Kenshi Miyabe

履歴
2018年9月　受理
2018年3月11日　投稿

タイトル
Muchnik degrees and Medvedev degrees of the randomness notions

種類
研究論文

国際会議と雑誌
ALC2015とALC2017の共同議事録
World Scientificから出版予定

Abstract
The main theme of this paper is computational power when a machine is allowed to access random sets.
The computability depends on the randomness notions and we compare them by Muchnik and Medvedev degrees.
The central question is whether, given an random oracle, one can compute a more random set.
The main result is that, for each Turing functional,
there exists a Schnorr random set whose output is not computably random.

ダウンロード
Muchnikdegrees

履歴
2017年2月1日　TOCS受理

タイトル
Coherence of reducibilities with randomness notions

種類
正論文

国際会議と雑誌
Theory of Computing Systems – October 2018, Volume 62, Issue 7, pp 1599–1619
DOI: https://doi.org/10.1007/s00224-017-9752-2

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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preprint