Effective Complexity and its Relation to Logical Depth

TL;DR

This paper formalizes the concept of effective complexity, explores its properties, and establishes a relationship with Bennett's logical depth, showing how complexity influences the depth of strings.

Contribution

It provides a rigorous formal definition of effective complexity and proves its fundamental properties, linking it to logical depth and clarifying their interplay.

Findings

Incompressible strings are effectively simple.

Strings with high effective complexity have large logical depth.

Effective complexity can be close to the string length.

Abstract

Effective complexity measures the information content of the regularities of an object. It has been introduced by M. Gell-Mann and S. Lloyd to avoid some of the disadvantages of Kolmogorov complexity, also known as algorithmic information content. In this paper, we give a precise formal definition of effective complexity and rigorous proofs of its basic properties. In particular, we show that incompressible binary strings are effectively simple, and we prove the existence of strings that have effective complexity close to their lengths. Furthermore, we show that effective complexity is related to Bennett's logical depth: If the effective complexity of a string $x$ exceeds a certain explicit threshold then that string must have astronomically large depth; otherwise, the depth can be arbitrarily small.