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68
A Combinatory Algebra for Sequential Functionals of Finite Type
 University of Utrecht
, 1997
"... It is shown that the type structure of finitetype functionals associated to a combinatory algebra of partial functions from IN to IN (in the same way as the type structure of the countable functionals is associated to the partial combinatory algebra of total functions from IN to IN), is isomorphic ..."
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It is shown that the type structure of finitetype functionals associated to a combinatory algebra of partial functions from IN to IN (in the same way as the type structure of the countable functionals is associated to the partial combinatory algebra of total functions from IN to IN), is isomorphic to the type structure generated by object N (the flat domain on the natural numbers) in Ehrhard's category of "dIdomains with coherence", or his "hypercoherences". AMS Subject Classification: Primary 03D65, 68Q55 Secondary 03B40, 03B70, 03D45, 06B35 Introduction PCF , "Godel's T with unlimited recursion", was defined in Plotkin's paper [16]. It is a simply typed calculus with a type o for integers and constants for basic arithmetical operations, definition by cases and fixed point recursion. More importantly, there is a special reduction relation attached to it which ensures (by Plotkin's "Activity Lemma") that all PCF definable highertype functionals have a sequential, i.e. nonparal...
Computational foundations of basic recursive function theory
 In Third IEEE Symposium on Logic in Computer Science
, 1988
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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.
Elimination of Skolem functions for monotone formulas in analysis
 ARCHIVE FOR MATHEMATICAL LOGIC
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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.
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
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A mathematical modeling of pure, recursive algorithms
 Logic at Botik ’89
, 1989
"... This paper follows previous work on the Formal Language of Recursion FLR and develops intensional (algorithmic) semantics for it: the intension of a term t on a structure A is a recursor, a set–theoretic object which represents the (abstract, recursive) algorithm defined by t on A. Main results are ..."
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This paper follows previous work on the Formal Language of Recursion FLR and develops intensional (algorithmic) semantics for it: the intension of a term t on a structure A is a recursor, a set–theoretic object which represents the (abstract, recursive) algorithm defined by t on A. Main results are the soundness of the reduction calculus of FLR (which models faithful, algorithm–preserving compilation) for this semantics, and the robustness of the class of algorithms assigned to a structure under algorithm adjunction. This is the second in a sequence of papers begun with [16] in which we develop a foundation for the theory of computation based on a mathematical modeling of recursive algorithms. The general features, aims and methodological assumptions of this program were discussed and illustrated by examples in the preliminary, expository report [15]. In [16] we studied the formal language of recursion FLR which is the main technical tool for this work, we developed several alternative denotational semantics for it and we established a key unique termination theorem for a reduction calculus which models faithful (algorithm–preserving) compilation. Here we will define the intensional semantics of FLR for structures with given (pure) recursors, the set–theoretic objects we use to model pure (side–effect–free) algorithms: the intension of a term t on each structure A is a recursor which models the algorithm expressed by t on A. Basic results of the paper During the preparation of this paper the author was partially supported by an NSF
On sequential functionals of type 3
 Math. Structures Comput. Sci
, 2006
"... We show that the extensional ordering of the sequential functionals of pure type 3, e.g. as defined via game semantics [2, 4], is not cpoenriched. This shows that this model does not equal Milner’s [9] fully abstract model for P CF. 1 ..."
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We show that the extensional ordering of the sequential functionals of pure type 3, e.g. as defined via game semantics [2, 4], is not cpoenriched. This shows that this model does not equal Milner’s [9] fully abstract model for P CF. 1
NONPRINCIPAL ULTRAFILTERS, PROGRAM EXTRACTION AND HIGHER ORDER REVERSE MATHEMATICS
"... Abstract. We investigate the strength of the existence of a nonprincipal ultrafilter over fragments of higher order arithmetic. Let (U) be the statement that a nonprincipal ultrafilter on N exists and let ACAω 0 be the higher order extension of ACA0. We show that ACAω 0 + (U) is Π1 2conservative ..."
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Abstract. We investigate the strength of the existence of a nonprincipal ultrafilter over fragments of higher order arithmetic. Let (U) be the statement that a nonprincipal ultrafilter on N exists and let ACAω 0 be the higher order extension of ACA0. We show that ACAω 0 + (U) is Π1 2conservative over ACAω 0 and thus that ACAω 0 + (U) is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly Π1 2 statement ∀f ∃g Aqf(f, g) in ACAω 0 + (U) a realizing term in Gödel’s system T can be extracted. This means that one can extract a term t ∈ T, such that ∀f Aqf(f, t(f)). In this paper we will investigate the strength of the existence of a nonprincipal ultrafilter over fragments of higher order arithmetic. We will classify the consequences of this statement in the spirit of reverse mathematics. Furthermore, we will provide a program extraction method. Let (U) be the statement that a nonprincipal ultrafilter on N exists. Let RCA ω 0, ACA ω 0 be the extensions of RCA0 resp. ACA0 to higher order arithmetic as introduced