Time-bounded kolmogorov complexity and solovay functions

Abstract

A Solovay function is a computable upper bound g for prefix-free Kolmogorov complexity K that is nontrivial in the sense that g agrees with K, up to some additive constant, on infinitely many places n. We obtain natural examples of Solovay functions by showing that for some constant c 0 and all computable functions t such that c 0 n≤t(n), the time-bounded version K t of K is a Solovay function. By unifying results of Bienvenu and Downey and of Miller, we show that a right-computable upper bound g of K is a Solovay function if and only if Ω g is Martin-Löf random. Letting Ω g =∑22-g(n), we obtain as a corollary that the Martin-Löf randomness of the various variants of Chaitin's Ω extends to the time-bounded case in so far as Ωktis Martin-Löf random for any t as above. As a step in the direction of a characterization of K-triviality in terms of jump-traceability, we demonstrate that a set A is K-trivial if and only if A is Og(n)-K(n)-jump traceable for all Solovay functions g, where the equivalence remains true when we restrict attention to functions g of the form K t , either for a single or all functions t as above. Finally, we investigate the plain Kolmogorov complexity C and its time-bounded variant C t of initial segments of computably enumerable sets. Our main theorem here is a dichotomy similar to Kummer's gap theorem and asserts that every high c.e. Turing degree contains a c.e. set B such that for any computable function t there is a constant c t >0 such that for all m it holds that Ct(Bm)≥ctm , whereas for any nonhigh c.e. set A there is a computable time bound t and a constant c such that for infinitely many m it holds that Ct(Bm)≤ log c. By similar methods it can be shown that any high degree contains a set B such that Ct(Bm)≥m/4 . The constructed sets B have low unbounded but high time-bounded Kolmogorov complexity, and accordingly we obtain an alternative proof of the result due to Juedes, Lathrop, and Lutz [JLL] that every high degree contains a strongly deep set. © 2009 Springer Berlin Heidelberg.