Results 1  10
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34
A DomainTheoretic Approach to Computability on the Real Line
, 1997
"... In recent years, there has been a considerable amount of work on using continuous domains in real analysis. Most notably are the development of the generalized Riemann integral with applications in fractal geometry, several extensions of the programming language PCF with a real number data type, and ..."
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Cited by 43 (8 self)
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In recent years, there has been a considerable amount of work on using continuous domains in real analysis. Most notably are the development of the generalized Riemann integral with applications in fractal geometry, several extensions of the programming language PCF with a real number data type, and a framework and an implementation of a package for exact real number arithmetic. Based on recursion theory we present here a precise and direct formulation of effective representation of real numbers by continuous domains, which is equivalent to the representation of real numbers by algebraic domains as in the work of StoltenbergHansen and Tucker. We use basic ingredients of an effective theory of continuous domains to spell out notions of computability for the reals and for functions on the real line. We prove directly that our approach is equivalent to the established Turingmachine based approach which dates back to Grzegorczyk and Lacombe, is used by PourEl & Richards in their found...
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.
Foundation of a Computable Solid Modelling
 Theoretical Computer Science
, 2002
"... Solid modelling and computational geometry are based on classical topology and geometry in which the basic predicates and operations, such as membership, subset inclusion, union and intersection, are not continuous and therefore not computable. But a sound computational framework for solids and g ..."
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Cited by 33 (13 self)
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Solid modelling and computational geometry are based on classical topology and geometry in which the basic predicates and operations, such as membership, subset inclusion, union and intersection, are not continuous and therefore not computable. But a sound computational framework for solids and geometry can only be built in a framework with computable predicates and operations. In practice, correctness of algorithms in computational geometry is usually proved using the unrealistic Real RAM machine model of computation, which allows comparison of real numbers, with the undesirable result that correct algorithms, when implemented, turn into unreliable programs. Here, we use a domaintheoretic approach to recursive analysis to develop the basis of an eective and realistic framework for solid modelling. This framework is equipped with a welldened and realistic notion of computability which reects the observable properties of real solids. The basic predicates and operations o...
An Extension Result for Continuous Valuations
, 1998
"... We show, by a simple and direct proof, that if a bounded valuation on a directed complete partial order (dcpo) is the supremum of a directed family of simple valuations then it has a unique extension to a measure on the Borel oealgebra of the dcpo with the Scott topology. It follows that every boun ..."
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Cited by 32 (4 self)
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We show, by a simple and direct proof, that if a bounded valuation on a directed complete partial order (dcpo) is the supremum of a directed family of simple valuations then it has a unique extension to a measure on the Borel oealgebra of the dcpo with the Scott topology. It follows that every bounded and continuous valuation on a continuous domain can be extended uniquely to a Borel measure. The result also holds for oefinite valuations, but fails for dcpo's in general. 1
Probabilistic Game Semantics
 Computer Science Society
, 2000
"... A category of HO/Nstyle games and probabilistic strategies is developedwhere the possible choices of a strategy are quantified so as to give a measure of the likelihood of seeing a given play. A 2sided die is shown to be universal in this category, in the sense that any strategy breaks down into a ..."
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Cited by 31 (1 self)
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A category of HO/Nstyle games and probabilistic strategies is developedwhere the possible choices of a strategy are quantified so as to give a measure of the likelihood of seeing a given play. A 2sided die is shown to be universal in this category, in the sense that any strategy breaks down into a composition between some deterministic strategy and that die. The interpretative power of the category is then demonstrated by delineating a Cartesian closed subcategory which provides a fully abstract model of a probabilistic extension of Idealized Algol.
Abstract versus concrete computation on metric partial algebras
 ACM Transactions on Computational Logic
, 2004
"... Data types containing infinite data, such as the real numbers, functions, bit streams and waveforms, are modelled by topological manysorted algebras. In the theory of computation on topological algebras there is a considerable gap between socalled abstract and concrete models of computation. We pr ..."
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Cited by 30 (19 self)
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Data types containing infinite data, such as the real numbers, functions, bit streams and waveforms, are modelled by topological manysorted algebras. In the theory of computation on topological algebras there is a considerable gap between socalled abstract and concrete models of computation. We prove theorems that bridge the gap in the case of metric algebras with partial operations. With an abstract model of computation on an algebra, the computations are invariant under isomorphisms and do not depend on any representation of the algebra. Examples of such models are the ‘while ’ programming language and the BCSS model. With a concrete model of computation, the computations depend on the choice of a representation of the algebra and are not invariant under isomorphisms. Usually, the representations are made from the set N of natural numbers, and computability is reduced to classical computability on N. Examples of such models are computability via effective metric spaces, effective domain representations, and type two enumerability. The theory of abstract models is stable: there are many models of computation, and
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...
The Convex Hull in a New Model of Computation
 In Proc. 13th Canad. Conf. Comput. Geom
, 2001
"... We present a new model of geometric computation which supports the design of robust algorithms for exact real number input as well as for input with uncertainty, i.e. partial input. In this framework, we show that the convex hull of N computable real points in R^d is indeed computable. We provide a ..."
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Cited by 12 (5 self)
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We present a new model of geometric computation which supports the design of robust algorithms for exact real number input as well as for input with uncertainty, i.e. partial input. In this framework, we show that the convex hull of N computable real points in R^d is indeed computable. We provide a robust algorithm which, given any set of N partial inputs, i.e. N dyadic or rational rectangles, approximating these points, computes the partial convex hull in time O(N log N) in 2d and 3d. As the rectangles are refined to the N points, the sequence of partial convex hulls converges effectively both in the Hausdorff metric and the Lebesgue measure to the convex hull of the N points.
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.