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DEMUTH’S PATH TO RANDOMNESS

Published online by Cambridge University Press:  15 September 2015

ANTONÍN KUČERA ,
ANDRÉ NIES  and
CHRISTOPHER P. PORTER
ANTONÍN KUČERA
Affiliation:
FACULTY OF MATHEMATICS AND PHYSICS CHARLES UNIVERSITY IN PRAGUE PRAGUE, CZECH REPUBLICE-mail: kucera@mbox.ms.mff.cuni.cz
ANDRÉ NIES
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF AUCKLAND AUCKLAND, NEW ZEALANDE-mail: andre@cs.auckland.ac.nz
CHRISTOPHER P. PORTER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF FLORIDA GAINESVILLE FL 32611-8105, USAE-mail: cp@cpporter.com
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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.

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 21 , Issue 3 , September 2015 , pp. 270 - 305
Copyright
Copyright © The Association for Symbolic Logic 2015 

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