On Kolmogorov Complexity of Random Very Long Braided Words

Posted on Posted in Theory


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. Surprisignly 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.

This is the distribution for the 1-pass reduction of Braid relations:

This is the 100-pass case i.e. the Braid relations were applied up to 100 times to reduce the long random word, and the curve for rate of change of Kolmogorov Complexity behaves like a Gamma distribution:

Basically the Braid relationships can be envisaged as the twists in the rubber band and propeller releases the additional complexity or entropy or energy but does so in integral quantized manner.

This is what the author has coined as Topological Oscillator vs. the usual primitive wave equations in complex numbers.


These braided forms are seen every day of our lives on surface of sun:


Source: http://spaceweather.com/images2013/01aug13/ch.jpg?PHPSESSID=au5hm6vncn2g428ub63920bui3

The white curves are indeed magnetic braids suspected to be the cause of immense heat on the surface of the sun, which is much hotter than its internals:


“A small NASA space telescope has revealed surprising magnetic braids of super-hot matter in the sun’s outer atmosphere, a find that may explain the star’s mysteriously hot corona, researchers say.”