Time-Bounded Kolmogorov Complexity and Solovay Functions

Abstract

Abstract.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 MartinLöf 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.

Solovay functions and their applications in algorithmic randomness

“…The characterization of weak Solovay functions in Theorem 2.2 holds also when relativized to any oracle by virtually the same proof. In fact, Hölzl, Kräling and Merkle [17] demonstrated the relativized version, as a joint generalization of, first, the already mentioned corresponding characterization of Solovay functions by Bienvenu and Downey [5], and, second, a remarkable characterization of the sequences that are weakly low for K by Miller [29]. For the sake of completeness, we shortly review the latter result, which now becomes a special case of Theorem 2.2.…”

“…This characterization extends trivially to the special case of Solovay functions in the sense that among all computable functions exactly the Solovay functions possess the property under consideration. Note that in case of the first property the latter characterization was shown in a conference article by the first two authors of this paper [5], whereas the corresponding characterization of weak Solovay functions was subsequently demonstrated by Hölzl, Kräling and Merkle [17], see Section 2.2 for further details.…”

“…The next theorem provides the fundamental characterization of Solovay functions in terms of the sum n 2 −f (n) . It was first proven in an earlier conference paper by Bienvenu and Downey [5] for computable functions, and then extended by Hölzl, Kräling and Merkle [17] to right-c.e. functions.…”

Randomness, Computation and Mathematics

“…(An observation of Hölzl, Kräling, and Merkle [40].) Building on earlier work of Gács, and of Miller and Yu, recently Merkle, Miller and Nies have proven that a set A is 1-random iff C(A n) ≥ n − g(n) − O(1) for any Solovay function g. In fact by themselves, Solovay functions characterize 1-randomness.…”

Kolmogorov complexity and computably enumerable sets

“…Note that log n is an upper bound of C(n). Hölzl, Kräling and Merkle [HKM09] observed that, in fact, for every c.e. set A there are infinitely many n such that C(A ↾ n ) is bounded by C(n) (plus an additive constant); this also holds for prefix-free complexity.…”