Exploring the abyss in Kleene's computability theory

TL;DR

This paper investigates the distinction between normal and non-normal functionals in Kleene's computability theory, introducing new non-normal functionals based on classical theorems and analyzing their computational complexity relative to the arithmetical hierarchy.

Contribution

It introduces new non-normal functionals derived from well-known theorems and examines their position within the computational hierarchy, highlighting the divide between different levels of the hyperarithmetical hierarchy.

Findings

New non-normal functionals are computable in  but not in weaker oracles.

Variations of these functionals can be computed in , crossing the abyss.

Examples are based on classical real analysis notions.

Abstract

Kleene's computability theory based on the S1-S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's 'machine model' which formalises computing with real numbers. A fundamental distinction in Kleene's framework is between normal and non-normal functionals where the former compute the associated Kleene quantifier $\exists^n$ and the latter do not. Historically, the focus was on normal functionals, but recently new non-normal functionals have been studied based on well-known theorems, the weakest among which seems to be the uncountability of the reals. These new non-normal functionals are fundamentally different from historical examples like Tait's fan functional: the latter is computable from $\exists^2$, while the former are computable in $\exists^3$ but not in weaker oracles. Of course, there is a great divide or abyss separating $\exists^2$ and $\exists^3$ and we identify slight variations of our new non-normal functionals that are again computable in $\exists^2$, i.e. fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, bounded variation, and semi-continuity from real analysis.