On the computational properties of ambivalent sets and functions

TL;DR

This paper explores the computational aspects of ambivalent functions, a class between semi-continuous and Baire 1 functions, establishing their computational properties and equivalences using Kleene's schemes.

Contribution

It introduces and analyzes the class of ambivalent functions, detailing their computational properties and establishing equivalences for standard operations within this class.

Findings

Computational equivalences for supremum and Baire 1 representations.

The structure functional $oldsymbol{\Omega_{f \Delta}}$ decides non-emptiness of ambivalent sets.

A selector for ambivalent sets is computable relative to $oldsymbol{\Omega_{f \Delta}}$ and $oldsymbol{\exists^{2}}$.

Abstract

Examples of discontinuous functions already appear in the work of Euler, Abel, Dirichlet, Fourier, and Bolzano. A ground-breaking discovery due to Baire was that many discontinuous functions are well-behaved in that they are the pointwise limit of a sequence of continuous functions; the latter form a class nowadays simply called `Baire 1'. We shall study a class strictly between the semi-continuous and Baire 1 functions, called the ambivalent fuctions. In particular, we investigate the computational properties of the class of ambivalent functions and sets, denoted $\bf \Delta$, working with Kleene's S1-S9 schemes. Computational equivalences for various standard operations (supremum, Baire 1 representation, \dots) on $\bf \Delta$ are established, including the structure functional $\Omega_{\bf \Delta}$ that decides if a given ambivalent set is non-empty. A selector is shown to be computable relative to $\Omega_{\bf \Delta}$ and Kleene's quantifier $\exists^{2}$.