**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.

**Download**

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.