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20
The Realizability Approach to Computable Analysis and Topology
, 2000
"... policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government. ..."
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Cited by 41 (19 self)
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policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government.
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 26 (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.
Topological and Limitspace subcategories of Countablybased Equilogical Spaces
, 2001
"... this paper we show that the two approaches are equivalent for a ..."
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Cited by 22 (4 self)
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this paper we show that the two approaches are equivalent for a
A Relationship between Equilogical Spaces and Type Two Effectivity
"... In this paper I compare two well studied approaches to topological semantics the domaintheoretic approach, exemplied by the category of countably based equilogical spaces, Equ, and Type Two Eectivity, exemplied by the category of Baire space representations, Rep(B ). These two categories are both ..."
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Cited by 16 (0 self)
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In this paper I compare two well studied approaches to topological semantics the domaintheoretic approach, exemplied by the category of countably based equilogical spaces, Equ, and Type Two Eectivity, exemplied by the category of Baire space representations, Rep(B ). These two categories are both locally cartesian closed extensions of countably based T 0 spaces. A natural question to ask is how they are related.
Comparing functional paradigms for exact realnumber computation
 in Proceedings ICALP 2002, Springer LNCS 2380
, 2002
"... Abstract. We compare the definability of total functionals over the reals in two functionalprogramming approaches to exact realnumber datatype of real numbers; and the intensional approach, in which one encodes real numbers using ordinary datatypes. We show that the type hierarchies coincide up to ..."
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Cited by 15 (3 self)
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Abstract. We compare the definability of total functionals over the reals in two functionalprogramming approaches to exact realnumber datatype of real numbers; and the intensional approach, in which one encodes real numbers using ordinary datatypes. We show that the type hierarchies coincide up to secondorder types, and we relate this fact to an analogous comparison of type hierarchies over the external and internal real numbers in Dana Scott’s category of equilogical spaces. We do not know whether similar coincidences hold at thirdorder types. However, we relate this question to a purely topological conjecture about the KleeneKreisel continuous functionals over the natural numbers. Finally, although it is known that, in the extensional approach, parallel primitives are necessary for programming total firstorder functions, we demonstrate that, in the intensional approach, such primitives are not needed for secondorder types and below. 1
Continuous Functionals of Dependent Types and Equilogical Spaces
, 2000
"... . We show that dependent sums and dependent products of continuous parametrizations on domains with dense, codense, and natural totalities agree with dependent sums and dependent products in equilogical spaces, and thus also in the realizability topos RT(P!). Keywords: continuous functionals, depen ..."
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Cited by 12 (8 self)
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. We show that dependent sums and dependent products of continuous parametrizations on domains with dense, codense, and natural totalities agree with dependent sums and dependent products in equilogical spaces, and thus also in the realizability topos RT(P!). Keywords: continuous functionals, dependent type theory, domain theory, equilogical spaces. 1 Introduction Recently there has been a lot of interest in understanding notions of totality for domains [3, 23, 4, 18, 21]. There are several reasons for this. Totality is the semantic analogue of termination, and one is naturally interested in understanding not only termination properties of programs but also how notions of program equivalence depend on assumptions regarding termination [21]. Another reason for studying totality on domains is to obtain generalizations of the nitetype hierarchy of total continuous functionals by Kleene and Kreisel [11], see [8] and [19] for good accounts of this subject. Ershov [7] showed how the Klee...
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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Cited by 12 (5 self)
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
The Meaning of Types  From Intrinsic to Extrinsic Semantics
 Department of Computer Science, University of Aarhus
, 2000
"... A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic " if all phrases have meanings that are independent of their typings, while typing ..."
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Cited by 10 (1 self)
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A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic " if all phrases have meanings that are independent of their typings, while typings represent properties of these meanings. For a simply typed lambda calculus, extended with recursion, subtypes, and named products, we give an intrinsic denotational semantics and a denotational semantics of the underlying untyped language. We then establish a logical relations theorem between these two semantics, and show that the logical relations can be "bracketed" by retractions between the domains of the two semantics. From these results, we derive an extrinsic semantics that uses partial equivalence relations.
What do Types Mean?  From Intrinsic to Extrinsic Semantics
, 2001
"... A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic" if all phrases have meanings that are independent of their typings, while typ ..."
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Cited by 7 (0 self)
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A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic" if all phrases have meanings that are independent of their typings, while typings represent properties of these meanings.