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12
Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability dist ..."
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Cited by 48 (10 self)
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We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...
The Realizability Approach to Computable Analysis and Topology
, 2000
"... policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government. ..."
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Cited by 41 (19 self)
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policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government.
Computable Banach Spaces via Domain Theory
 Theoretical Computer Science
, 1998
"... This paper extends the ordertheoretic approach to computable analysis via continuous domains to complete metric spaces and Banach spaces. We employ the domain of formal balls to define a computability theory for complete metric spaces. For Banach spaces, the domain specialises to the domain of clos ..."
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Cited by 15 (2 self)
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This paper extends the ordertheoretic approach to computable analysis via continuous domains to complete metric spaces and Banach spaces. We employ the domain of formal balls to define a computability theory for complete metric spaces. For Banach spaces, the domain specialises to the domain of closed balls, ordered by reversed inclusion. We characterise computable linear operators as those which map computable sequences to computable sequences and are effectively bounded. We show that the domaintheoretic computability theory is equivalent to the wellestablished approach by PourEl and Richards. 1 Introduction This paper is part of a programme to introduce the theory of continuous domains as a new approach to computable analysis. Initiated by the various applications of continuous domain theory to modelling classical mathematical spaces and performing computations as outlined in the recent survey paper by Edalat [6], the authors started this work with [9] which was concerned with co...
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 8 (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.
Admissible Domain Representations of Topological Spaces
 Department of Mathematics, Uppsala University
, 2005
"... In this paper we consider admissible domain representations of topological spaces. A domain representation D of a space X is λadmissible if, in principle, all other λbased domain representations E of X can be reduced to D via a continuous function from E to D. We present a characterisation theorem ..."
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Cited by 6 (1 self)
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In this paper we consider admissible domain representations of topological spaces. A domain representation D of a space X is λadmissible if, in principle, all other λbased domain representations E of X can be reduced to D via a continuous function from E to D. We present a characterisation theorem of when a topological space has a λadmissible and κbased domain representation. We also prove that there is a natural cartesian closed category of countably based and countably admissible domain representations. These results are generalisations of [Sch02]. 1
RZ: A tool for bringing constructive and computable mathematics closer to programming practice
 CiE 2007: Computation and Logic in the Real World, volume 4497 of LNCS
, 2007
"... Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Obje ..."
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Cited by 5 (2 self)
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Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools. 1
Fundamentals of Computing I
 Logic, Problem Solving, Programs, & Computers
, 1992
"... on topological spaces via domain representations ..."
Motivation and Background
"... P, also called the enumeration operators, are characterized by the equation F (x) = [ \Phi F (y) fi fi y x \Psi 1 for all x 2 P, where y x means that y is a finite subset of x. It follows that a continuous function is completely determined by the relation n 2 F (y) for n 2 N and y N: (1 ..."
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P, also called the enumeration operators, are characterized by the equation F (x) = [ \Phi F (y) fi fi y x \Psi 1 for all x 2 P, where y x means that y is a finite subset of x. It follows that a continuous function is completely determined by the relation n 2 F (y) for n 2 N and y N: (1) We say a continuous function F : P ! P is computable when the corresponding relation (1) is recursively enumerable. Relation (1) can also be identified with sets in P using suitable encodings. Whenever a topological space X is represented as a subspace of<F8.35