Random noise increases Kolmogorov complexity and Hausdorff dimension

by   Gleb Posobin, et al.

Consider a binary string x of length n whose Kolmogorov complexity is α n for some α<1. We want to increase the complexity of x by changing a small fraction of bits in x. This is always possible: Buhrman, Fortnow, Newman and Vereshchagin (2005) showed that the increase can be at least δ n for large n (where δ is some positive number that depends on α and the allowed fraction of changed bits). We consider a related question: what happens with the complexity of x when we randomly change a small fraction of the bits (changing each bit independently with some probability τ)? It turns out that a linear increase in complexity happens with high probability, but this increase is smaller than in the case of arbitrary change. We note that the amount of the increase depends on x (strings of the same complexity could behave differently), and give an exact lower and upper bounds for this increase (with o(n) precision). The proof uses the combinatorial and probabilistic technique that goes back to Ahlswede, Gács and Körner (1976). For the reader's convenience (and also because we need a slightly stronger statement) we provide a simplified exposition of this technique, so the paper is self-contained.


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