Extracting kolmogorov complexity with applications to dimension zero-one laws

Abstract

We apply recent results on extracting randomness from independent sources to "extract" Kolmogorov complexity, For any α, ε > 0, given a string x with K(x) > α|x|, we show how to use a constant number of advice bits to efficiently compute another string y, |y| = Ω(|x|), with K(y) > (1 - ε)|y|. This result holds for both classical and space-bounded Kolmogorov complexity. We use the extraction procedure for space-bounded complexity to establish zero-one laws for polynomial-space strong dimension, Our results include: (i) If Dim pspace (E) > 0, then Dim pspace (E/O(1)) = 1. (ii) Dim(E/O(1) | ESPACE) is either 0 or 1. (iii) Dim(E/poly | ESPACE) is either 0 or 1. In other words, from a dimension standpoint and with respect to a small amount of advice, the exponential-time class E is either minimally complex or maximally complex within ESPACE. © Springer-Verlag Berlin Heidelberg 2006.