Towards a Universal Measure of Complexity

TL;DR

This paper proposes a new universal measure of complexity based on a non-linear transformation of entropy, addressing limitations of entropy as a complexity indicator and applicable to dynamic and critical systems.

Contribution

It introduces a novel complexity measure grounded in Gell-Mann's view, emphasizing the distance from less complex states and supporting diverse system analyses.

Findings

Most complex states are highly mixed, combining regularity and disorder.

The measure is flexible and extendable for various systems.

It effectively analyzes critical behavior and dynamic complexity.

Abstract

Recently it has been argued that entropy can be a direct measure of complexity, where the smaller value of entropy indicates lower system complexity, while its larger value indicates higher system complexity. We dispute this view and propose a universal measure of complexity based on the Gell-Mann's view of complexity. Our universal measure of complexity bases on a non-linear transformation of time-dependent entropy, where the system state with the highest complexity is the most distant from all the states of the system of lesser or no complexity. We have shown that the most complex is optimally mixed states consisting of pure states i.e., of the most regular and most disordered which the space of states of a given system allows. A parsimonious paradigmatic example of the simplest system with a small and a large number of degrees of freedom, is shown to support this methodology. Several important features of this universal measure are pointed out, especially its flexibility (i.e., its openness to extensions), ability to the analysis of a system critical behavior, and ability to study the dynamic complexity.