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

Notre Dame Journal of Formal Logic 50 (4):381-391 (2009)
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
Keywords Martin-Lof randomness   prefix-free Kolmogorov complexity
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DOI 10.1215/00294527-2009-017
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David Diamondstone (2012). Low Upper Bounds in the LR Degrees. Annals of Pure and Applied Logic 163 (3):314-320.

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