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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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policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government.
Equilogical Spaces
, 1998
"... It is well known that one can build models of full higherorder dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relation ..."
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Cited by 33 (12 self)
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It is well known that one can build models of full higherorder dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures as well. In particular, we can easily dene the category of ERs and equivalencepreserving continuous mappings over the standard category Top 0 of topological T 0 spaces; we call these spaces (a topological space together with an ER) equilogical spaces and the resulting category Equ. We show that this categoryin contradistinction to Top 0 is a cartesian closed category. The direct proof outlined here uses the equivalence of the category Equ to the category PEqu of PERs over algebraic lattices (a full subcategory of Top 0 that is well known to be cartesian closed from domain theory). In another paper with Carboni and Rosolini (cited herein) a more abstract categorical generalization shows why many such categories are cartesian closed. The category Equ obviously contains Top 0 as a full subcategory, and it naturally contains many other well known subcategories. In particular, we show why, as a consequence of work of Ershov, Berger, and others, the KleeneKreisel hierarchy of countable functionals of nite types can be naturally constructed in Equ from the natural numbers object N by repeated use in Equ of exponentiation and binary products. We also develop for Equ notions of modest sets (a category equivalent to Equ) and assemblies to explain why a model of dependent type theory is obtained. We make some comparisons of this model to other, known models. 1
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 17 (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.
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.
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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. 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...
Mathematical Logic Quarterly
"... Abstract. In this paper I compare two well studied approaches to topological semanticsâ€” ..."
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Abstract. In this paper I compare two well studied approaches to topological semanticsâ€”
Topology and Computability
, 1998
"... In this thesis I will relate standard computability to computability of realsand related spacesand link these notions with denotational semantics of computer languages. The framework for the proposed study was worked out in collaboration with Steve Awodey, Lars Birkedal and Dana Scott. The first ..."
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In this thesis I will relate standard computability to computability of realsand related spacesand link these notions with denotational semantics of computer languages. The framework for the proposed study was worked out in collaboration with Steve Awodey, Lars Birkedal and Dana Scott. The first section gives a motivation for our setup and indicates how standard computability fits in it. The subsequent sections discuss applications of the ideas to examples in analysis, topology, and domain theory. They are meant to demonstrate the nature of the work that I intend to accomplish. 1 Motivation and Background We take as a canonical example of a space the powerset of natural numbers P = PN ordered by inclusion `. It is an algebraic lattice and a countably based T0space under the socalled Scott topology, i.e., the weak topology rather than the compact Hausdorff topology on 2N. The family RE of recursively enumerable sets of natural numbers is a subspace of P, and an element x 2 P is computable when x 2 RE. We next want to extend this well known idea of computability of elements to computability of functions. 1 The continuous functions F: P! P, also called the enumeration operators, are characterized by the equation F (x) = [ \Phi F (y) fifi y o / x\Psi for all x 2 P, where y o / x means that y is a finite subset of x. It follows that a continuous function is completely determined by the relation n 2 F (y) for n 2 N and y o / N: (1) We say a continuous function F: P! Pis computable when the corresponding relation (1) is recursively enumerable. Relation (1) can also be identified with sets in P using suitable encodings. Whenever a topological space X is represented as a subspace of P, we consider its computable elements to be the elements of X &quot; RE. Before we can assess whether this is useful, the first question is, which topological spaces are subspaces of P? Theorem 1.1 (Embedding Theorem) A topological space is homeomorphic to a subspace of P if, and only if, it is a countably based T0space.
Abstract Equilogical Spaces
"... It is well known that one can build models of full higherorder dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relation ..."
Abstract
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It is well known that one can build models of full higherorder dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures