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5 - Relativization in randomness

Published online by Cambridge University Press:  07 May 2020

By
Johanna N. Y. Franklin
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

We discuss the different contexts in which relativization occurs in randomness and the effect that the relativization chosen has on the results we can obtain. We study several characterizations of the K-trivials in terms of concepts ranging from cuppability to density, and we consider a uniform relativization for randomness that gives us more natural results for computable randomness, Schnorr randomness, and Kurtz randomness than the classical relativization does (the relativization for Martin-Löf randomness is unaffected by this change). We then evaluate the relativizations we have considered and suggest some avenues for further work.

Keywords

algorithmic randomnesscomputable randomnessK-trivialKurtz randomnesslownessMartin-Löf randomnessrelativizationSchnorr randomnessuniform relativizationvan Lambalgen's Theorem
Type
Chapter
Information
Algorithmic Randomness
Progress and Prospects
 , pp. 134 - 174
Publisher: Cambridge University Press
Print publication year: 2020

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