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Coherence of Reducibilities with Randomness Notions

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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.

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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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  1. Department of Mathematics, School of Science and Technology Meiji University, Kawasaki-shi, Kanagawa, 214-8571, Japan

    Kenshi Miyabe

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  1. Kenshi Miyabe
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Miyabe, K. Coherence of Reducibilities with Randomness Notions. Theory Comput Syst 62, 1599–1619 (2018). https://doi.org/10.1007/s00224-017-9752-2

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