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.

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

In this note, we prove that having unique head-normal forms is a sufficient condition on partial combinatory algebras to be completable. As application, we show that the pca of strongly normalizing CL-terms as well as the pca of natural numbers with partial recursive function application can be extended to total combinatory algebras. 1.

. We present a model construction for the Calculus of Constructions (CC) where all dependencies are carried out in a set-theoretical 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 Cartesian-closed categories having small Cartesia...

Proceedings

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

Abstract not found

These are the lecture notes for a tutorial at FMCS 2004 in Kananaskis. The aim is to give a first introduction to Partial Combinatory Algebras and the construction of Realizability Toposes. The first part, where Partial Combinatory Algebras are discussed, requires no specific background (except for some of the examples perhaps), although familiarity with combinatory logic and lambda calculus will not hurt. The second part on realizability toposes presupposes some knowledge of category theory; more specifically, we will assume that the reader knows what a topos is. Apart from that the material is self-contained. 1 Partial Combinatory Algebras We give the basic definitions and properties of Partial Combinatory Algebras in the first subsection. Next, we discuss some of the important examples. Finally, we touch upon the theory of Partial Combinatory Algebras. 1.1 Partial Applicative Structures and Combinatory Completeness We first introduce the basic concept of a Partial Applicative Structure, which may be viewed as a universe for computation. Then look at terms over an applicative structure, we formulate

. We introduce a simple translation from -calculus to combinatory logic (cl) such that: A is an sn -term iff the translation result of A is an sn term of cl (the reductions are fi-reduction in -calculus and weak reduction in cl). None of the conventional translations from -calculus to cl satisfy the above property. Our translation provides a simpler sn proof of Godel's -calculus by the ordinal number assignment method. By using our translation, we construct a homomorphism from a conditionally partial combinatory algebra which arises over sn -terms to a partial combinatory algebra which arises over sn cl-terms. 1 Introduction We often find some translations from -calculus to combinatory logic (cl) provide a pleasing viewpoint in the study of -calculus. The most typical example can be found in the study of the equational theories and the model theories of -calculus. The translations from -calculus to cl have been investigated comprehensively by Curry school [8], and we come to know ...

Abstract not found