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5 - Relativization in randomness
- By Johanna N. Y. Franklin
Edited by Johanna N. Y. Franklin, Hofstra University, New York, Christopher P. Porter, Drake University, Iowa
Book:

Algorithmic Randomness

Published online:

07 May 2020

Print publication:

07 May 2020, pp 134-174
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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.

STRONG JUMP-TRACEABILITY
NOAM GREENBERG, DAN TURETSKY
Journal:

Bulletin of Symbolic Logic / Volume 24 / Issue 2 / June 2018

Published online by Cambridge University Press:

07 August 2018, pp. 147-164

Print publication:

June 2018
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We review the current knowledge concerning strong jump-traceability. We cover the known results relating strong jump-traceability to randomness, and those relating it to degree theory. We also discuss the techniques used in working with strongly jump-traceable sets. We end with a section of open questions.

Low upper bounds of ideals
Antonín Kučera, Theodore A. Slaman
Journal:

The Journal of Symbolic Logic / Volume 74 / Issue 2 / June 2009

Published online by Cambridge University Press:

12 March 2014, pp. 517-534

Print publication:

June 2009
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We show that there is a low T-upper bound for the class of K-trivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in T-degrees for which there is a low T-upper bound.