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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