Garden-of-Eden theorem

Garden-of-Eden theorem

Edward F. Moore’s

Garden-of-Eden theorem

was the first

rigorous

statement of cellular automata theory.
Though originally stated for cellular automata on the plane,
it works in arbitrary dimension.

Theorem 1

(Moore, 1962)

Let

A

=

⟨

Q

,

N

,

f

⟩

be a

cellular automaton

on

Z

d

with finitely many states.
If

A

has two mutually erasable patterns,
then it has an orphan pattern.

In other words,

surjective

d

-dimensional cellular automata are pre-injective.

The same year, John Myhill proved the converse implication.

Theorem 2

(Myhill, 1962)

Let

A

=

⟨

Q

,

N

,

f

⟩

be a cellular automaton on

Z

d

with finitely many states.
If

A

has an orphan pattern,
then it has two mutually erasable patterns.

In other words,
pre-injective

d

-dimensional cellular automata are surjective.

Corollary 1

An

injective

d

-dimensional cellular automaton is surjective.

Theorems

1

and

2

hold because of the following statement.

Lemma 1

Let

a

>

0

,

d

≥

1

,

r

≥

1

,

k

≥

1

.
For every sufficiently large

n

,

(

a

k

d

-

1

)

n

d

<

a

(

k

⁢

n

-

2

⁢

r

)

d

(1)

Observe that,
if

a

=

|

Q

|

and

𝒩

=

{

x

∈

ℤ

d

∣

|

x

i

|

≤

r

⁢

∀

i

∈

{

1

,

…

,

d

}

}

,

(2)

then

(

a

k

⁢

n

)

d

is the number of possible hypercubic patterns of side

k

⁢

n

,
while

(

a

k

⁢

n

-

2

⁢

r

)

d

is the maximum number of their images via

f

.

Proof.

The inequality (

1

) is in fact satisfied
precisely by those

n

that

satisfy

log

a

⁡

(

a

k

d

-

1

)

<

(

k

-

2

⁢

r

n

)

d

,

which is true for

n

large enough since

log

a

⁡

(

a

k

d

-

1

)

<

k

d

while

lim

n

→

∞

⁡

(

k

-

2

⁢

r

/

n

)

d

=

k

d

.

□

For the rest of this entry, we will suppose
that the

neighborhood

index

𝒩

has the form (

2

).

Proof of Moore’s theorem.

Suppose

𝒜

has two mutually erasable patterns

p

1

,

p

2

:
it is not restrictive to suppose that their common

support

is a

d

-hypercube of side

k

.
Define an

equivalence relation

ρ

on patterns of side

n

by stating that

p

⁢

ρ

⁢

q

if and only if

f

⁢

(

p

)

=

f

⁢

(

q

)

:
since

p

1

and

p

2

are mutually erasable,

ρ

has at most

|

Q

|

k

d

-

1

equivalence classes

.

By the same criterion,
we define a family

{

ρ

n

}

n

≥

1

of equivalence relations between hypercubic patterns of side

k

⁢

n

.
Each such pattern can be subdivided
into

n

d

subpatterns of side

k

:
and it is clear that,
if two patterns are in

relation

ρ

n

,
then each of those subpatterns is in relation

ρ

.

But then, the relation

ρ

n

cannot have more than

(

|

Q

|

k

d

-

1

)

n

d

equivalence classes:
consequently, at most

(

|

Q

|

k

d

-

1

)

n

d

patterns of side

k

⁢

n

-

2

⁢

r

can have a

preimage

.
By Lemma

1

with

a

=

|

Q

|

,
for

n

large enough, some patterns of side

k

⁢

n

-

2

⁢

r

must be orphan.

□

Proof of Myhill’s theorem.

Let

p

be an orphan pattern for

𝒜

.
Again, it is not restrictive to suppose
that the support of

p

is a hypercube of side

k

.
For

n

≥

1

let

ν

n

be the number of patterns of side

k

⁢

n

that are

not

orphan.
Split each pattern of side

k

⁢

n

into

n

d

patterns of side

k

, as we did before.
Then

ν

n

cannot exceed the number of those
that do not contain a copy of

p

,
which in turn is at most

(

|

Q

|

k

d

-

1

)

n

d

.

Take

n

so large that (

1

) is satisfied:
for a fixed state

q

0

∈

Q

, and for

q

1

=

f

⁢

(

q

0

,

…

,

q

0

)

,
there are more

configurations

that take value

q

0

outside the hypercube

{

0

,

…

,

k

⁢

n

-

2

⁢

r

-

1

}

d

than there are configurations that take value

q

1

outside the hypercube

{

-

r

,

…

,

k

⁢

n

-

1

}

d

and are not

Gardens of Eden

.
As the configurations of the second type
are the images of those the first type,
there must be two with the same image:
those configurations correspond to two mutually erasable patterns.

□

Theorems

1

and

2

have been shown

[

2

]

to hold for cellular automata on amenable groups.
Moore’s theorem, in fact,

characterizes

amenable groups
(cf.

[

1

]

):
whether Myhill’s theorem also does,
is an

open problem

.

References

1

Bartholdi, L. (2010)
Gardens of Eden and amenability on cellular automata.

J. Eur. Math. Soc.

12(1)

, 141–148.
Preprint:

arXiv:0709.4280v1

2

Ceccherini-Silberstein, T., Machì, A. and Scarabotti, F. (1999)
Amenable groups and cellular automata.

Annales de l’Institut Fourier, Grenoble

49(2)

, 673–685.

3

Moore, E.F. (1962)

Machine

models of self-reproduction.

Proc. Symp. Appl. Math.

14

, 17–33.

4

Myhill, J. (1962)
The

converse

of Moore’s Garden-of-Eden theorem.

Proc. Amer. Mat. Soc.

14

, 685–686.

Title

Garden-of-Eden theorem

Canonical name

GardenofEdenTheorem

Date of creation

2013-03-22 19:22:07

Last modified on

2013-03-22 19:22:07

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

9

Author

Ziosilvio (18733)

Entry type

Theorem

Classification

msc 68Q80

Classification

msc 37B15

Generated on Sat Feb 10 12:49:31 2018 by

LaTeXML