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6 - Aspects of Chaitin’s Omega

Published online by Cambridge University Press:  07 May 2020

By
George Barmpalias
Edited by
Johanna N. Y. Franklin  and
Christopher P. Porter
Johanna N. Y. Franklin
Affiliation:
Hofstra University, New York
Christopher P. Porter
Affiliation:
Drake University, Iowa
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Summary

The halting probability of a Turing machine was introduced by Chaitin, who also proved that it is an algorithmically random real number and named it Omega. Since his seminal work, many popular expositions have appeared, mainly focusing on the metamathematical or philosophical significance of this number (or debating against it). At the same time, a rich mathematical theory exploring the properties of Chaitin's Omega has been brewing in various technical papers, which quietly reveals the significance of this number to many aspects of contemporary algorithmic information theory. The purpose of this survey is to expose these developments and tell a story about Omega which outlines its multi-faceted mathematical properties and roles in algorithmic randomness.

Keywords

algorithmic informationalgorithmic randomnessChaitin's Omegacomputability theoryhalting probability
Type
Chapter
Information
Algorithmic Randomness
Progress and Prospects
 , pp. 175 - 205
Publisher: Cambridge University Press
Print publication year: 2020

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