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4 - Algorithmic randomness and layerwise computability

Published online by Cambridge University Press:  07 May 2020

Johanna N. Y. Franklin
Affiliation:
Hofstra University, New York
Christopher P. Porter
Affiliation:
Drake University, Iowa
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Summary

Algorithmic randomness lies at the intersection between computability theory and probability theory. In order to fully explore this interaction, one naturally needs a computable version of measurable functions. While several such notions appear in the literature, most of them do not interact well with algorithmic randomness because they are only defined up to a null set. Therefore, we need a computable notion of measurable function which is well defined on algorithmically random points, and this is what layerwise computability precisely does. This article is a survey about this notion. We give the main definitions, the most important properties, and several applications of this notion. We prioritize motivating this framework and explaining its salient features.

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Chapter
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Algorithmic Randomness
Progress and Prospects
, pp. 115 - 133
Publisher: Cambridge University Press
Print publication year: 2020

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