On Time-Bounded Incompressibility of Compressible Strings and Sequences

Abstract

For every total recursive time bound $t$, a constant fraction of all compressible (low Kolmogorov complexity) strings is $t$-bounded incompressible (high time-bounded Kolmogorov complexity); there are uncountably many infinite sequences of which every initial segment of length $n$ is compressible to $\log n$ yet $t$-bounded incompressible below ${1/4}n - \log n$; and there are countable infinitely many recursive infinite sequence of which every initial segment is similarly $t$-bounded incompressible. These results are related to, but different from, Barzdins's lemma.Comment: 9 pages, LaTeX, no figures, submitted to Information Processing Letters. Changed and added a Barzdins-like lemma for infinite sequences with different quantification oreder, a fixed constant, and uncountably many sequence