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Domain Representations of Topological Spaces
, 2000
"... A domain representation of a topological space X is a function, usually a quotient map, from a subset of a domain onto X . Several different classes of domain representations are introduced and studied. It is investigated when it is possible to build domain representations from existing ones. It is, ..."
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Cited by 25 (9 self)
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A domain representation of a topological space X is a function, usually a quotient map, from a subset of a domain onto X . Several different classes of domain representations are introduced and studied. It is investigated when it is possible to build domain representations from existing ones. It is, for example, discussed whether there exists a natural way to build a domain representation of a product of topological spaces from given domain representations of the factors. It is shown that any T 0 topological space has a domain representation. These domain representations are very large. However, smaller domain representations are also constructed for large classes of spaces. For example, each second countable regular Hausdorff space has a domain representation with a countable base. Domain representations of functions and function spaces are also studied.
Density Theorems for the DomainsWithTotality Semantics of Dependent Types
 Applied Categorical Structures
, 2000
"... . We study a semantics of dependent types and universe operators based on parametrized domains with totality. The main results are generalizations of the Kleene/Kreisel density theorem for the continuous functionals. This continues work of E. Palmgren and V. Stoltenberg{Hansen on the domain interpre ..."
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Cited by 9 (0 self)
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. We study a semantics of dependent types and universe operators based on parametrized domains with totality. The main results are generalizations of the Kleene/Kreisel density theorem for the continuous functionals. This continues work of E. Palmgren and V. Stoltenberg{Hansen on the domain interpretation of dependent types, and of D. Normann on universes of wellfounded types with density. Key words: Continuous functionals, Domains, Totality, Dependent types, Universes 1. Introduction In Mathematical Logic and Computer Science there is growing interest in constructive type theories as developed by Martin{Lof [8]. This paper is concerned with a semantics of such theories within the realm of Ershov{Scott domains [5] with totality [10]. Erik Palmgren and Viggo Stoltenberg{Hansen [15], [17] developed a semantics for a partial type theory (modelling partial functions and functionals) based on the notion of a parametrization, i.e. a domain depending on parameters. Since this semantics wa...
Continuous Functionals of Dependent and Transfinite Types
, 1995
"... this paper we study some extensions of the KleeneKreisel continuous functionals [7, 8] and show that most of the constructions and results, in particular the crucial density theorem, carry over from nite to dependent and transnite types. Following an approach of Ershov we dene the continuous functi ..."
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Cited by 9 (2 self)
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this paper we study some extensions of the KleeneKreisel continuous functionals [7, 8] and show that most of the constructions and results, in particular the crucial density theorem, carry over from nite to dependent and transnite types. Following an approach of Ershov we dene the continuous functionals as the total elements in a hierarchy of ErshovScottdomains of partial continuous functionals. In this setting the density theorem says that the total functionals are topologically dense in the partial ones, i.e. every nite (compact) functional has a total extension. We will extend this theorem from function spaces to dependent products and sums and universes. The key to the proof is the introduction of a suitable notion of density and associated with it a notion of codensity for dependent domains with totality. We show that the universe obtained by closing a given family of basic domains with totality under some quantiers has a dense and codense totality provided the totalities on the basic domains are dense and codense and the quantiers preserve density and codensity. In particular we can show that the quantiers and have this preservation property and hence, for example, the closure of the integers and the booleans (which are dense and codense) under and has a dense and codense totality. We also discuss extensions of the density theorem to iterated universes, i.e. universes closed under universe operators. From our results we derive a dependent continuous choice principle and a simple ordertheoretic characterization of extensional equality for total objects. Finally we survey two further applications of density: Waagb's extension of the KreiselLacombeShoeneldTheorem showing the coincidence of the hereditarily eectively continuous hierarchy...
Density and Choice for Total Continuous Functionals
 About and Around Georg Kreisel
, 1996
"... this paper is to give complete proofs of the density theorem and the choice principle for total continuous functionals in the natural and concrete context of the partial continuous functionals [Ers77], essentially by specializing more general treatments in the literature. The proofs obtained are rel ..."
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Cited by 8 (3 self)
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this paper is to give complete proofs of the density theorem and the choice principle for total continuous functionals in the natural and concrete context of the partial continuous functionals [Ers77], essentially by specializing more general treatments in the literature. The proofs obtained are relatively short and hopefully perspicious, and may contribute to redirect attention to the fundamental questions Kreisel originally was interested in. Obviously this work owes much to other sources. In particular I have made use of work by Scott [Sco82] (whose notion of an information system is taken as a basis to introduce domains), Roscoe [Ros87], Larsen and Winskel [LW84] and Berger [Ber93]. The paper is organized as follows. Section 1 treats information systems, and in section 2 it is shown that the partial orders defined by them are exactly the (Scott) domains with countable basis. Section 3 gives a characterization of the continuous functions between domains, in terms of approximable mappings. In section 4 cartesian products and function spaces of domains and information systems are introduced. In section 5 the partial and total continuous functionals are defined. Section 6 finally contains the proofs of the two theorems above; it will be clear that the same proofs also yield effective versions of these theorems.
Full Abstraction, Totality and PCF
 Math. Structures Comput. Sci
, 1997
"... ion, Totality and PCF Gordon Plotkin Abstract Inspired by a question of Riecke, we consider the interaction of totality and full abstraction, asking whether full abstraction holds for Scott's model of cpos and continuous functions if one restricts to total programs and total observations. ..."
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Cited by 8 (1 self)
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ion, Totality and PCF Gordon Plotkin Abstract Inspired by a question of Riecke, we consider the interaction of totality and full abstraction, asking whether full abstraction holds for Scott's model of cpos and continuous functions if one restricts to total programs and total observations. The answer is negative, as there are distinct operational and denotational notions of totality. However, when two terms are each total in both senses then they are totally equivalent operationally iff they are totally equivalent in the Scott model. Analysing further, we consider sequential and parallel versions of PCF and several models: Scott's model of continuous functions, Milner's fully abstract model of PCF and their effective submodels. We investigate how totality differs between these models. Some apparently rather difficult open problems arise, essentially concerning whether the sequential and parallel versions of PCF have the same expressive power, in the sense of total equivale...
A Mahlouniverse of effective domains
, 1996
"... We construct a typed hierarchy of effective algebraic domains with totality of hight the first recursively Mahlo ordinal. The hierarchy is based on the empty type and the domains for singleton, boolean values and natural numbers, and it is closed under dependent sums and products of effectivly p ..."
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We construct a typed hierarchy of effective algebraic domains with totality of hight the first recursively Mahlo ordinal. The hierarchy is based on the empty type and the domains for singleton, boolean values and natural numbers, and it is closed under dependent sums and products of effectivly parameterised families of types, and under universes closed under any continuous operator that generates a universe.
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"... Our aim in this thesis is to study a uniform method to introduce computability on large, usually uncountable, mathematical structures. The method we choose is domain representations using ScottErshov domains. Domain theory is a theory of approximations and incorporates a natural computability theor ..."
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Our aim in this thesis is to study a uniform method to introduce computability on large, usually uncountable, mathematical structures. The method we choose is domain representations using ScottErshov domains. Domain theory is a theory of approximations and incorporates a natural computability theory. This provides us with a uniform way to introduce computability on structures that have computable domain representations, by computations on the approximations of the structure. It is shown that large classes of topological spaces have satisfactory domain representations. In particular, all metric spaces are domain representable. It is also shown that the space of compact subsets of a complete metric space can be given a domain representation uniformly from a domain representation of the metric space. Several other classes of topological spaces are shown to have domain representations, although not all of them are suitable for introducing computability.