Time-Bounded Kolmogorov Complexity and Solovay Functions

Abstract

A Solovay function is an 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 computable 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 =∑2^−g(n) is Martin-Löf random. 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 $\Omega _{ \textnormal{K}^{t}}$ 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 computable Solovay functions g. Furthermore, this equivalence remains true when the universal quantification over all computable Solovay functions in the second statement is restricted either to all functions of the form K^t for some function t as above or to a single function K^t of this form.

Finally, we investigate into the plain Kolmogorov complexity C and its time-bounded variant C^t of initial segments of computably enumerable sets. Our main theorem here 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 C^t(B↾m)≥c _t ⋅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)≤logm+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 et al. (Theor. Comput. Sci. 132(1–2):37–70, 1994) that every high degree contains a strongly deep set.