Demuth's path to randomness

TL;DR

This paper explores Osvald Demuth's contributions to constructive analysis and algorithmic randomness, highlighting his pioneering ideas, key results, and the connections he established between these fields and computability theory.

Contribution

It provides a comprehensive analysis of Demuth's work, emphasizing his early constructivist roots and his innovative development of randomness notions and their relation to constructive analysis.

Findings

Demuth introduced effective null sets leading to major randomness notions.

He proved results connecting constructive analysis with algorithmic randomness.

Demuth independently discovered key concepts in algorithmic randomness.

Abstract

Osvald Demuth (1936--1988) studied constructive analysis from the viewpoint of the Russian school of constructive mathematics. In the course of his work he introduced various notions of effective null set which, when phrased in classical language, yield a number of major algorithmic randomness notions. In addition, he proved several results connecting constructive analysis and randomness that were rediscovered only much later. In this paper, we trace the path that took Demuth from his constructivist roots to his deep and innovative work on the interactions between constructive analysis, algorithmic randomness, and computability theory. We will focus specifically on (i) Demuth's work on the differentiability of Markov computable functions and his study of constructive versions of the Denjoy alternative, (ii) Demuth's independent discovery of the main notions of algorithmic randomness, as well as the development of Demuth randomness, and (iii) the interactions of truth-table reducibility, algorithmic randomness, and semigenericity in Demuth's work.