Two-Dimensional Kolmogorov Complexity and Validation of the Coding Theorem Method by Compressibility

TL;DR

This paper introduces a new measure of n-dimensional algorithmic complexity based on Turing machines and algorithmic probability, validating it through experiments and applying it to classify cellular automata.

Contribution

It presents a novel n-dimensional complexity measure using deterministic Turing machines and validates it against compression methods and cellular automata.

Findings

The measure is stable under formalism changes.

Results align with lossless compression estimates.

Successfully classifies cellular automata complexity.

Abstract

We propose a measure based upon the fundamental theoretical concept in algorithmic information theory that provides a natural approach to the problem of evaluating $n$-dimensional complexity by using an $n$-dimensional deterministic Turing machine. The technique is interesting because it provides a natural algorithmic process for symmetry breaking generating complex $n$-dimensional structures from perfectly symmetric and fully deterministic computational rules producing a distribution of patterns as described by algorithmic probability. Algorithmic probability also elegantly connects the frequency of occurrence of a pattern with its algorithmic complexity, hence effectively providing estimations to the complexity of the generated patterns. Experiments to validate estimations of algorithmic complexity based on these concepts are presented, showing that the measure is stable in the face of some changes in computational formalism and that results are in agreement with the results obtained using lossless compression algorithms when both methods overlap in their range of applicability. We then use the output frequency of the set of 2-dimensional Turing machines to classify the algorithmic complexity of the space-time evolutions of Elementary Cellular Automata.