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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...
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...
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...
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.
Effective Metric Spaces and Representations of the Reals
 THEORETICAL COMPUTER SCIENCE, 2002 (CCA'99 SPECIAL ISSUE
, 2000
"... Based on standard notions of classical recursion theory, a natural model of approximate computability for partial functions between effective metric spaces is presented. It generalizes the KoFriedman approach to computability of real functions by means of oracle Turing machines and avoids the ex ..."
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Cited by 5 (1 self)
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Based on standard notions of classical recursion theory, a natural model of approximate computability for partial functions between effective metric spaces is presented. It generalizes the KoFriedman approach to computability of real functions by means of oracle Turing machines and avoids the explicit use of representations. The topological arithmetical hierarchy is introduced and shown to be strict if the underlying space contains an effectively discrete sequence. The domains of computable functions are just the \Pi 2 sets of that hierarchy if the space admits a finitary stratification. Finally, this framework is used to investigate and characterize the standard representations of the real numbers. They are just those functions from the name space onto the reals which have both computable extensions and inversions that are computable as relations.
Fundamentals of Computing I
 Logic, Problem Solving, Programs, & Computers
, 1992
"... on topological spaces via domain representations ..."
DomainTheoretic Methods for Program Synthesis
"... formal proofs. A recent outcome of this analysis is the development of computer systems for automated or interactive theorem proving that can for instance be used for computer aided program verication. An example of such a system is the interactive theorem prover Minlog developed by the logic group ..."
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formal proofs. A recent outcome of this analysis is the development of computer systems for automated or interactive theorem proving that can for instance be used for computer aided program verication. An example of such a system is the interactive theorem prover Minlog developed by the logic group at the University of Munich (7). As a former member of this group I was mainly involved in the theoretical background steering the implementation of the system. The system also exploits the socalled proofsasprograms paradigm as a logical approach to correct software development: from a formal proof that a certain specication has a solution one fully automatically extracts a program that provably meets the specication. We carried out a number of extended case studies extracting programs from proofs in areas such as arithmetic (6), graph theory (7), innitary combinatorics (7), and lambda calculus (1,2). Special emphasis has been put on an ecient implemen
Approximate Decidability in Euclidean Spaces
"... We study concepts of decidability (recursivity) for subsets of Euclidean spaces R k within the framework of approximate computability (type two theory of effectivity) . A new notion of approximate decidability is proposed and discussed in some detail. It is an effective variant of F. Hausdorff' ..."
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We study concepts of decidability (recursivity) for subsets of Euclidean spaces R k within the framework of approximate computability (type two theory of effectivity) . A new notion of approximate decidability is proposed and discussed in some detail. It is an effective variant of F. Hausdorff's concept of resolvable sets and modifies and generalizes notions of recursivity known from computable analysis, formerly used for open or closed sets only, to more general types of sets. Approximate decidability of a set can equivalently be expressed by computability of the characteristic function by means of an appropriately working oracle Turing machine. The notion fulfills some further natural requirements and is invariant under canonical embeddings of sets into spaces of higher dimensions. However, it is not closed under binary union or intersection of sets. We also show how the framework of resolvability and approximate decidability can be applied to investigate concepts of reducibility for subsets of Euclidean spaces.