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## Prefix and plain Kolmogorov complexity characterizations of 2-randomness: simple proofs

Archive for Mathematical Logic. 2015. Vol. 54. No. 5. P. 615-629.

Joseph Miller [16] and independently Andre Nies, Frank Stephan and Sebastiaan Terwijn [18] gave a complexity characterization of 2-random sequences in terms of plain Kolmogorov complexity C: they are sequences that have infinitely many initial segments with O(1)-maximal plain complexity (among the strings of the same length). Later Miller [17] showed that prefix complexity K can also be used in a similar way: a sequence is 2-random if and only if it has infinitely many initial segments with O(1)-maximal prefix complexity (which is n + K (n) for strings of length n). The known proofs of these results are quite involved; in this paper we provide simple direct proofs for both of them. In [16] Miller also gave a quantitative version of the first result: the 0'-randomness deficiency of a sequence {\omega} equals lim inf [n - C ({\omega}1 . . . {\omega}n)] + O(1). (Our simplified proof can also be used to prove this.) We show (and this seems to be a new result) that a similar quantitative result is also true for prefix complexity: 0'-randomness deficiency equals lim inf [n + K (n) -- K ({\omega}1 . . . {\omega}n)] + O(1). Prefix and plain Kolmogorov complexity characterizations of 2-randomness: simple proofs (PDF Download Available). Available from: http://www.researchgate.net/publication/258082399_Prefix_and_plain_Kolmogorov_complexity_characterizations_of_2-randomness_simple_proofs [accessed Oct 20, 2015].