### Coherence of reducibilities with randomness notions

News
1 Feb 2016, accepted by TOCS

Title
Coherence of reducibilities with randomness notions

Type
Full paper

Journal
TOCS, post proceedings of CCR2016

Abstract
Loosely speaking, when $A$ is “more random” than $B$ and $B$ is “random”,
then $A$ should be random.
The theory of algorithmic randomness has some formulations of “random” sets
and “more random” sets.
In this paper, we study which pairs $(R,r)$ of randomness notions $R$
and reducibilities $r$ have the follwing property:
if $A$ is $r$-reducible to $B$ and $A$ is $R$-random,
then $B$ should be $R$-random.
The answer depends on the notions $R$ and $r$.
The implications hold for most pairs, but not for some.
We also give characterizations of $n$-randomness via complexity.

preprint

### A way to judge using probability

News
13 Dec, 2016 the slide file was uploaded

Title
A way to judge using probability

Type
A lecture in a high school

slide in Japanese

### 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
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.