Spectral Representations and Global Maps of Cellular Automata Dynamics

TL;DR

This paper introduces a spectral Fourier-based representation of cellular automata that enables global mapping of their dynamics, potentially improving computational efficiency and revealing recursive structures.

Contribution

It presents a novel spectral encoding scheme for cellular automata of any topology, facilitating analysis and visualization of their global dynamics.

Findings

Spectral representation applicable to arbitrary CA topologies

Global maps reveal recursive structures in CA dynamics

Potential for analog computation with reduced heat production

Abstract

We present a spectral representation of any computation performed by a Cellular Automaton (CA) of arbitrary topology and dimensionality via an appropriate coding scheme in Fourier space that can be implemented in an analog machine ideally circumventing part of the overall waste heat production. We explore further consequences of this encoding and we provide a simple example based on the Game-of-Life where we find global maps for small lattices indicating an interesting underlying recursive structure.