The non-normal abyss in Kleene's computability theory

TL;DR

This paper explores the distinction between normal and non-normal functionals in Kleene's computability theory, revealing new non-normal functionals that challenge traditional boundaries and are based on classical mathematical concepts.

Contribution

It introduces and analyzes new non-normal functionals related to well-known theorems, showing they can differ significantly in computational complexity from classical examples.

Findings

New non-normal functionals are computable only in higher levels of the arithmetical hierarchy.

Certain non-normal functionals based on classical theorems fall on different sides of the normal/non-normal divide.

The study connects classical mathematical notions with computational complexity in Kleene's framework.

Abstract

Kleene's computability theory based on his 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 like 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 only computable in $\exists^{3}$. While there is a great divide separating $\exists^{2}$ and $\exists^{3}$, we identify certain closely related non-normal functionals that fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, and semi-continuity.