Kleene's Two Kinds of Recursion

TL;DR

This paper explains Kleene's two recursion theorems, highlighting their differences, applications, and how Rogers' result helps unify their use in defining recursive functions.

Contribution

It clarifies the distinction between the First and Second Recursion Theorems and shows how Rogers' result bridges their applicability in recursion theory.

Findings

FRT produces least fixed points, unlike SRT.

Both theorems are applicable in some cases, with FRT being stronger.

Rogers' result enables standardization of higher-order functionals.

Abstract

This is an elementary expository article regarding the application of Kleene's Recursion Theorems in making definitions by recursion. Whereas the Second Recursion Theorem (SRT) is applicable in a first-order setting, the First Recursion Theorem (FRT) requires a higher-order setting. In some cases both theorems are applicable, but one is stronger than the other: the FRT always produces least fixed points, but this is not always the case with the SRT. Nevertheless, an old result by Rogers allows us to bridge this gap by subtly redefining the implementation of a higher-order functional in order to bring it to a `standard form.'