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24
Exhaustible sets in highertype computation
 Logical Methods in Computer Science
"... Abstract. We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no examp ..."
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Cited by 13 (12 self)
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Abstract. We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela–Ascoli type characterization of compact subsets of function spaces. We also show that, in the nonempty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications. 1.
Effective Domain Representations of H(X), the space of compact subsets
, 1999
"... This paper gives effective domain representations of spaces H(X) of nonempty compact subsets of effective complete metric spaces X. The domain representation of H(X) is constructed from a domain representation of X using the Plotkin power domain construction. As an application of the representation ..."
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Cited by 13 (5 self)
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This paper gives effective domain representations of spaces H(X) of nonempty compact subsets of effective complete metric spaces X. The domain representation of H(X) is constructed from a domain representation of X using the Plotkin power domain construction. As an application of the representation an effective version of a fundamental theorem on IFS (iterated function system) is shown.
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 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.
Partial Continuous Functions and Admissible Domain Representations
 the Journal of Logic and Computation
, 2007
"... It is well known that to be able to represent continuous functions between domain representable spaces it is critical that the domain representations of the spaces we consider are dense. In this article we show how to develop a representation theory over a category of domains with morphisms partial ..."
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Cited by 7 (2 self)
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It is well known that to be able to represent continuous functions between domain representable spaces it is critical that the domain representations of the spaces we consider are dense. In this article we show how to develop a representation theory over a category of domains with morphisms partial continuous functions. The raison d’être for introducing partial continuous functions is that by passing to partial maps, we are free to consider totalities which are not dense. We show that the category of admissibly representable spaces with morphisms functions which are representable by a partial continuous function is Cartesian closed. Finally, we consider the question of effectivity. Key words. Domain theory, domain representations, computability theory, computable analysis. 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
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.
Streams, Stream Transformers and Domain Representations
 Prospects for Hardware Foundations, Lecture Notes in Computer Science
, 1998
"... We present a general theory for the computation of stream transformers of the form F: (R B) (T A), where time T and R, and data A and B, are discrete or continuous. We show how methods for representing topological algebras by algebraic domains can be applied to transformations of continuous ..."
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Cited by 3 (3 self)
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We present a general theory for the computation of stream transformers of the form F: (R B) (T A), where time T and R, and data A and B, are discrete or continuous. We show how methods for representing topological algebras by algebraic domains can be applied to transformations of continuous streams. A stream transformer is continuous in the compactopen topology on continuous streams if and only if it has a continuous lifting to a standard algebraic domain representation of such streams. We also examine the important problem of representing discontinuous streams, such as signals T A, where time T is continuous and data A is discrete.
Fundamentals of Computing I
 Logic, Problem Solving, Programs, & Computers
, 1992
"... on topological spaces via domain representations ..."