Cellular Automata/Equivalence Classes

Introduction[edit | edit source]

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[edit | edit source]

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[edit | edit source]

or different symmetries

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

Equivalence classes[edit | edit source]

References[edit | edit source]

mix[edit | edit source]

Cellular Automata/Equivalence Classes

Contents

Introduction[edit | edit source]

Definiton of equivalence[edit | edit source]

Clustering rules[edit | edit source]

Equivalence classes[edit | edit source]

References[edit | edit source]

mix[edit | edit source]

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Cellular Automata/Equivalence Classes

Introduction[edit | edit source]

Definiton of equivalence[edit | edit source]

Clustering rules[edit | edit source]

Equivalence classes[edit | edit source]

References[edit | edit source]

mix[edit | edit source]

Navigation menu

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