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Weak L^1-computability and Limit L^1-computability

The result in this paper was included in L1-computability, layerwise computability and Solovay reducibility

News
Mar 12, 2012, Draft

Title
Weak L^1-computability and Limit L^1-computability

Type
Extended abstract

Journal
in preparation

Abstract
The class of the differences between two integral tests for Schnorr ran- domness is an important class related to Schnorr randomness. In this paper we study other randomness versions. We also claim that Solovay reducibility for lower semicomputable functions generalizes layerwise com- putability.

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in preparation

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L^1-computability and weak L^1-computability

Title
L^1-computability and weak L^1-computability

Type
Talk at “Kyoto computable analysis Symposium 2012” on 24-27 Feb 2012.

Abstract
Computable functions are simple functions and
in Weihrauch approach computable functions are always continuous.
However there are some simple discontinuous functions such as the
floor function.
Therefore we need another mathematical notion to measure simplicity.
One candidate is L^1-computability, which was introduced by Pour-El
and Richard 1989.
In this talk I show you that more effectivised version of L^1-computability
has a strong connection with Schnorr randomness
and that some weaker versions of L^1-computability has connections
with some stronger randomness notions.

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slides

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Schnorr Layerwise Computability

Title
Schnorr Layerwise Computability

Type
Talk at “Workshop on Proof Theory and Computability Theory 2012” on 20-23 Feb 2012.
Workshop website

Abstract
In order to formalize the notion of randomness mathematically,
the theory of algorithmic randomness uses computability theory.
Recent researches show that some notions in algorithmic randomness conversely
are useful to study computable analysis.
One example is layerwise computability defined by Hoyrup and Rojas 2009.
In this talk I introduce Schnorr layerwise computability,
which is a Schnorr-randomness version,
and explain why this is a more natural notion.

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slides

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An integral test for Schnorr randomness and its application

The result in this paper was included in L1-computability, layerwise computability and Solovay reducibility

News
Jan 26, 2012, Submitted to a conference

Title
An integral test for Schnorr randomness and its application

Type
Fullpaper

Journal
Submitted

Abstract
The author proposed in the previous paper that a characterization of a randomness notion by integral tests is a useful tool to study the relation between algorithmic randomness and computable analysis. In this paper we give a version of Schnorr randomness. With this result we show the connection between L1-computability and Schnorr layerwise computability. Finally we apply them to study the points on which two Radon-Nikodym derivatives are equal.

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preprint