Notions of computability at higher types II

In Part I of this series of papers [Lon01a] we gave a historical survey of the study of notions of higher-type computability. In the present paper and its sequel [Lon01b], we undertake a more systematic exposition of notions of higher-type computability, showing how many of the existing ideas and results can be fitted into a coherent framework. In Part II we will restrict our attention to notions of computable functional—that is, the objects of type σ → τ that we consider will always be (total or partial) functions from objects of type σ to objects of type τ . For such notions, the overall picture is easy to grasp with the help of some fairly concrete and naive definitions. A more general picture, embracing non-functional notions of computability and requiring slightly more sophisticated conceptual apparatus, will be presented in Part III. In Section 1 we develop a general theory of type structures which, although simple, furnishes a useful framework for understanding particular notions of computable functional and their interrelationships. In Section 2 we survey the various known notions of hereditarily total computable functional within this framework, and in Section 3 the known notions of hereditarily partial computable functional. For the most part we consider the total and partial cases separately, and this is perhaps a limitation of our present treatment. In Section 4 we go some way towards rectifying this by considering how total and partial notions may be related or combined.