Computability and Analysis, a Historical Approach

TL;DR

This paper explores the historical relationship between computability theory and analysis, focusing on how the computability of theorems in analysis has evolved from early pioneers to modern classifications in the Weihrauch lattice.

Contribution

It provides a historical overview and discusses recent advances in classifying the computational content of analysis theorems within the Weihrauch lattice.

Findings

Early work linked computability with analysis concepts.

Recent classification of theorems in the Weihrauch lattice.

Historical insights into the development of computable analysis.

Abstract

The history of computability theory and and the history of analysis are surprisingly intertwined since the beginning of the twentieth century. For one, \'Emil Borel discussed his ideas on computable real number functions in his introduction to measure theory. On the other hand, Alan Turing had computable real numbers in mind when he introduced his now famous machine model. Here we want to focus on a particular aspect of computability and analysis, namely on computability properties of theorems from analysis. This is a topic that emerged already in early work of Turing, Specker and other pioneers of computable analysis and eventually leads us to the very recent project of classifying the computational content of theorems in the Weihrauch lattice.