On Kolmogorov Complexity of Random Very Long Braided Words

TL;DR

This paper investigates how Braid group relations significantly reduce the Kolmogorov Complexity of random words, revealing distinct reduction bands and suggesting a universal distribution pattern across algebraic structures.

Contribution

It introduces a novel empirical analysis of Kolmogorov Complexity reduction via Braid relations and proposes the universality of these distribution patterns beyond Braid groups.

Findings

Braid relations cause substantial complexity reduction in random words.

Distinct bands of complexity reduction are observed with gaps in between.

Empirical evidence suggests universality of these reduction distributions.

Abstract

Any positive word comprised of random sequence of tokens form a finite alphabet can be reduced (without change of length) using an appropriate size Braid group relationships. Surprisingly the Braid relations dramatically reduce the Kolmogorov Complexity of the original random word and do so in distinct bands of (rate of change) values with gaps in between. Distribution of these bands are estimated and empirical statistics collected by actually coding approximations to the Kolmogorov Complexity (in Mathematica 9.0). Lempel-Ziv-Welch lossless compression algorithm techniques used to estimate the distribution for gaped bands. Evidence provided that such distributions of reduction in Kolmogorov Complexity based upon Braid groups are universal i.e. they can model more general algebraic structures other than Braid groups.