Using Almost-Everywhere Theorems from Analysis to Study Randomness

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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Randomness notions in Muchnik and Medvedev degrees

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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abstract
slide

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

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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slide

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

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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slide

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Variants of layerwise computability

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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Randomness and computability

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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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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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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