# Cellular Automata/Equivalence Classes

## Introduction

Stephen Wolfram was one of the first to focus on the complete set of CA rules. He observed all the rules that can be created using binary cells $k=2$ and a three cell neighborhood $m=2$. There are $k^{k^m}=2^{2^3}=256$ such rules. The number of rules grows exponentially with the number of cell states and even faster with the neighbourhood size. This vast number of rules is a limiting factor when observing the whole set of rules.

## Definiton of equivalence

In this definition a more formal name for rule is used, the local definition function.

The local definition function $f_1$ is equal to the local definition function $f_2$, if and ony if there exists a homomorfism $g$ of global CA states that

$f_1 f_2$

## Clustering rules

or different symmetries

1. input complement
2. output complement
3. reflection symmetry
4. rotation smetry (2D) ?can be produced by reflection no it can not
5. equivalence class