us with a uniform way to introduce computability on structures that have computable domain representations, by computations on the approximations of the structure. It is shown that large classes of topological spaces have satisfactory domain representations. In particular, all metric spaces are domain representable. It is also shown that the space of compact subsets of a complete metric space can be given a domain representation uniformly from a domain representation of the metric space. Several other classes of topological spaces are shown to have domain representations, although not all of them are suitable for introducing computability. Domain
Archive for Mathematical Logic manuscript No. (will be inserted by the editor)
"... Abstract. A strong reducibility relation between partial numberings is introduced which is such that the reduction function transfers exactly the numbers which are indices under the numbering to be reduced into corresponding indices of the other numbering. The degrees of partial numberings of a give ..."
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Abstract. A strong reducibility relation between partial numberings is introduced which is such that the reduction function transfers exactly the numbers which are indices under the numbering to be reduced into corresponding indices of the other numbering. The degrees of partial numberings of a given set with respect to this relation form an upper semilattice. In addition, Ershov’s completion construction for total numberings is extended to the partial case: every partially numbered set can be embedded in a set which results from the given set by adding one point and which is enumerated by a total and complete numbering. As is shown, the degrees of complete numberings of the extended set also form an upper semilattice. Moreover, both semilattices are isomorphic. This is not so in the case of the usual, weaker reducibility relation for partial numberings which allows the reduction function to transfer arbitrary numbers into indices. 1.
An Enquiry Concerning Categories Effective Continuous Cpos
"... Introduction Consider the following easy game. Person A chooses an integer between I and 1000. Person B asks yesorno questions to A until she knows what number he is thinking of. The goal for B is to ask as few questions as possible. Soon after playing the game for a few rounds, B realises that ..."
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Introduction Consider the following easy game. Person A chooses an integer between I and 1000. Person B asks yesorno questions to A until she knows what number he is thinking of. The goal for B is to ask as few questions as possible. Soon after playing the game for a few rounds, B realises that it is always sucient to ask A 10 questions, but that 9 questions sometimes does not give B enough information. B gets bored and tries to find an easy set of questions to ask A. She finds that the following question can be asked over and over again: Is the number in the lower half of the interval that one can deduce it must be in after the previous question? The answers B gets to her questions is an approximating sequence of the number that A has chosen. She knows with certainty that the number is in the given interval, and each question gives her a better approximation of the number. Given enough questions, the approximation will consist of only one number. The sequence converges to that nu