Impossibility of independence amplification in Kolmogorov complexity theory

The paper studies randomness extraction from sources with bounded independence and the issue of independence amplification of sources, using the framework of Kolmogorov complexity. The dependency of strings $x$ and $y$ is ${\rm dep}(x,y) = \max\{C(x) - C(x \mid y), C(y) - C(y\mid x)\}$, where $C(\cdot)$ denotes the Kolmogorov complexity. It is shown that there exists a computable Kolmogorov extractor $f$ such that, for any two $n$-bit strings with complexity $s(n)$ and dependency $\alpha(n)$, it outputs a string of length $s(n)$ with complexity $s(n)- \alpha(n)$ conditioned by any one of the input strings. It is proven that the above are the optimal parameters a Kolmogorov extractor can achieve. It is shown that independence amplification cannot be effectively realized. Specifically, if (after excluding a trivial case) there exist computable functions $f_1$ and $f_2$ such that ${\rm dep}(f_1(x,y), f_2(x,y)) \leq \beta(n)$ for all $n$-bit strings $x$ and $y$ with ${\rm dep}(x,y) \leq \alpha(n)$, then $\beta(n) \geq \alpha(n) - O(\log n)$

Counting dependent and independent strings

The paper gives estimations for the sizes of the the following sets: (1) the set of strings that have a given dependency with a fixed string, (2) the set of strings that are pairwise \alpha independent, (3) the set of strings that are mutually \alpha independent. The relevant definitions are as follows: C(x) is the Kolmogorov complexity of the string x. A string y has \alpha -dependency with a string x if C(y) - C(y|x) \geq \alpha. A set of strings {x_1, \ldots, x_t} is pairwise \alpha-independent if for all i different from j, C(x_i) - C(x_i | x_j) \leq \alpha. A tuple of strings (x_1, \ldots, x_t) is mutually \alpha-independent if C(x_{\pi(1)} \ldots x_{\pi(t)}) \geq C(x_1) + \ldots + C(x_t) - \alpha, for every permutation \pi of [t]