Non-approximability of the Randomness Deficiency Function

Abstract

Let x be a binary string of length n. Consider the set P _x of all pairs of integers (a,b) such that the randomness deficiency of x in a finite set S of Kolmogorov complexity at most a is at most b. The paper [4] proves that there is no algorithm that for every given x upper semicomputes the minimal deficiency function β _x (a) = min{b | (a,b) ∈ p _x }with precision n/log⁴ n. We strengthen this result in two respects. First, we improve the precision to n/4. Second, we show that there is no algorithm that for every given x enumerates a set at distance at most n/4 from P _x , which is easier than to upper semicompute the minimal deficiency function of x with the same accuracy.