Notre Dame Journal of Formal Logic

Van Lambalgen's Theorem and High Degrees

Johanna N.Y. Franklin and Frank Stephan

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We show that van Lambalgen's Theorem fails with respect to recursive randomness and Schnorr randomness for some real in every high degree and provide a full characterization of the Turing degrees for which van Lambalgen's Theorem can fail with respect to Kurtz randomness. However, we also show that there is a recursively random real that is not Martin-Löf random for which van Lambalgen's Theorem holds with respect to recursive randomness.

Article information

Notre Dame J. Formal Logic Volume 52, Number 2 (2011), 173-185.

First available in Project Euclid: 28 April 2011

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03D32: Algorithmic randomness and dimension [See also 68Q30]

van Lambalgen's Theorem Schnorr randomness recursive randomness Kurtz randomness


Franklin, Johanna N.Y.; Stephan, Frank. Van Lambalgen's Theorem and High Degrees. Notre Dame J. Formal Logic 52 (2011), no. 2, 173--185. doi:10.1215/00294527-1306181.

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