Recursion in Higher Types*

Abstract

Recursion in higher types is an extension of the theory of recursive functions on the integers. This chapter presents an exposition of the basic notions and facts of this theory. It develops higher-type recursion in the context of the general theory of inductive definability. The study is based on Moschovakis approach. The chapter is divided into two parts—functional induction and recursion in higher types. The functional inductions includes monotone operators on partial functions, recursion in type 2 objects and quantifiers, the stage comparison theorem, semirecursive relations, and the enumeration theorem. The topics discussed under recursion in higher types are normality and enumeration in higher type recursion, the original definition of Kleene, substitution theorems of Kleene, sections and envelopes, inductive analysis of semirecursive sets, and closure under higher existential quantification.