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32
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...
Computing with Real Numbers  I. The LFT Approach to Real Number Computation  II. A Domain Framework for Computational Geometry
 PROC APPSEM SUMMER SCHOOL IN PORTUGAL
, 2002
"... We introduce, in Part I, a number representation suitable for exact real number computation, consisting of an exponent and a mantissa, which is an in nite stream of signed digits, based on the interval [ 1; 1]. Numerical operations are implemented in terms of linear fractional transformations ( ..."
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Cited by 16 (1 self)
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We introduce, in Part I, a number representation suitable for exact real number computation, consisting of an exponent and a mantissa, which is an in nite stream of signed digits, based on the interval [ 1; 1]. Numerical operations are implemented in terms of linear fractional transformations (LFT's). We derive lower and upper bounds for the number of argument digits that are needed to obtain a desired number of result digits of a computation, which imply that the complexity of LFT application is that of multiplying nbit integers. In Part II, we present an accessible account of a domaintheoretic approach to computational geometry and solid modelling which provides a datatype for designing robust geometric algorithms, illustrated here by the convex hull algorithm.
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.
Quantum Domain Theory  Definitions and Applications
 Proceedings of CCA’03
, 2003
"... Domain theory is a branch of classical computer science. It has proven to be a rigourous mathematical structure to describe denotational semantics for programming languages and to study the computability of partial functions. In this paper, we study the extension of domain theory to the quantum sett ..."
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Cited by 7 (0 self)
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Domain theory is a branch of classical computer science. It has proven to be a rigourous mathematical structure to describe denotational semantics for programming languages and to study the computability of partial functions. In this paper, we study the extension of domain theory to the quantum setting. By defining a quantum domain we introduce a rigourous definition of quantum computability for quantum states and operators. Furthermore we show that the denotational semantics of quantum computation has the same structure as the denotational semantics of classical probabilistic computation introduced by Kozen [23]. Finally, we briefly review a recent result on the application of quantum domain theory to quantum information processing. 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
Induction and recursion on the partial real line with applications to Real PCF
 Theoretical Computer Science
, 1997
"... The partial real line is an extension of the Euclidean real line with partial real numbers, which has been used to model exact real number computation in the programming language Real PCF. We introduce induction principles and recursion schemes for the partial unit interval, which allow us to verify ..."
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Cited by 6 (1 self)
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The partial real line is an extension of the Euclidean real line with partial real numbers, which has been used to model exact real number computation in the programming language Real PCF. We introduce induction principles and recursion schemes for the partial unit interval, which allow us to verify that Real PCF programs meet their specification. They resemble the socalled Peano axioms for natural numbers. The theory is based on a domainequationlike presentation of the partial unit interval. The principles are applied to show that Real PCF is universal in the sense that all computable elements of its universe of discourse are definable. These elements include higherorder functions such as integration operators. Keywords: Induction, coinduction, exact real number computation, domain theory, Real PCF, universality. Introduction The partial real line is the domain of compact real intervals ordered by reverse inclusion [28,21]. The idea is that singleton intervals represent total rea...
Characteristic Properties of MajorantComputability over the Reals
 Proc. of CSL'98, LNCS, 1584
"... . Characteristic properties of majorantcomputable realvalued functions are studied. A formal theory of computability over the reals which satisfies the requirements of numerical analysis used in Computer Science is constructed on the base of the definition of majorantcomputability proposed in [13 ..."
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Cited by 5 (5 self)
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. Characteristic properties of majorantcomputable realvalued functions are studied. A formal theory of computability over the reals which satisfies the requirements of numerical analysis used in Computer Science is constructed on the base of the definition of majorantcomputability proposed in [13]. A modeltheoretical characterization of majorantcomputability realvalued functions and their domains is investigated. A theorem which connects the graph of a majorantcomputable function with validity of a finite formula on the set of hereditarily finite sets on IR, HF( IR) (where IR is a proper elementary enlargement of the standard reals) is proven. A comparative analysis of the definition of majorantcomputability and the notions of computability earlier proposed by Blum et al., Edalat, Sunderhauf, PourEl and Richards, StoltenbergHansen and Tucker is given. Examples of majorantcomputable realvalued functions are presented. 1 Introduction In the recent time, attention to the prob...
The Computational Dimension of a Topological Space
 Computability and Complexity in Analysis, volume 2064 of Lecture Notes in Computer Science
, 2000
"... When a topological space X can be embedded into the space ! ?;n of n?sequences of , then we can define the corresponding computational notion over X because a machine with n + 1 heads on each tape can input/output sequences in ! ?;n . This means that the least number n such that X can be topologica ..."
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Cited by 4 (3 self)
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When a topological space X can be embedded into the space ! ?;n of n?sequences of , then we can define the corresponding computational notion over X because a machine with n + 1 heads on each tape can input/output sequences in ! ?;n . This means that the least number n such that X can be topologically embedded into ! ?;n serves as a degree of complexity of the space. We show that this number, which we call the computational dimension of the space, is equal to the topological dimension for separable metric spaces. As a corollary, the 2dimensional Euclidean space IR 2 can be embedded in f0; 1g ! ?;2 but not in ! ?;1 for any character set , and infinite dimensional spaces like the set of closed/open/compact subsets of IR n and the set of continuous functions from IR n to IR m can be embedded in ! ? but not in ! ?;n for any n.
Compact Metric Spaces as MinimalLimit Sets in Domains of Bottomed Sequences
, 2003
"... It is shown that every compact metric space X is homeomorphically embedded in an !algebraic domain D as the set of minimal limit elements. ..."
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Cited by 4 (3 self)
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It is shown that every compact metric space X is homeomorphically embedded in an !algebraic domain D as the set of minimal limit elements.