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Theory of Computing Systems

, Volume 62, Issue 7, pp 1599–1619 | Cite as

Coherence of Reducibilities with Randomness Notions

  • Kenshi Miyabe
Article
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Abstract

Loosely speaking, when A is “more random” than B and B is “random”, then A should be random. The theory of algorithmic randomness has some formulations of “random” sets and “more random” sets. In this paper, we study which pairs (R, r) of randomness notions R and reducibilities r have the follwing property: if A is r-reducible to B and A is R-random, then B should be R-random. The answer depends on the notions R and r. The implications hold for most pairs, but not for some. We also give characterizations of n-randomness via complexity.

Keywords

Schnorr reducibility Van Lambalgen reducibility Decidable machine 

Notes

Acknowledgments

This work was supported by JSPS KAKENHI, Grant-in-Aid for Young Scientists (B), Grant Number 26870143. The author appreciates Frank Stephan for a useful comment to initial work of this paper at CCR2014 at Singapore, and André Nies for some comments to a later manuscript. The author also appreciates the reviewers for careful reading and some advice.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsSchool of Science and Technology Meiji UniversityKawasaki-shiJapan

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