cellular automaton

cellular automaton

A

d

-dimensional

cellular automaton

is a triple

𝒜

=

⟨

Q

,

𝒩

,

f

⟩

where:

1.

Q

is a

nonempty set

, called

alphabet

or

set of states

.

2.

𝒩

=

{

ν

1

,

…

,

ν

|

𝒩

|

}

is a

finite, nonempty

subset of

ℤ

d

,
called

neighborhood

index

.

3.

f

:

Q

𝒩

→

Q

is the

local

function

.

The local function of the cellular automaton

𝒜

induces, by synchronous application at all points,
a

global function

F

𝒜

on

d

-dimensional

configurations

c

:

ℤ

d

→

Q

:

F

𝒜

⁢

(

c

)

⁢

(

z

)

=

f

⁢

(

c

⁢

(

z

+

ν

1

)

,

…

,

c

⁢

(

z

+

ν

|

𝒩

|

)

)

=

f

⁢

(

c

|

z

+

𝒩

)

.

(1)

Observe that the global function is

continuous

in the

product topology

.

Cellular automata are a Turing-complete model of computation.
In fact, given a

Turing machine

,
it is straightforward to construct a cellular automaton
that emulates it in real time.
On the other hand,
given a one-dimensional cellular automaton,
there exists a Turing machine
that emulates it in linear time
with respect to the size of the input.

Cellular automata make good tools for

qualitative

analysis

.
In fact, given the description of a cellular automaton,
it is straightforward to write a

computer

program

that implements its features.
Also, several phenomena in different fields of sciences
— from physics to biology to sociology —
can be described in terms of finite-range interactions between

discrete

agents.

On the other hand,
retrieving the

properties

of the global dynamics from the local description
is usually a very difficult task,
and may depend on the

dimension

.
As an example,
reversibility of the global function
is

decidable

in dimension

1

but not in dimension

2

(or higher).

The

definition

above can be generalized to the case
of a

generic

group

G

acting by

translations

on the space

Q

G

of configurations.
If

L

g

⁢

(

x

)

=

g

⋅

x

⁢

∀

x

∈

G

,

(2)

then the local function

f

:

Q

𝒩

→

Q

(

𝒩

being a finite nonempty subset of

G

)
induces

F

𝒜

:

Q

G

→

Q

G

via the

relation

F

𝒜

⁢

(

c

)

⁢

(

g

)

=

f

⁢

(

(

c

∘

L

g

)

|

𝒩

)

=

f

⁢

(

c

|

g

⁢

𝒩

)

⁢

∀

c

:

G

→

Q

.

(3)

Observe that

L

=

{

L

g

}

g

∈

G

is a

left

action,
while

(

c

,

g

)

↦

c

∘

L

g

is a

right

action.
It is of course possible to only use left action
by defining

F

𝒜

⁢

(

c

)

⁢

(

g

)

as

f

⁢

(

(

c

∘

L

g

-

1

)

|

𝒩

)

instead:
since this is just a

reparametrization

of the family

L

,
the bulk of the theory does not change.

More to come …

Title

cellular automaton

Canonical

name

CellularAutomaton

Date of creation

2013-03-22 19:21:58

Last modified on

2013-03-22 19:21:58

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

5

Author

Ziosilvio (18733)

Entry type

Definition

Classification

msc 37B15

Classification

msc 68Q80

Defines

neighborhood (cellular automaton)

Defines

neighborhood index (cellular automaton)

Defines

local function (cellular automaton)

Defines

global function (cellular automaton)

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