Muchnik degrees and Medvedev degrees of the randomness notions

履歴
2018年9月 受理
2018年3月11日 投稿

タイトル
Muchnik degrees and Medvedev degrees of the randomness notions

種類
査読ありの事後会議録

国際会議と雑誌
ALC2015とALC2017の共同議事録
World Scientificから出版

Proceedings of the 14th and 15th Asian Logic Conferences, pp. 108-128 (2019) January

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.

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Muchnikdegrees

Coherence of reducibilities with randomness notions

履歴
2017年2月1日 TOCS受理

タイトル
Coherence of reducibilities with randomness notions

種類
正論文

国際会議と雑誌
Theory of Computing Systems – October 2018, Volume 62, Issue 7, pp 1599–1619
DOI: https://doi.org/10.1007/s00224-017-9752-2

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.

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preprint

Relation between the rate of convergence of strong law of large numbers and the rate of concentration of Bayesian prior in game-theoretic probability

履歴
2017年8月8日 オンライン
2017年7月28日 SPA受理

タイトル
Relation between the rate of convergence of strong law of large numbers and the rate of concentration of Bayesian prior in game-theoretic probability
(with R. Sato and A. Takemura)

種類
正論文

国際会議と雑誌
Stochastic Processes and their Applications
Volume 128, Issue 5, May 2018, Pages 1466-1484
The page at SPA

Abstract
We study the behavior of the capital process of a continuous Bayesian mixture of fixed proportion
betting strategies in the one-sided unbounded forecasting game in game-theoretic probability. We
establish the relation between the rate of convergence of the strong law of large numbers in the selfnormalized
form and the rate of divergence to infinity of the prior density around the origin. In
particular we present prior densities ensuring the validity of Erdos–Feller–Kolmogorov–Petrowsky ˝
law of the iterated logarithm.

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arXiv

Randomness and Solovay degrees

履歴
2018年3月19日 オンライン
2018年3月5日 JLA受理

タイトル
Randomness and Solovay degrees
(with A. Nies and F. Stephan)

種類
正論文

国際会議と雑誌
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.

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coromandel_JLA_final

計算可能測度論

履歴
2018年3月11日提出

タイトル
計算可能測度論

種類
総説論文

ジャーナル
数理解析研究所講究録
RIMS共同研究(公開型)証明論と証明活動
2017年12月25日〜27日

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rims