Reducibilities relating to Schnorr randomness

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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Mass problems for randomness notions

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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slide

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On the notions of randomness and probability

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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slide

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Using Almost-Everywhere Theorems from Analysis to Study Randomness

News
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
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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CCR in a TV program!!

It is really a pity that I couldn’t go CCR 2016 in Hawaii,
but it’s an honor to have such attention from media.

http://thinktechhawaii.com/bounding-rationality-with-computation/

https://www.youtube.com/watch?v=1kTtBlYbk9s

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Mass problems for randomness notions

News
13 Feb 2016, the slide was uploaded

Title
Mass problems for randomness notions

Type
In a seminar at Meiji University
The slide and the talk is in Japanese.

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slide

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What is random?

News
12 Dec 2015, the slide was uploaded

Title
What is random?

Type
Ikura salon at Meiji University

The slide and the talk is in Japanese.

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Slide

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Separation of randomness notions in Weihrauch degrees

News
2 Oct 2015, the abstract was uploaded

Title
Separation of randomness notions in Weihrauch degrees

Type
Short talk on black boards at Dagstuhl Seminar

Abstract
If someone says that a function is “computable”, it sometimes means that it is programmable with a programming language with a random generator. Computability with a random set can be paid more attension. In this talk we will consider whether we can make a random set more random. In other words, we will consider randomness notions in Weihrauch degrees and see some separation of them.
This is a joint work with Rupert Hölzl.

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Characterizations of 3-randomness via complexity

News
8 Sep 2015, the slide file was uploaded

Title
Characterizations of 3-randomness via complexity

Type
Talk at MSJ Autumn Meeting 2015

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miyabe-mathsoc2015Sep

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Reducibilities as refinements of the randomness hierarchy

News
8 Sep 2015, the slide file was uploaded

Title
Reducibilities as refinements of the randomness hierarchy

Type
Talk at CTFM2015

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miyabe-ctfm2015

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