Effective Choice and Boundedness Principles in Computable Analysis

TL;DR

This paper introduces a new framework using Weihrauch reducibility to classify the computational content of mathematical theorems in analysis, providing finer distinctions than previous approaches.

Contribution

It develops a novel classification method for theorems based on their computational realizers and introduces separation techniques and metatheorems for analyzing their Weihrauch degrees.

Findings

Classified core theorems like the Intermediate Value Theorem and Banach Inverse Mapping Theorem within the Weihrauch degree structure.

Identified choice principles as key components in the computational classification of theorems.

Compared and refined existing classifications from constructive and reverse mathematics.

Abstract

In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem and others. We also explore how existing classifications of the Hahn-Banach Theorem and Weak K"onig's Lemma fit into this picture. We compare the results of our classification with existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective theorems. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example.