Notre Dame Journal of Formal Logic

An Extension of van Lambalgen's Theorem to Infinitely Many Relative 1-Random Reals

Kenshi Miyabe


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 considering 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 $\textrm{high}_n$ for some universal Turing machine M by using the extended van Lambalgen's Theorem.

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Notre Dame J. Formal Logic Volume 51, Number 3 (2010), 337-349.

First available in Project Euclid: 18 August 2010

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Zentralblatt MATH identifier

Primary: 03D32: Algorithmic randomness and dimension [See also 68Q30]
Secondary: 03D25: Recursively (computably) enumerable sets and degrees

van Lambalgen's Theorem martingale high Omega operator


Miyabe, Kenshi. An Extension of van Lambalgen's Theorem to Infinitely Many Relative 1-Random Reals. Notre Dame J. Formal Logic 51 (2010), no. 3, 337--349. doi:10.1215/00294527-2010-020.

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