Time-Bounded Kolmogorov Complexity and Solovay Functions
Lecture Notes in Computer Science
A Solovay function is a computable upper bound g for prefixfree 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 c0 and all computable functions t such that c0n ≤ 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 Ωg is Martin-Löf random. Letting Ωg = 2 −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 Ω K t is 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 O(g(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 ct > 0 such that for all m it holds that C t (B m) ≥ ct · m, 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 C t (A m) ≤ log m + c. By similar methods it can be shown that any high degree contains a set B such that C t (B m) ≥ + 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.