Notre Dame Journal of Formal Logic

The K-Degrees, Low for K Degrees,and Weakly Low for K Sets

Joseph S. Miller

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We call A weakly low for K if there is a c such that $K^A(\sigma)\geq K(\sigma)-c$ for infinitely many σ; in other words, there are infinitely many strings that A does not help compress. We prove that A is weakly low for K if and only if Chaitin's Ω is A-random. This has consequences in the K-degrees and the low for K (i.e., low for random) degrees. Furthermore, we prove that the initial segment prefix-free complexity of 2-random reals is infinitely often maximal. This had previously been proved for plain Kolmogorov complexity.

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Notre Dame J. Formal Logic Volume 50, Number 4 (2009), 381-391.

First available in Project Euclid: 11 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 68Q30
Secondary: 03D30 03D80

Martin-Lof randomness prefix-free Kolmogorov complexity


Miller , Joseph S. The K -Degrees, Low for K Degrees,and Weakly Low for K Sets. Notre Dame Journal of Formal Logic 50 (2009), no. 4, 381--391. doi:10.1215/00294527-2009-017.

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