Counting dependent and independent strings

TL;DR

This paper provides estimations for the sizes of sets of strings based on their Kolmogorov complexity and various notions of dependency and independence among strings, advancing understanding in information theory.

Contribution

It introduces new estimations for the sizes of string sets with specific dependency and independence properties using Kolmogorov complexity.

Findings

Estimations for the size of strings with fixed dependency.

Bounds on the number of pairwise -independent strings.

Results on the size of mutually -independent string sets.

Abstract

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].