Dimensionality and randomness

TL;DR

This paper investigates how arranging random data into multi-dimensional structures affects their Kolmogorov complexity, revealing a relationship between dimensionality and complexity loss.

Contribution

It provides a quantitative and characterizing analysis of complexity loss when organizing random strings into arrays and trees, linking to negligible classes.

Findings

Complexity decreases with increasing dimensions.

Quantitative measures of complexity loss are established.

Relationship between array structure and negligible classes is characterized.

Abstract

Arranging the bits of a random string or real into k columns of a two-dimensional array or higher dimensional structure is typically accompanied with loss in the Kolmogorov complexity of the columns, which depends on k. We quantify and characterize this phenomenon for arrays and trees and its relationship to negligible classes.