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42
Computational Foundations of Basic Recursive Function Theory
 Theoretical Computer Science
, 1988
"... The theory of computability, or basic recursive function theory as it is often called, is usually motivated and developed using Church's Thesis. Here we show that there is an alternative computability theory in which some of the basic results on unsolvability become more absolute, results on complet ..."
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Cited by 20 (7 self)
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The theory of computability, or basic recursive function theory as it is often called, is usually motivated and developed using Church's Thesis. Here we show that there is an alternative computability theory in which some of the basic results on unsolvability become more absolute, results on completeness become simpler, and many of the central concepts become more abstract. In this approach computations are viewed as mathematical objects, and the major theorems in recursion theory may be classified according to which axioms about computation are needed to prove them. The theory is a typed theory of functions over the natural numbers, and there are unsolvable problems in this setting independent of the existence of indexings. The unsolvability results are interpreted to show that the partial function concept, so important in computer science, serves to distinguish between classical and constructive type theories (in a different way than does the decidability concept as expressed in the ...
On the NoCounterexample Interpretation
 J. SYMBOLIC LOGIC
, 1997
"... In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive functi ..."
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Cited by 18 (10 self)
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In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive functionals \Phi A of order type ! " 0 which realize the Herbrand normal form A of A. Subsequently more
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.
Elimination of Skolem functions for monotone formulas in analysis
 ARCHIVE FOR MATHEMATICAL LOGIC
"... ..."
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.
1996], Computation on abstract data types. The extensional approach, with an application to streams
 Annals of Pure and Applied Logic
"... In this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a form of generalized recursion theory which uses schemata for explicit definition, conditional definiti ..."
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Cited by 7 (2 self)
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In this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a form of generalized recursion theory which uses schemata for explicit definition, conditional definition and least fixed point (LFP) recursion in functionals of type level ≤ 2 over any appropriate structure. It is applied here to the case of potentially infinite (and more general partial) streams as an abstract data type. 1
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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Cited by 6 (0 self)
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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
Symmetry and Interactivity in Programming
 Bulletin of Symbolic Logic
, 2001
"... We recall some of the early occurrences of the notions of interactivity and symmetry in the operational and denotational semantics of programming languages. We suggest some connections with ludics. ..."
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Cited by 5 (1 self)
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We recall some of the early occurrences of the notions of interactivity and symmetry in the operational and denotational semantics of programming languages. We suggest some connections with ludics.
The history and concept of computability
 in Handbook of Computability Theory
, 1999
"... We consider the informal concept of a “computable ” or “effectively calculable” function on natural numbers and two of the formalisms used to define it, computability” and “(general) recursiveness. ” We consider their origin, exact technical definition, concepts, history, how they became fixed in th ..."
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Cited by 5 (1 self)
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We consider the informal concept of a “computable ” or “effectively calculable” function on natural numbers and two of the formalisms used to define it, computability” and “(general) recursiveness. ” We consider their origin, exact technical definition, concepts, history, how they became fixed in their present roles, and how