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 $\mathrm{\Omega}_g=\sum 2^{-g(n)}$, 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 $\mathrm{\Omega}_{K^t}$ 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 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 C$^t(B\upharpoonright m) \geq c_t \cdot m$, 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$^t(A\upharpoonright m) \leq \log m + c$. By similar methods it can be shown that any high degree contains a set B such that C$^t(B\upharpoonright m) \geq^+ m/4$. 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.