Time-Bounded Kolmogorov Complexity and Solovay Functions

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 Kt 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-L¨of random. Letting Ωg = 2−g(n) , we obtain as a corollary that the Martin-L¨of randomness of the various variants of Chaitin’s Ω extends to the time-bounded case in so far as ΩKt is MartinL¨of 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 So...

• Type: Conference
• Year: 2009
• Where: MFCS
• Authors: Rupert Hölzl, Thorsten Kräling, Wolfgang Merkle
• Keywords: Computable Function, Kolmogorov Complexity, MFCS 2009, Solovay Functions