## Notre Dame Journal of Formal Logic

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

Kenshi Miyabe

#### 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 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.

#### Article information

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

Dates
First available in Project Euclid: 18 August 2010

http://projecteuclid.org/euclid.ndjfl/1282137986

Digital Object Identifier
doi:10.1215/00294527-2010-020

Zentralblatt MATH identifier
05773615

Mathematical Reviews number (MathSciNet)
MR2675686

#### Citation

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. http://projecteuclid.org/euclid.ndjfl/1282137986.

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