Monthly Archives: June 2010

An extension of van Lambalgen’s Theorem to infinitely many relative 1-random reals

Title
An extension of van Lambalgen’s Theorem to infinitely many relative 1-random reals

Type
Fullpaper

Journal
Notre Dame Journal of Formal Logic, 51(3):337-349, 2010.
Permanent link to this document in this Journal.
Received November 30, 2009; accepted December 8, 2009; printed June 16, 2010

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preprint
Japanese summary

Abstract
Van Lambalgen’s Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen’s Theorem by considerint the join of infinitely many reals which are random relative to each other.
In addition, we study computability of the reals in the range of Omega operators. It is known that $\Omega^{\phi’}$ is high. We extend this result to that $\Omega^{\phi^{(n)}}$ is $\textrm{high}_n$. We also prove that there exists A such that, for each n, the real $\Omega^A_M$ is highn for some universal Turing machine M by using the extended van Lambalgen’s Theorem.

Cited by
@article{bienvenu2010ergodic,
title={{Ergodic-type characterizations of algorithmic randomness}},
author={Bienvenu, L. and Day, A. and Mezhirov, I. and Shen, A.},
journal={Programs, Proofs, Processes},
pages={49–58},
year={2010},
publisher={Springer}
}
@article{bienvenu2011constructive,
title={A constructive version of Birkhoffʼs ergodic theorem for Martin–L{\”o}f random points},
author={Bienvenu, L. and Day, A. and Hoyrup, M. and Mezhirov, I. and Shen, A.},
journal={Information and Computation},
year={2011},
publisher={Elsevier}
}
@article{yu2011characterizing,
title={Characterizing strong randomness via Martin-L{\”o}f randomness},
author={Yu, L.},
journal={Annals of Pure and Applied Logic},
year={2011},
publisher={Elsevier}
}

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Algorithmic randomness

Title
Algorithmic randomness

Type
Talk

This talk in RIMS colloquium in Kyoto in Japan on Jun 16, 2010.
This talk is a survey talk about algorithmic randomness.

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slide
Attached slides are in Japanese.

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Algorithmic randomness over general spaces

Title
Algorithmic randomness over general spaces

Type
Talk

This talk in RIMS CS seminar in Kyoto in Japan on Jun 10, 2010.
This talk is based on the paper Algorithmic randomness over general spaces.

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slide
Attached slides are in English.

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