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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.
Relative and Modified Relative Realizability
 Annals of Pure and Applied Logic
, 2001
"... this paper) and it was described by means of tripos theory right from the beginnings of that theory, see, e.g., [17, Section 1.5, item (ii)]. Recently there has been a renewed interest in Relative Realizability, both in Thomas Streicher's "Topos for Computable Analysis" [18] and in [2 ..."
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this paper) and it was described by means of tripos theory right from the beginnings of that theory, see, e.g., [17, Section 1.5, item (ii)]. Recently there has been a renewed interest in Relative Realizability, both in Thomas Streicher's "Topos for Computable Analysis" [18] and in [2, 1, 4]. The idea is, that instead of doing realizability with one partial combinatory algebra A one uses an inclusion of partial combinatory algebras A ] ` A (such that there are combinators k; s 2 A ] which also serve as combinators for A), the principal point being that "(A ] ) computable" functions may also act on data (in A) that need not be computable
A Simple Model Construction for the Calculus of Constructions
 Types for Proofs and Programs, International Workshop TYPES'95
, 1996
"... . We present a model construction for the Calculus of Constructions (CC) where all dependencies are carried out in a settheoretical setting. The Soundness Theorem is proved and as a consequence of it Strong Normalization for CC is obtained. Some other applications of our model constructions are: sh ..."
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. We present a model construction for the Calculus of Constructions (CC) where all dependencies are carried out in a settheoretical setting. The Soundness Theorem is proved and as a consequence of it Strong Normalization for CC is obtained. Some other applications of our model constructions are: showing that CC + Classical logic is consistent (by constructing a model for it) and showing that the Axiom of Choice is not derivable in CC (by constructing a model in which the type that represents the Axiom of Choice is empty). 1 Introduction In the literature there are many investigations on the semantics of polymorphic calculus with dependent types (see for example [12, 11, 10, 1, 5, 13]). Most of the existing models present a semantics for systems in which the inhabitants of the impredicative universe (types) are "lifted" to inhabitants of the predicative universe (kinds) (see [16]). Such systems are convenient to be modeled by locally Cartesianclosed categories having small Cartesia...
Completing Partial Combinatory Algebras with Unique HeadNormal Forms
, 1996
"... In this note, we prove that having unique headnormal forms is a sufficient condition on partial combinatory algebras to be completable. As application, we show that the pca of strongly normalizing CLterms as well as the pca of natural numbers with partial recursive function application can be exte ..."
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In this note, we prove that having unique headnormal forms is a sufficient condition on partial combinatory algebras to be completable. As application, we show that the pca of strongly normalizing CLterms as well as the pca of natural numbers with partial recursive function application can be extended to total combinatory algebras. 1.
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 Proceedings of TACAS’98
, 1998
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Extending Partial Combinatory Algebras
, 1999
"... Introduction Consider a structure A = hA; s; k; \Deltai, where A is some set containing the distinguished elements s; k, equipped with a binary operation \Delta on A, called application, which may be partial. Notation 1.1. 1 Instead of a \Delta b we write ab; and in writing applicative expression ..."
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Introduction Consider a structure A = hA; s; k; \Deltai, where A is some set containing the distinguished elements s; k, equipped with a binary operation \Delta on A, called application, which may be partial. Notation 1.1. 1 Instead of a \Delta b we write ab; and in writing applicative expressions, the usual convention of association to the left is employed. So for elements a; b; c 2 A, the expression aba(ac) is short for ((a \Delta b) \Delta a) \Delta (a \Delta c). 2 ab # will mean that ab is defined; ab " means that ab is not defined. Obviously, an applicative expression
(In)consistency of extensions of Higher Order Logic and Type Theory
"... Abstract. It is wellknown, due to the work of Girard and Coquand, that adding polymorphic domains to higher order logic, HOL, or its type theoretic variant λHOL, renders the logic inconsistent. This is known as Girard’s paradox, see [7]. But there is also a another presentation of higher order logi ..."
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Abstract. It is wellknown, due to the work of Girard and Coquand, that adding polymorphic domains to higher order logic, HOL, or its type theoretic variant λHOL, renders the logic inconsistent. This is known as Girard’s paradox, see [7]. But there is also a another presentation of higher order logic, in its type theoretic variant called λPREDω, to which polymorphic domains can be added safely, Both λHOL and λPREDω are wellknown type systems and in this paper we study why λHOL with polymorphic domains is inconsistent and why nd λPREDω with polymorphic domains remains consistent. We do this by describing a simple model for the latter and we show why this can not be a model of the first. 1
SN Combinators and Partial Combinatory Algebras
"... . We introduce an intersection typing system for combinatory logic, such that a term of combinatory logic is typeable iff it is sn. We then prove the soundness and completeness for the class of partial combinatory algebras. Let F be the class of nonempty filters which consist of types. Then F is an ..."
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. We introduce an intersection typing system for combinatory logic, such that a term of combinatory logic is typeable iff it is sn. We then prove the soundness and completeness for the class of partial combinatory algebras. Let F be the class of nonempty filters which consist of types. Then F is an extensional nontotal partial combinatory algebra. Furthermore, it validates the strongest consistent equality of the set of sn terms of combinatory logic. By F , we can solve BethkeKlop's question; "find a suitable representation of the finally collapsed partial combinatory algebra of P ". Here, P is a partial combinatory algebra, and is the set of closed sn terms of combinatory logic modulo the inherent equality. Our solution is the following: the finally collapsed partial combinatory algebra of P is representable in F . To be more precise, it is isomorphically embeddable into F . 1 Introduction Combinatory logic (cl, for short) is a simple rewriting system where the terms (clterms, fo...