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39
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
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 27 (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 25 (4 self)
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this paper we show that the two approaches are equivalent for a
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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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
Infinite sets that admit fast exhaustive search
 In Proceedings of the 22nd Annual IEEE Symposium on Logic In Computer Science
, 2007
"... Abstract. Perhaps surprisingly, there are infinite sets that admit mechanical exhaustive search in finite time. We investigate three related questions: What kinds of infinite sets admit mechanical exhaustive search in finite time? How do we systematically build such sets? How fast can exhaustive sea ..."
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Abstract. Perhaps surprisingly, there are infinite sets that admit mechanical exhaustive search in finite time. We investigate three related questions: What kinds of infinite sets admit mechanical exhaustive search in finite time? How do we systematically build such sets? How fast can exhaustive search over infinite sets be performed? Keywords. Highertype computability and complexity, Kleene–Kreisel functionals, PCF, Haskell, topology. 1.
EXHAUSTIBLE SETS IN HIGHERTYPE COMPUTATION
, 2008
"... 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 C ..."
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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.
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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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...