**News**

13 Dec, 2016 the slide file was uploaded

**Title**

A way to judge using probability

**Type**

A lecture in a high school

**Download**

slide in Japanese

宮部賢志（ミヤベケンシ）

**News**

13 Dec, 2016 the slide file was uploaded

**Title**

A way to judge using probability

**Type**

A lecture in a high school

**Download**

slide in Japanese

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**News**

10 Oct 2016, published online

29 Feb 2016, Accepted by BSL

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**

The Bulletin of Symbolic Logic, Volume 22, Issue 3

September 2016, pp. 305-331

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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**News**

20 Sep, 2016, the slide file was uploaded

**Title**

Randomness notions in Muchnik and Medvedev degrees

**Type**

Talk at a meeting of CTFM2016

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**News**

2 Sep, 2016, the slide file was uploaded

**Title**

Computability of reals on the space where the triangle inequality does not hold

**Type**

Talk at a meeting of Mathematical Society of Japan

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**News**

25 July, 2016, the slide file was uploaded

**Title**

Computability of reals on the space where the triangle inequality does not hold

**Type**

Talk at a lab seminar

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**News**

6 June 2016, the slide file was uploaded

**Title**

Variants of layerwise computability

**Type**

Talk at Thirteenth International Conference on Computability and Complexity in Analysis (CCA2016)

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slide(cca2016)

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**News**

3 June 2016, the memo was uploaded

**Title**

Randomness and computability

**Type**

Colloquium at Tsukuba University

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tsukuba

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**News**

22 Sep 2014. Accepted to publish in TOCS

24 Mar 2014. Submitted

**Title**

Reducibilities relating to Schnorr randomness

**Type**

Full paper

**Journal**

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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preprint

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**News**

7 Mar 2016, the slide was uploaded

**Title**

Mass problems for randomness notions

**Type**

The mathematical Society of Japan in University of Tsukuba

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**News**

6 Mar 2016, the slide was uploaded

**Title**

On the notions of randomness and probability

**Type**

A talk in The Third Meeting of Quantum Foundation Club

The slide and the talk is in Japanese.

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