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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 ..."
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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.
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.
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.
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.
DomainTheoretic Methods for Program Synthesis
"... formal proofs. A recent outcome of this analysis is the development of computer systems for automated or interactive theorem proving that can for instance be used for computer aided program verication. An example of such a system is the interactive theorem prover Minlog developed by the logic group ..."
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formal proofs. A recent outcome of this analysis is the development of computer systems for automated or interactive theorem proving that can for instance be used for computer aided program verication. An example of such a system is the interactive theorem prover Minlog developed by the logic group at the University of Munich (7). As a former member of this group I was mainly involved in the theoretical background steering the implementation of the system. The system also exploits the socalled proofsasprograms paradigm as a logical approach to correct software development: from a formal proof that a certain specication has a solution one fully automatically extracts a program that provably meets the specication. We carried out a number of extended case studies extracting programs from proofs in areas such as arithmetic (6), graph theory (7), innitary combinatorics (7), and lambda calculus (1,2). Special emphasis has been put on an ecient implemen
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