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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.
Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations
, 1994
"... Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations Ralph Loader, of St. Hugh's College, Oxford. Thesis submitted for the Degree of D.Phil. Michaelmas term, 1994. T his thesis is an investigation into models of typed calculi and of linear l ..."
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Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations Ralph Loader, of St. Hugh's College, Oxford. Thesis submitted for the Degree of D.Phil. Michaelmas term, 1994. T his thesis is an investigation into models of typed calculi and of linear logic. The models we investigate are denotational in nature; we construct various categories, in which types (or formulae) are interpreted by objects, and terms (proofs) by morphisms. The results we investigate compare particular properties of the syntax and the semantics of a calculus, by trying to use syntax to characterise features of a model, or vice versa. There are four chapters in the thesis, one each on linear logic and the simply typed calculus, and two on inductive datatypes. In chapter one, we look at some models of linear logic, and prove a full completeness result for multiplicative linear logic. We form a model, the linear logical predicates , by abstracting a little the structure ...
Equational Theories for Inductive Types
 Annals of Pure and Applied Logic
, 1997
"... This paper provides characterisations of the equational theory of the per model of a typed lambda calculus with inductive types. The characterisation may be cast as a full abstraction result; in other words we show that the equations between terms valid in this model coincides with a certain synt ..."
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This paper provides characterisations of the equational theory of the per model of a typed lambda calculus with inductive types. The characterisation may be cast as a full abstraction result; in other words we show that the equations between terms valid in this model coincides with a certain syntactically defined equivalence relation. Along the way we give other characterisations of this equivalence; from below, from above, and from a domain model; a version of the KreiselLacombeShoenfield theorem allows us to transfer the result from the domain model to the per model. 0 Introduction This paper concerns a typed calculus with inductive types which correspond semantically to initial algebras of (covariant) functors; the calculus lies between Godel's T and Girard's F in prooftheoretic strength. The goal of the paper is to analyse the structure of the model of this calculus given by the category PER of partial equivalence relations over the natural numbers. We shall show that ...
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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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.