Using Almost-Everywhere Theorems from Analysis to Study Randomness

News
May 2015, Resubmitted.
3 Nov 2014. Submitted

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

Type
Full paper

Journal
Submitted
arXiv
The latest version is here.

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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Schnorr triviality and its equivalent notions

News
13 Sep 2013, Accepted by TOCS
23 Mar 2013, Submitted

Title
Schnorr triviality and its equivalent notions

Type
Full paper

Journal
Theory of Computing Systems
Volume 56, Issue 3 , pp 465-486
DOI: 10.1007/s00224-013-9506-8

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preprint

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Unified Characterizations of Lowness Properties via Kolmogorov Complexity

News
19 Jan 2014, submitted
24 Mar 2015, published

Title
Unified Characterizations of Lowness Properties via Kolmogorov Complexity
(with T. Kihara)

Type
Full paper

Journal
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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The randomness hierarchy and reducibilities as its refinements

News
23 Feb 2015, the slide file was uploaded

Title
The randomness hierarchy and reducibilities as its refinements

Type
Joint seminar at TMU (in Japanese)

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TMU-slide in Japanese

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Total-machine reducibility and randomness notions

News
4 Jan 2015, the slide file was uploaded

Title
Total-machine reducibility and randomness notions

Type
Asian Logic Conference 2015

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slide

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Derandomization in game-theoretic probability

News
13 Nov 2014, the slide file was uploaded

Title
Derandomization in game-theoretic probability

Type
GTP 2014: Fifth Workshop on Game-Theoretic Probability and Related Topics

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miyabe-gtp2014

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Mathematical Seminar, Oct 2014

An article was published in Mathematical Seminar Oct 2014.

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NII Shonan Meeting “Algorithmic Randomness and Complexity”, 8-12 Sep, 2014

NII Shonan Meeting “Algorithmic Randomness and Complexity”

The webpage for participants.

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Betting game and mathematics

News
17 Aug 2014, the slide file was uploaded

Title
Betting game and mathematics

Type
Summer Seminar of Meiji University for high school students

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miyabe-summer-seminar-2014

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Derandomization in Game-Theoretic Probability

News
27 Sep 2014, Online
3 Aug 2014, Accepted in SPA
12 Feb 2014. Submitted

Title
Derandomization in Game-Theoretic Probability
(with A. Takemura)

Type
Full paper

Journal
Stochastic Processes and their Applications 125, 39-59, 2015

Abstract
We give a general method for constructing a deterministic strategy
of Reality from a randomized strategy in game-theoretic probability.
The construction can be seen as derandomization in game-theoretic probability.

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preprint

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