Everywhere complex sequences and probabilistic method

Abstract

The main subject of the paper is everywhere complex sequences. An everywhere complex sequence is a sequence that does not contain substrings of Kolmogorov complexity less than $\alpha n-O(1)$ where $n$ is the length of substring and $\alpha$ is a constant between 0 and 1. First, we prove that no randomized algorithm can produce everywhere complex sequence with positive probability. On the other hand, for weaker notions of everywhere complex sequences the situation is different. For example, there is a probabilistic algorithm that produces (with probability~1) sequences whose substrings of length $n$ have complexity $\sqrt{n}-O(1)$. Finally, one may replace the complexity of a substring (in the definition of everywhere complex sequence) by its conditional complexity when the position is given. This gives a stronger notion of everywhere complex sequence, and no randomized algorighm can produce (with positive probability) such a sequence even if $\alpha n$ is replaced by $\sqrt{n}$, $\log^* n $ or any other monotone unbounded computable function.