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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.
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...
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. The ..."
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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...
Equational Theories for Inductive Types
 Annals of Pure and Applied Logic
, 1997
"... This paper provides characterisations of the equational theory of the per model of a typed lambda calculus with inductive types. The characterisation may be cast as a full abstraction result; in other words we show that the equations between terms valid in this model coincides with a certain synt ..."
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Cited by 7 (0 self)
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This paper provides characterisations of the equational theory of the per model of a typed lambda calculus with inductive types. The characterisation may be cast as a full abstraction result; in other words we show that the equations between terms valid in this model coincides with a certain syntactically defined equivalence relation. Along the way we give other characterisations of this equivalence; from below, from above, and from a domain model; a version of the KreiselLacombeShoenfield theorem allows us to transfer the result from the domain model to the per model. 0 Introduction This paper concerns a typed calculus with inductive types which correspond semantically to initial algebras of (covariant) functors; the calculus lies between Godel's T and Girard's F in prooftheoretic strength. The goal of the paper is to analyse the structure of the model of this calculus given by the category PER of partial equivalence relations over the natural numbers. We shall show that ...
Categories of Domains With Totality
 Preprint Series, Inst. Math. Univ. Oslo
, 2000
"... We investigate domains with totality where density in general does not hold. We define three categories of domains X with totality X satisfying certain structural properties. We then define the ordered set of evaluation structures. These will induce domains with totality. We show that the set of ..."
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Cited by 6 (0 self)
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We investigate domains with totality where density in general does not hold. We define three categories of domains X with totality X satisfying certain structural properties. We then define the ordered set of evaluation structures. These will induce domains with totality. We show that the set of evaluation structures in a natural way is closed under dependent sums and products and under direct limits.
Limit Spaces and Transfinite Types
, 1998
"... We give a characterisation of an extension of the KleeneKreisel continuous functionals to objects of transfinite types using limit spaces of transfinite types. ..."
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Cited by 2 (0 self)
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We give a characterisation of an extension of the KleeneKreisel continuous functionals to objects of transfinite types using limit spaces of transfinite types.