On Time-Bounded Incompressibility of Compressible Strings and Sequences
E.G. Daylight (Univ. Amsterdam), W.M. Koolen (CWI), P.M.B. Vitanyi, (CWI, Univ Amsterdam)

TL;DR
This paper investigates the properties of compressible strings and sequences under time-bounded Kolmogorov complexity, showing many are incompressible within certain time bounds, with implications for complexity theory.
Contribution
It establishes that a significant fraction of low complexity strings are time-bounded incompressible and constructs sequences with specific incompressibility properties under time constraints.
Findings
A constant fraction of compressible strings are t-bounded incompressible.
Existence of uncountably many infinite sequences with initial segments that are compressible but t-bounded incompressible.
Countably many recursive sequences have all initial segments t-bounded incompressible.
Abstract
For every total recursive time bound , a constant fraction of all compressible (low Kolmogorov complexity) strings is -bounded incompressible (high time-bounded Kolmogorov complexity); there are uncountably many infinite sequences of which every initial segment of length is compressible to yet -bounded incompressible below ; and there are countable infinitely many recursive infinite sequence of which every initial segment is similarly -bounded incompressible. These results are related to, but different from, Barzdins's lemma.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Algorithms and Data Compression · Complexity and Algorithms in Graphs
