Natural Complexity, Computational Complexity and Depth

TL;DR

This paper explores the concept of depth as a measure of complexity in natural systems, linking it to computational complexity and emphasizing its significance in systems with embedded computation.

Contribution

It introduces depth as a complexity measure for natural systems, compares it with other measures, and discusses its properties and implications.

Findings

Depth quantifies the shortest parallel computation for system states.

Large depth indicates systems with embedded computation.

Depth differs from other complexity measures in key properties.

Abstract

Depth is a complexity measure for natural systems of the kind studied in statistical physics and is defined in terms of computational complexity. Depth quantifies the length of the shortest parallel computation required to construct a typical system state or history starting from simple initial conditions. The properties of depth are discussed and it is compared to other complexity measures. Depth can only be large for systems with embedded computation.