On Time-Bounded Incompressibility of Compressible Strings and Sequences

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 $t$, 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.