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.

ntroduce some simple general theory to allow us to talk about notions of higher-type computable functional. The following definitions (with minor variations) appear frequently in the literature. Definition 1.1 (Weak partial type structures) A weak partial type structure, or weak PTS A [over a set X], consists of the following data: . for each type #, a set A # of elements of type # [equipped with a canonical bijection A 0 # = X], . for each #, # , a partial application function ## : A ### A # # A # . We usually omit type subscripts from application operations, and often write x y simply as xy. By convention, w

We consider a version of PCF, and prove, using both syntactic and semantic means, that the operational equivalences of the base language are preserved when the language is extended with sum and product types, with polymorphic types, and with recursive types. These theorems show that the additions to the type systems are orthogonal to the original language. 1 Introduction Type systems for programming languages are rarely monolithic: although a type system may be composed of many parts, each part can usually be understood on its own. Consider, for instance, the programming language Standard ML (SML) [22]. SML's type system includes base types of integers, reals, strings, and characters, and type constructors for lists, functions, tuples, references, exceptions, user-defined recursive datatypes, and polymorphism. On a syntactic level, the type rules of the parts do not interfere with one another: the type-checking rule for application, for example, uses only the fact that the operator is...