カテゴリー: Publication

投稿日:

Using Almost-Everywhere Theorems from Analysis to Study Randomness

履歴
2016年10月10日　オンライン出版
2016年2月29日　BSL受理
2015年5月　再投稿
2014年11月3日　投稿

タイトル
Using Almost-Everywhere Theorems from Analysis to Study Randomness
(with Jing Zhang and Andre Nies)

種類
正論文

国際会議と雑誌
The Bulletin of Symbolic Logic, Volume 22, Issue 3
September 2016, pp. 305-331
arXiv
最新版.

Abstract
We study algorithmic randomness notions via effective versions of almost-everywhere theorems from analysis and ergodic theory. The effectivization is in terms of objects described by a computably enumerable set, such as lower semicomputable functions. The corresponding randomness notions are slightly stronger than Martin-Lo ̈f (ML) randomness. We establish several equivalences. Given a ML-random real z, the additional randomness strengths needed for the following are equivalent.
(1) all effectively closed classes containing z have density 1 at z.
(2) all nondecreasing functions with uniformly left-c.e. increments are differentiable at z.
(3) z is a Lebesgue point of each lower semicomputable integrable function.
We also consider convergence of left-c.e. martingales, and convergence in the sense of Birkhoff’s pointwise ergodic theorem. Lastly we study randomness notions for density of $\Pi^0_n$ and $\Sigma^1_1$ classes.

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Reducibilities relating to Schnorr randomness

履歴
2014年9月22日　受理
2014年3月24日　投稿

タイトル
Reducibilities relating to Schnorr randomness

種類
正論文

雑誌
Theory of Computing Systems, 58(3), 441-462, 2016.
DOI: 10.1007/s00224-014-9583-3

Abstract
Some measures of randomness have been introduced for Martin- L ̈of randomness such as K-reducibility, C-reducibility and vL-reducibility. In this paper we study Schnorr-randomness versions of these reducibilities. In particular, we characterize the computably-traceable reducibility via relative Schnorr randomness, which was asked in Nies’ book (Problem 8.4.22). We also show that Schnorr reducibility implies uniform-Schnorr-randomness version of vL-reducibility, which is the Schnorr-randomness version of the result that K-reducibility implies vL-reducibility.

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投稿日:

Unified Characterizations of Lowness Properties via Kolmogorov Complexity

履歴
2014年1月19日投稿
2015年3月24日出版確認

タイトル
Unified Characterizations of Lowness Properties via Kolmogorov Complexity
(with T. Kihara)

種類
正論文

国際会議と雑誌
Archive for Mathematical Logic: Volume 54, Issue 3 (2015), Page 329-358
DOI: 10.1007/s00153-014-0413-8

Abstract
Consider a randomness notion $\mathcal C$.
A uniform test in the sense of $\mathcal C$ is a total computable procedure that each oracle $X$ produces a test relative to $X$ in the sense of $\mathcal C$.
We say that a binary sequence $Y$ is $\mathcal C$-random uniformly relative to $X$ if $Y$ passes all uniform $\mathcal C$ tests relative to $X$.

Suppose now we have a pair of randomness notions $\mathcal C$ and $\mathcal D$ where $\mathcal{C}\subseteq \mathcal{D}$, for instance Martin-L\”of randomness and Schnorr randomness. Several authors have characterized classes of the form Low($\mathcal C, \mathcal D$) which consist of the oracles $X$ that are so feeble that $\mathcal C \subseteq \mathcal D^X$. Our goal is to do the same when the randomness notion $\mathcal D$ is relativized uniformly: denote by Low$^\star$($\mathcal C, \mathcal D$) the class of oracles $X$ such that every $\mathcal C$-random is uniformly $\mathcal D$-random relative to $X$.

(1) We show that $X\in{\rm Low}^\star({\rm MLR},{\rm SR})$ if and only if $X$ is c.e.~tt-traceable if and only if $X$ is anticomplex if and only if $X$ is Martin-L\”of packing measure zero with respect to all computable dimension functions.

(2) We also show that $X\in{\rm Low}^\star({\rm SR},{\rm WR})$ if and only if $X$ is computably i.o.~tt-traceable if and only if $X$ is not totally complex if and only if $X$ is Schnorr Hausdorff measure zero with respect to all computable dimension functions.

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