### Randomness and Solovay degrees

News
19 Mar 2018, online
5 Mar 2018, accepted by JLA

Title
Randomness and Solovay degrees
(with A. Nies and F. Stephan)

Type
Full paper

Journal
Journal of Logic and Analysis, Vol 10, pp.1–13, 2018.
Open access

Abstract
We consider the behaviour of Schnorr randomness, a randomness notion weaker than Martin-L\”of’s, for left-r.e. reals under Solovay reducibility. Contrasting with results on Martin-L\”of-randomenss, we show that Schnorr randomness is not upward closed in the Solovay degrees. Next, some left-r.e. Schnorr random $\alpha$ is the sum of two left-r.e. reals that are far from random. We also show that the left-r.e. reals of effective dimension $>r$, for some rational $r$, form a filter in the Solovay degrees.

coromandel_JLA_final

### Computable Measure Theory

News
11 Mar 2018, Submitted

Title
Computable Measure Theory

Type
Survey in Japanese

Journal
RIMS Kokyuroku 25 – 27 Dec 2017

rims

### Muchnik degrees and Medvedev degrees of the randomness notions

News
11 Mar 2018, submitted to a Journal

Title
Muchnik degrees and Medvedev degrees of the randomness notions

Type
Full paper

Journal
TBA

Abstract
The main theme of this paper is computational power when a machine is allowed to access random sets.
The computability depends on the randomness notions and we compare them by Muchnik and Medvedev degrees.
The central question is whether, given an random oracle, one can compute a more random set.
The main result is that, for each Turing functional,
there exists a Schnorr random set whose output is not computably random.