Complexity, parallel computation and statistical physics

TL;DR

This paper introduces the concept of depth as a measure of complexity in natural systems, linking the amount of parallel computation needed to simulate a system with its physical complexity.

Contribution

It formalizes the notion of depth as an objective measure of complexity, applicable to systems in statistical physics, and demonstrates its relevance through various examples.

Findings

Depth correlates with physical complexity in statistical physics systems.

Systems with greater depth require more parallel computational steps to simulate.

Depth serves as a useful proxy for understanding complexity in natural systems.

Abstract

The intuition that a long history is required for the emergence of complexity in natural systems is formalized using the notion of depth. The depth of a system is defined in terms of the number of parallel computational steps needed to simulate it. Depth provides an objective, irreducible measure of history applicable to systems of the kind studied in statistical physics. It is argued that physical complexity cannot occur in the absence of substantial depth and that depth is a useful proxy for physical complexity. The ideas are illustrated for a variety of systems in statistical physics.