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## $L^1$-computability, layerwise computability and Solovay reducibility

**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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preprint

**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.

## Unpredictability of initial points

**News**

25 Dec 2013, the slide file was uploaded

**Title**

Unpredictability of initial points

**Type**

RIMS Workshop: Dynamical Systems and Computation

**Download**

miyabe-DSC

## Variants of Layerwise Computability

**News**

9 Dec 2013, the slide file was uploaded.

2 Dec 2013, the talk was given.

**Title**

Variants of Layerwise Computability

**Type**

ARGENTINA-JAPAN-NEW ZEALAND WORKSHOP, UNIVERSITY OF AUCKLAND

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slide

## Uniform Kurtz randomness

**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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preprint

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

**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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preprint

**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.

## Almost uniform weak n-randomness

**News**

27 Sep 2013, the slide file was uploaded.

**Title**

Almost uniform weak n-randomness

**Type**

Eighth International Conference on Computability, Complexity and Randomness (CCR 2013)

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ccr-cs

## Being a Lebesgue point for integral tests

**News**

18 Sep 2013, the slide file was uploaded.

**Title**

Being a Lebesgue point for integral tests

**Type**

Asian Logic Conference 2013

**Download**

slide

## $L^1$-computability and Schnorr randomness

**News**

26 August 2013, the slide file was uploaded.

**Title**

$L^1$-computability and Schnorr randomness

**Type**

In a seminar at Universität der Bundeswehr München

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slide

## The emergence of probability from randomness and games

**News**

22 August 2013, the slides were uploaded.

**Title**

The emergence of probability from randomness and games

(with A. Takemura)

**Type**

Invited talk at Modeling Market Dynamics and Equilibrium – New Challenges, New Horizons

**Download**

Slide