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49
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 59 (13 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 52 (20 self)
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policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government.
Properly injective spaces and function spaces
 TO APPEAR IN TOPOLOGY AND ITS APPLICATIONS
, 1997
"... Given an injective space D (a continuous lattice endowed with the Scott topology) and a subspace embedding j: X → Y, Dana Scott asked whether the higherorder function [X → D] → [Y → D] which takes a continuous map f: X → D to its greatest continuous extension ¯ f: Y → D along j is Scott continuous ..."
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Cited by 36 (12 self)
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Given an injective space D (a continuous lattice endowed with the Scott topology) and a subspace embedding j: X → Y, Dana Scott asked whether the higherorder function [X → D] → [Y → D] which takes a continuous map f: X → D to its greatest continuous extension ¯ f: Y → D along j is Scott continuous. In this case the extension map is a subspace embedding. We show that the extension map is Scott continuous iff D is the trivial onepoint space or j is a proper map in the sense of Hofmann and Lawson. In order to avoid the ambiguous expression “proper subspace embedding”, we refer to proper maps as finitary maps. We show that the finitary sober subspaces of the injective spaces are exactly the stably locally compact spaces. Moreover, the injective spaces over finitary embeddings are the algebras of the upper power space monad on the category of sober spaces. These coincide with the retracts of upper power spaces of sober spaces. In the full subcategory of locally compact sober spaces, these are known to be the continuous meetsemilattices. In the full subcategory of stably locally compact spaces these are again the continuous lattices. The above characterization of the injective spaces over finitary embeddings is an instance of a general result on injective objects in posetenriched categories with the structure of a KZmonad established in this paper, which we also apply to various full subcategories closed under the upper power space construction and to the upper and lower power locale monads. The above results also hold for the injective spaces over dense subspace embeddings (continuous Scott domains). Moreover, we show that every sober space has a smallest finitary dense sober subspace (its support). The support always contains the subspace of maximal points, and in the stably locally compact case (which includes densely injective spaces) it is the subspace of maximal points iff that subspace is compact.
The Measurement Process in Domain Theory
 Proceedings of the 27 th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science
, 2000
"... We introduce the measurement idea in domain theory and then apply it to establish two fixed point theorems. The first is an extension of the Scott fixed point theorem which applies to nonmonotonic mappings. The second is a contraction principle for monotone maps that guarantees the existence of uniq ..."
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Cited by 30 (20 self)
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We introduce the measurement idea in domain theory and then apply it to establish two fixed point theorems. The first is an extension of the Scott fixed point theorem which applies to nonmonotonic mappings. The second is a contraction principle for monotone maps that guarantees the existence of unique fixed points. 1
An Application of MartinLöf Randomness to Effective Probability Theory
 COMPUTABILITY IN EUROPE (CIE 2009), HEIDELBERG: GERMANY
, 2009
"... In this paper we provide a framework for computable analysis of measure, probability and integration theories. We work on computable metric spaces with computable Borel probability measures. We introduce and study the framework of layerwise computability which lies on MartinLöf randomness and the ..."
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Cited by 26 (8 self)
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In this paper we provide a framework for computable analysis of measure, probability and integration theories. We work on computable metric spaces with computable Borel probability measures. We introduce and study the framework of layerwise computability which lies on MartinLöf randomness and the existence of a universal randomness test. We then prove characterizations of effective notions of measurability and integrability in terms of layerwise computability. On the one hand it gives a simple way of handling effective measure theory, on the other hand it provides powerful tools to study MartinLöf randomness, as illustrated in a sequel paper.
PCF extended with real numbers: a domaintheoretic approach to higherorder exact real number computation
, 1996
"... ..."
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 12 (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...
Domain Representations of Partial Functions, with Applications to Spatial Objects and Constructive Volume Geometry
, 2000
"... A partial spatial object is a partial map from space to data. Data types of partial spatial objects are modelled by topological algebras of partial maps and are the foundation for a high level approach to volume graphics called constructive volume geometry (CVG), where space and data are subspaces o ..."
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Cited by 11 (4 self)
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A partial spatial object is a partial map from space to data. Data types of partial spatial objects are modelled by topological algebras of partial maps and are the foundation for a high level approach to volume graphics called constructive volume geometry (CVG), where space and data are subspaces of # dimensional Euclidean space. We investigate the computability of partial spatial object data types, in general and in volume graphics, using the theory of effective domain representations for topological algebras. The basic mathematical problem considered is to classify which partial functions between topological spaces can be represented by total continuous functions between given domain representations of the spaces. We prove theorems about partial functions on regular Hausdorff spaces and their domain representations, and apply the results to partial spatial objects and CVG algebras.
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.
Integration in real PCF
 Information and Computation
, 1996
"... Real PCF is an extension of the programming language PCF with a data type for real numbers. Although a Real PCF definable real number cannot be computed in finitely many steps, it is possible to compute an arbitrarily small rational interval containing the real number in a sufficiently large number ..."
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Cited by 8 (4 self)
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Real PCF is an extension of the programming language PCF with a data type for real numbers. Although a Real PCF definable real number cannot be computed in finitely many steps, it is possible to compute an arbitrarily small rational interval containing the real number in a sufficiently large number of steps. Based on a domaintheoretic approach to integration, we show how to define integration in Real PCF. We propose two approaches to integration in Real PCF. One consists in adding integration as primitive. The other consists in adding a primitive for function maximization and then recursively defining integration from maximization. In both cases we have a computational adequacy theorem for the corresponding extension of Real PCF. Moreover, based on previous work on Real PCF definability, we show that Real PCF extended with the maximization operator is universal. 1