A note on cellular automata

TL;DR

This paper introduces a new algebraic perspective on cellular automata by defining a monoid structure on uniformly continuous functions, providing an alternative proof of the Curtis-Hedlund theorem.

Contribution

It establishes an isomorphism between uniformly continuous functions and cellular automata monoids, offering a novel definition and proof approach.

Findings

Monoid structure on uniformly continuous functions is isomorphic to cellular automata monoid.

Provides a new, simpler proof of the Curtis-Hedlund theorem.

Offers an alternative algebraic characterization of cellular automata.

Abstract

For an arbitrary group $G$ and arbitrary set $A$, we define a monoid structure on the set of all uniformly continuous functions $A^G\to A$ and then we show that it is naturally isomorphic to the monoid of cellular automata $\mathrm{CA}(G, A)$. This gives a new equivalent definition of a cellular automaton over the group $G$ with alphabet set $A$. We use this new interpretation to give a simple proof of the theorem of Curtis-Hedlund.

Equations

$$T(x)(g)=\mu ((g^{-1}\cdot x{)}_{{|}_S}),$$

$$(f_1\ast f_2)(x)=f_1((f_2(g^{-1}\cdot x){)}_{g\in G}).$$

$$f_T(x)=T(x)(1).$$

$$W_S=\{(x,y)\in A^G\times A^G:x_{{|}_S}=y_{{|}_S}\}.$$

$$(f_T\times f_T)(x,y)=(f_T(x),f_T(y)).$$

$$x_{{|}_S}=y_{{|}_S}\Rightarrow T(x)(1)=T(y)(1),$$

$$(f_T\times f_T{)}^{-1}({\Delta }_A)\in \mathcal{U},$$

$$f_{T_1\circ T_2}(x)$$

$$T(h\cdot x)(g)$$

$$\{W_{\Omega }:\Omega \subseteq G,|\Omega |<\infty \}$$

$$x_{{|}_S}=y_{{|}_S}\Rightarrow f(x)=f(y)\Rightarrow T(x)(1)=T(y)(1).$$

$$W_{S\cap S^{\prime }}\subseteq W_S\circ W_{S^{\prime }}\subseteq V\circ V\subseteq V.$$

$$(f(g^{-1}\cdot x){)}_{g\in G}\in X,$$

$$(f_1\ast f_2)(x)=f_1((f_2(g^{-1}\cdot x){)}_{g\in G}).$$

Full text

Let $G$ be a group and $A$ be a non-empty set. The set $A^{G}$ consists of all configurations $x:G\to A$. We suppose that the sets $A^{G}$ and $A$ are equipped with the pro-discrete and discrete uniform structures, respectively (see [1] for basic definitions). We also suppose that the group $G$ acts on $A^{G}$ by shift: $(g\cdot x)(h)=x(g^{-1}h)$, for all $g,h\in G$ and $x\in A^{G}$. Recall that a cellular automaton over $G$ with alphabet set $A$ is a map $T:A^{G}\to A^{G}$ such that there exists a finite subset $S\subseteq G$ and a function $\mu:A^{S}\to A$ with the following property: for all $x\in A^{G}$ and all $g\in G$, we have

[TABLE]

where ${|_{S}}$ denotes the restriction. Any such a set $S$ is a called a memory set and $\mu$ is called a local defining function. Let $\mathrm{CA}(G,A)$ be the set of all such cellular automata. This set is a monoid with the ordinary composition of mappings. The reader may see [2] for detailed discussion.

Let $\mathcal{U}(A^{G},A)$ be the set of all uniformly continuous functions from $A^{G}$ to $A$. Define a binary operation $\ast$ on this set by

[TABLE]

We soon will see that $\mathcal{U}(A^{G},A)$ is a monoid. The identity of this monoid is the projection map $p_{1}:A^{G}\to A$ defined by $p_{1}(x)=x(1)$, where $1$ is the identity of $G$. For any $T\in\mathrm{CA}(G,A)$, we define a new map $f_{T}:A^{G}\to A$ by

[TABLE]

Theorem A. *The map $T\mapsto f_{T}$ is an isomorphism between the monoids $\mathrm{CA}(G,A)$ and $\mathcal{U}(A^{G},A)$. *

[TABLE]

Let $\mathcal{U}$ denote the pro-discrete uniform structure over $A^{G}$. We know that $W_{S}\in\mathcal{U}$. Consider the map $f_{T}\times f_{T}:A^{G}\times A^{G}\to A\times A$ defined by

[TABLE]

We have the implication

[TABLE]

and this means that $W_{S}\subseteq(f_{T}\times f_{T})^{-1}(\Delta_{A})$, where $\Delta_{A}$ is the diagonal of $A\times A$. This shows that

[TABLE]

and hence $f_{T}$ is uniformly continuous. Now, for arbitrary automata $T_{1}$ and $T_{2}$, we have

[TABLE]

This shows that the map $T\mapsto f_{T}$ is a homomorphism. Note that if $f_{T_{1}}=f_{T_{2}}$, then for any $x\in A^{G}$, we have $T_{1}(x)(1)=T_{2}(x)(1)$, and since $T_{1}$ and $T_{2}$ are $G$-equivariant, so $T_{1}=T_{2}$, proving that the map is injective.

Now, suppose that $f\in\mathcal{U}(A^{G},A)$. Define a map $T:A^{G}\to A^{G}$ by $T(x)(g)=f(g^{-1}\cdot x)$. First, note that $T$ is $G$-equivariant: for any $h\in G$, we have

[TABLE]

Since $f$ is uniformly continuous, we have $(f\times f)^{-1}(\Delta_{A})\in\mathcal{U}$. On the other hand, we know that the set

[TABLE]

is a basis for the pro-discrete uniform structure over $A^{G}$. Hence there exists a finite subset $S\subseteq G$ such that $W_{S}\subseteq(f\times f)^{-1}(\Delta_{A})$. In other words

[TABLE]

This shows that $T$ is a cellular automaton with the memory set $S$. Clearly $f_{T}=f$ and this shows that the map $T\mapsto f_{T}$ is surjective. Therefore, we proved that $\mathcal{U}(A^{G},A)$ is a monoid and it is isomorphic to $CA(G,A)$. ∎

As a result, we now have a very easy definition of a cellular automaton: any uniformly continuous map $f:A^{G}\to A$ is a cellular automaton! As an application, we reprove the theorem of Curtis and Hedlund (see [2]):

Corollary A. *Let $A$ be finite and $T:A^{G}\to A^{G}$ be continuous and $G$-equivariant. Then $T$ is a cellular automaton. *

As another application, we prove the existence of the minimal memory set for a cellular automaton.

Corollary B. *Let $T$ be a cellular automaton and $S$ and $S^{\prime}$ be two memory sets for $T$. Then $S\cap S^{\prime}$ is also a memory set. *

[TABLE]

This proves that $S\cap S^{\prime}$ is a memory set. ∎

Finally, we must say that a similar statement is true for any arbitrary subshift $X\leq A^{G}$, after a small modification: Let $\mathcal{U}_{0}(X,A)$ be the set of all uniformly continuous functions $f:X\to A$, with the further property that

[TABLE]

for all $x\in X$. Then we can define the binary operation

[TABLE]

on the set $\mathcal{U}_{0}(X,A)$ and it becomes a monoid again. We can prove then the next theorem.

Theorem B. There is a natural isomorphism between $\mathcal{U}_{0}(X,A)$ and the monoid of all cellular automata $X\to X$.