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Computability Over the Partial Continuous Functionals
, 1998
"... We show that to every recursive total continuous functional there is a representative of in the hierearchy of partial continuous funcriohals such that is S1  S9 computable over the hierarchy of partial continuous functionals. Equivalently, the representative will be PCFdefinable over the parti ..."
Abstract

Cited by 13 (3 self)
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We show that to every recursive total continuous functional there is a representative of in the hierearchy of partial continuous funcriohals such that is S1  S9 computable over the hierarchy of partial continuous functionals. Equivalently, the representative will be PCFdefinable over the partial continuous functionals, where PCF is Plotkin's programming language for computable functionals.
Selection Functions, Bar Recursion and Backward Induction
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE, 20, PP 127168
, 2010
"... ..."
On the ubiquity of certain total type structures
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2007
"... It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel co ..."
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Cited by 4 (2 self)
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It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel continuous functionals, its effective substructure C eff, and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often nontrivial, and it is not immediately clear why these particular type structures should arise so ubiquitously. In this paper we present some new results which go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, C eff or HEO (as appropriate). We obtain versions of our results for both the “standard” and “modified” extensional collapse constructions. The proofs make essential use of a technique due to Normann. Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the above three type structures are highly canonical mathematical objects.
Exact Real Number Computations Relative to Hereditary Total Functions
 Theoretical Computer Science
, 2000
"... We show that the continuous existential quantifier is not definable in Escard6's RealPCF from all functionals equivalent to a given total one in a uniform way. ..."
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Cited by 4 (0 self)
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We show that the continuous existential quantifier is not definable in Escard6's RealPCF from all functionals equivalent to a given total one in a uniform way.
Sequential Games and Optimal Strategies
"... This article gives an overview of recent work on the theory of selection functions. We explain the intuition behind these highertype objects, and define a general notion of sequential game whose optimal strategies can be computed via a certain product of selection functions. Several instances of th ..."
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Cited by 3 (3 self)
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This article gives an overview of recent work on the theory of selection functions. We explain the intuition behind these highertype objects, and define a general notion of sequential game whose optimal strategies can be computed via a certain product of selection functions. Several instances of this game are considered in a variety of areas such as fixed point theory, topology, game theory, highertype computability, and proof theory. These examples are intended to illustrate how the fundamental construction of optimal strategies based on products of selection functions permeates several research areas.
unknown title
"... 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.
Representation Theorems for Transfinite Computability And Definability
, 1996
"... this paper we will pursue some of the ideas in Kreisel's approach and show that transfinite versions of the continuous functionals can be used to represent complex properties. We will be more precise later ..."
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this paper we will pursue some of the ideas in Kreisel's approach and show that transfinite versions of the continuous functionals can be used to represent complex properties. We will be more precise later