
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 and arbitrary set , we define a monoid structure on the set of all uniformly continuous functions and then we show that it is naturally isomorphic to the monoid of cellular automata . This gives a new equivalent definition of a cellular automaton over the group with alphabet set . We use this new interpretation to give a simple proof of the theorem of Curtis-Hedlund.
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Taxonomy
TopicsCellular Automata and Applications · semigroups and automata theory · Computability, Logic, AI Algorithms
A note on cellular automata
M. Shahryari
M. Shahryari
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
Abstract.
For an arbitrary group and arbitrary set , we define a monoid structure on the set of all uniformly continuous functions and then we show that it is naturally isomorphic to the monoid of cellular automata . This gives a new equivalent definition of a cellular automaton over the group with alphabet set . We use this new interpretation to give a simple proof of the theorem of Curtis-Hedlund.
Let be a group and be a non-empty set. The set consists of all configurations . We suppose that the sets and are equipped with the pro-discrete and discrete uniform structures, respectively (see [1] for basic definitions). We also suppose that the group acts on by shift: , for all and . Recall that a cellular automaton over with alphabet set is a map such that there exists a finite subset and a function with the following property: for all and all , we have
[TABLE]
where denotes the restriction. Any such a set is a called a memory set and is called a local defining function. Let 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 be the set of all uniformly continuous functions from to . Define a binary operation on this set by
[TABLE]
We soon will see that is a monoid. The identity of this monoid is the projection map defined by , where is the identity of . For any , we define a new map by
[TABLE]
Theorem A. *The map is an isomorphism between the monoids and . *
Proof.
We first show that is uniformly continuous. Suppose is a memory set for and put
[TABLE]
Let denote the pro-discrete uniform structure over . We know that . Consider the map defined by
[TABLE]
We have the implication
[TABLE]
and this means that , where is the diagonal of . This shows that
[TABLE]
and hence is uniformly continuous. Now, for arbitrary automata and , we have
[TABLE]
This shows that the map is a homomorphism. Note that if , then for any , we have , and since and are -equivariant, so , proving that the map is injective.
Now, suppose that . Define a map by . First, note that is -equivariant: for any , we have
[TABLE]
Since is uniformly continuous, we have . On the other hand, we know that the set
[TABLE]
is a basis for the pro-discrete uniform structure over . Hence there exists a finite subset such that . In other words
[TABLE]
This shows that is a cellular automaton with the memory set . Clearly and this shows that the map is surjective. Therefore, we proved that is a monoid and it is isomorphic to . ∎
As a result, we now have a very easy definition of a cellular automaton: any uniformly continuous map is a cellular automaton! As an application, we reprove the theorem of Curtis and Hedlund (see [2]):
Corollary A. *Let be finite and be continuous and -equivariant. Then is a cellular automaton. *
Proof.
Define a map by . Note that , so it is continuous. Since is compact, so is uniformly continuous and hence . This shows that there exists a cellular automaton such that . But since is -equivariant, it can be easily seen that , proving that is a cellular automaton. ∎
As another application, we prove the existence of the minimal memory set for a cellular automaton.
Corollary B. *Let be a cellular automaton and and be two memory sets for . Then is also a memory set. *
Proof.
Let be the corresponding uniformly continuous mapping and . We know that a finite set is a memory set for if and only if . Clearly . Let and choose such that and . This shows that . Hence
[TABLE]
This proves that is a memory set. ∎
Finally, we must say that a similar statement is true for any arbitrary subshift , after a small modification: Let be the set of all uniformly continuous functions , with the further property that
[TABLE]
for all . Then we can define the binary operation
[TABLE]
on the set and it becomes a monoid again. We can prove then the next theorem.
Theorem B. There is a natural isomorphism between and the monoid of all cellular automata .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] N. Bourbaki. Elements of Mathematics: General Topology (Part 1) . English translation, Springer.
- 2[2] T. Ceccherini-Silberstein, M. Coornaert. Cellular Automata and Groups . Springer, 2010.
