Category: Publication

News
13 Sep 2013, Accepted by TOCS
23 Mar 2013, Submitted

Title
Schnorr triviality and its equivalent notions

Type
Full paper

Journal
Theory of Computing Systems (2015)
Volume 56, Issue 3 , pp 465-486
DOI: 10.1007/s00224-013-9506-8

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KURENAI

News
19 Jan 2014, submitted
24 Mar 2015, published

Title
Unified Characterizations of Lowness Properties via Kolmogorov Complexity
(with T. Kihara)

Type
Full paper

Journal
Archive for Mathematical Logic: Volume 54, Issue 3 (2015), Page 329-358
DOI: 10.1007/s00153-014-0413-8

Abstract
Consider a randomness notion $\mathcal C$.
A uniform test in the sense of $\mathcal C$ is a total computable procedure that each oracle $X$ produces a test relative to $X$ in the sense of $\mathcal C$.
We say that a binary sequence $Y$ is $\mathcal C$-random uniformly relative to $X$ if $Y$ passes all uniform $\mathcal C$ tests relative to $X$.

Suppose now we have a pair of randomness notions $\mathcal C$ and $\mathcal D$ where $\mathcal{C}\subseteq \mathcal{D}$, for instance Martin-L\”of randomness and Schnorr randomness. Several authors have characterized classes of the form Low($\mathcal C, \mathcal D$) which consist of the oracles $X$ that are so feeble that $\mathcal C \subseteq \mathcal D^X$. Our goal is to do the same when the randomness notion $\mathcal D$ is relativized uniformly: denote by Low$^\star$($\mathcal C, \mathcal D$) the class of oracles $X$ such that every $\mathcal C$-random is uniformly $\mathcal D$-random relative to $X$.

(1) We show that $X\in{\rm Low}^\star({\rm MLR},{\rm SR})$ if and only if $X$ is c.e.~tt-traceable if and only if $X$ is anticomplex if and only if $X$ is Martin-L\”of packing measure zero with respect to all computable dimension functions.

(2) We also show that $X\in{\rm Low}^\star({\rm SR},{\rm WR})$ if and only if $X$ is computably i.o.~tt-traceable if and only if $X$ is not totally complex if and only if $X$ is Schnorr Hausdorff measure zero with respect to all computable dimension functions.

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An article was published in Mathematical Seminar Oct 2014.

News
27 Sep 2014, Online
3 Aug 2014, Accepted in SPA
12 Feb 2014. Submitted

Title
Derandomization in Game-Theoretic Probability
(with A. Takemura)

Type
Full paper

Journal
Stochastic Processes and their Applications 125, 39-59, 2015

Abstract
We give a general method for constructing a deterministic strategy
of Reality from a randomized strategy in game-theoretic probability.
The construction can be seen as derandomization in game-theoretic probability.

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News
7 May 2014. Published online
Dec 2013. Accepted
May, 2012. Resubmit
Sep, 2011. Resubmitted a revised version
25 May, 2010. Submitted to a Journal

Title
Algorithmic randomness over general spaces

Type
Fullpaper

Journal
Math. Log. Quart. 60, No. 3, 184–204 (2014)
DOI 10.1002/malq.201200051

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Abstract
The study of Martin-Löf randomness on a computable metric space with a computable measure has had much progress recently.
In this paper we study Martin-Löf randomness on a more general space, that is, a computable topological space with a computable measure.
On such a space, Martin-Löf randomness may not be a natural notion because there is not a universal test, and Martin-Löf randomness and complexity randomness (defined in this paper) do not coincide in general. We show that SCT3 is a sufficient condition for the existence and the coincidence and study how much we can weaken the condition.