Author: miyabe

News
July 16, 2011, Accepted.
June 16, 2011, Resubmitted to a Journal
Mar 29, 2011, Submitted to a Journal

Title
The diifference between optimality and universality

(Former)Degree of non-randomness and uniform Solovay reducibility

Type
Fullpaper

Journal
Logic Journal of the IGPL (2012) 20 (1): 222-234.
abstract page

Abstract
We introduce a degree of non-randomness using a test concept.
The degree implies a notion of an optimal Martin-L\”of test, which is different from a universal test.
In the latter half we generalize Solovay reducibility.
Solovay reducibility is a measure of relative randomness between two reals.
We introduce uniform solovay reducibility, which is a measure of relative randomness between two sequences of reals.
Finally we prove that a sequence is uniform Solovay complete iff it is the sequence of measures of an optimal Martin-L\”of test.

The content of the paper will be merged into Algorithmic randomness over general spaces.

News
June 13, 2011, Rejected
Sep 21, 2010, Submitted to a Journal

Title
A computable topological space of measures

Type
Fullpaper

Journal
submitted

Download
Japanese summary

Abstract
We show that the space of bounded non-negative Borel measures on
a computable topological space is also a computable topological space
with A-topology. Then we dene computable measures which may not
be probabilistic and may be even innite. We also study randomness for
non-negative Borel measures which may not be probabilistic.

News
Mar 19, 2011. Added citation information
Mar 7, 2011. Published online
May 18, 2010. The paper “Truth-table Schnorr randomness and truth-table reducible randomness” is accepted by Mathematical Logic Quarterly on May 18, 2010.

Title
Truth-table Schnorr randomness and truth-table reducible randomness

Type
Fullpaper

Journal
Mathematical Logic Quarterly 57(3):323-338, 2011
DOI 10.1002/malq.200910128
Journal Page

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preprint
Japanese summary

Abstract
Schnorr randomness and computably randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness (defined by Franklin and Stephan too only by martingales) and truth-table reducible randomness, for which we prove that van Lambalgen’s Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen’s Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness.

Cited by
@article{franklin2009van,
title={{van Lambalgen’s Theorem and high degrees}},
author={Franklin, J.N.Y. and Stephan, F.},
year={2009},
publisher={Submitted}
}
@misc{bienvenucharacterizing,
title={Characterizing lowness for Demuth randomness},
author={Bienvenu, L. and Downey, R. and Greenberg, N. and Nies, A. and Turetsky, D.},
publisher={Submitted}
}