Category: Publication

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.

News
17 July 2013, published
27 Mar 2013, accepted
19 Sep 2012, submitted

Title
L1-computability, layerwise computability and Solovay reducibility

Type
Full paper

Journal
Computability, 2:15-29, 2013.

Abstract
We propose a hierarchy of classes of functions that corresponds to the hierarchy of randomness notions. Each class of functions converges at the corresponding random points. We give various characterizations of the classes, that is, characterizations via integral tests, L1-computability and layerwise computability. Furthermore, the relation among these classes is formulated using Solovay reducibility for lower semicomputable functions.

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Correction
Proposition 2.3.
Let $\mu$ be a computable measure on a computable metric space.
Then there exists a computable sequence $\{r_n\}$ such that $\mu(\overline{B}(\alpha_i,r_j)\setminus B(\alpha_i, r_j))$ for all $i$ and $j$.

This statement should be the following.
Proposition 2.3.
Let $\mu$ be a computable measure on a computable metric space.
Then there exists a computable sequence $\{r_n\}$ such that
$\{ r_0,r_1, … \}$ is dense in the interval $(0 , \infty)$ and $\mu(\overline{B}(\alpha_i,r_j)\setminus B(\alpha_i, r_j))$ for all $i$ and $j$.

This problem was pointed out by K. Weihrauch on 19 Jan 2014. I appreciate his notice.

News
4 Nov 2013, Published online
16 May 2013, Submitted to a Journal

Title
Uniform Kurtz randomness
(with Takayuki Kihara)

Type
Fullpaper

Journal
Journal of Logic and Computation, 24 (4): 863-882, 2014
doi: 10.1093/logcom/ext054

Abstract
We propose studying uniform Kurtz randomness, which is the uni- form relativization of Kurtz randomness. This notion has more natural properties than the usual relativization. For instance, van Lambalgen’s theorem holds for uniform Kurtz randomness while not for (the usual relativization of) Kurtz randomness. Another advantage is that lowness for uniform Kurtz randomness has many characterizations, such as those via complexity, martingales, Kurtz tt-traceability, and Kurtz dimensional measure.

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News
Oct, 2013. Online
Sep 3, 2011. Accepted
June 7, 2011. Submitted

Title
An optimal superfarthingale and its convergence over a computable topological space

Type
Conference paper

Conference
Solomonoff 85th Memorial Conference at Melbourne, Australia.

Lecture Notes in Computer Science
Volume 7070 2013
Algorithmic Probability and Friends. Bayesian Prediction and Artificial Intelligence
Papers from the Ray Solomonoff 85th Memorial Conference, Melbourne, VIC, Australia, November 30 – December 2, 2011
http://link.springer.com/book/10.1007/978-3-642-44958-1

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Abstract
We generalize the convergenece of an optimal semimeasure
to a real probability in algorithmic probability by using game-theoretic
probability theory and the theory of computable topology. Two lemmas
in the proof give as corollary the existence of an optimal test and an
optimal integral test, which are important from the point of view of
algorithmic randomness. We only consider an SCT3 space, where we
can approximate the measure of an open set. Our proof of almost-sure
convergence to the real probability by a superfarthingale indicates why
the convergence in Martin-L¨of sense does not hold.

News
19 Mar 2013, Accepted
25 Aug 2012, Submitted to a Journal

Title
The law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges
(with Akimichi Takemura)

Type
Fullpaper

Journal
Stochastic Processes and their Applications, 123, 3132-3152, 2013.

Abstract
We prove both the validity and the sharpness of the law of the iter- ated logarithm in game-theoretic probability with quadratic and stronger hedges.

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