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

Miller (J Symb Log 69(3):907–913, 2004, http://projecteuclid.org/euclid.jsl/1096901774) and independently Nies et al. (J Symb Log 70(2):515–535, 2005) gave a complexity characterization of 2-random sequences in terms of plain Kolmogorov complexity $${C(\cdot)}$$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 (Notre Dame J Form Log 50(4):381–391, 2009) showed that prefix complexity $${K(\cdot)}$$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 elementary proofs. Miller (J Symb Log 69(3):907–913, 2004, http://projecteuclid.org/euclid.jsl/1096901774) also gave a quantitative version of the first result: the $${\mathbf{0}'}$$0′-randomness deficiency of a sequence $${\omega}$$ω equals lim inf$${_n [n - C(\omega_1\dots\omega_n)] + O(1)}$$n[n-C(ω1⋯ωn)]+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: $${\mathbf{0}'}$$0′-randomness deficiency equals lim inf$${_n [n + KP(n) - K(\omega_1\dots\omega_n)]+ O(1)}$$n[n+KP(n)-K(ω1⋯ωn)]+O(1).