Cellular Automata

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1 Δεκ 2013 (πριν από 3 χρόνια και 8 μήνες)

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Cellular
Automata

Grid of cells, connected to neighbors

Spatial
organization.
Typically
1
or
2
dimensional

Time and space are both discrete

Each
cell
has
a
state

Cell’
s
state
at
t+1
depends
only
on
states
of
its
neighbors
and
itself
at
t.
Behavior
is
determined
locally
One-dimensional
Cellular
Automata
Transition Rules
Time
Wolfram’
s
Classification
Scheme

I: Steady state at end

II
Repetitive
cycle

III: Random-like behavior

Rule
30

Cannot
compress
behavior
(other
than
by
using
Rule
30)

IV: Complex patterns with local structures that move through
space/time

Edge
of
Chaos?
(Langton,
Crutchfield,
Kauffman)

Langton’
s
Lambda
parameter


Number of rules producing a live cell/Total number of rules

Not
too
rigid
and
not
too
fluid

Information
can
be
effectively
transmitted
Type
1:
Steady-state
Patterns
Type
2:
Repetitive
Cycles
Type
3:
Random-like
patterns
Type
4:
Local
Structures
that
Move
Langton’
s
Lambda
Parameter

=10/32, Type II

=12/32, Type IV

=14/32, Type III
Rule
30
(Wolfram,
2002)
This
rule
produces
complex
patterns
with
even
the
simplest
initial
condition
(one

on”

cell)
Sensitivity
to
initial
conditions
Changing
one
cell
in
initial
seed
pattern
causes
a
cascade
of
changes
Rule 30
Rule 22
Cellular
Automata
Terminology

Cell-space: define a lattice structure with maximum
extent of n columns and m rows

Moore neighborhood: N, S, E, W and diagonal neighbors

Von Neumann: only N, S, E, W cells

L

(
i
,
j
)
|
i
,
j

N
,
0

i

n
,
0

j

m



N
i
,
j

k
,
l



L
k

i

1
and
l

j

1



N
i
,
j

k
,
l



L
k

i

l

j

1


n
m
Cellular
Automata
Terminology

Totalistic rules

the state of the next state cell is only dependent upon the sum of the
states of the neighbor cells

Reversible rules

No application of the rules loses any information

For every obtainable state there is only state that can produce it

Atypical, because these do not incorporate cell interactions

Sometimes applied in modeling physical systems (e.g. billiard balls)
Cellular
Automata
Broadened

Mobile automata

A single active cell, which updates its position and state

Turing Machines

The active cell has a state, and states determine which transition rule is
applied

Substitution Systems

On each iteration, each cells is replaced with a set of cells

Tag systems

Remove cells from left, and add to the right depending on removed cells

Continuous state systems

On each iteration, each cells is replaced with a set of cells

Asynchronously updating systems
Mobile
automata
Turing
Machines
Substitution
System
Cantor’s
Set
Fractals

Self-similarity at multiple scales

Formed by iteration

Fractional dimensionality

The Cantor set: replace every 1 pattern with 101 with same length

Cantor set = the points remaining 1 when this is applied infinite times

Infinite number of points, but no length

A = measure of a measuring device

An object has N units of measure A

N

1
A






D
D=
dimensionality

D

log(
N
)
log
1
A






If
A=
1/3
and
D
=2,
N=9
If
A=
1/3
and
D
=1,
N=3
Cantor’s
Set
A = 1/3, N= 2, so D=log(2)/log(3)
A = 1/9, N= 4, so D=log(4)/log(9)

A

1
3
T






,

N

2
T
,
D

log
2
T


log
3
T



T
log(
2
)
T
log(
3
)

0.6309
Dimensionality is between 0 and 1
Hilbert’s
Space
Filling
Curve

Dimensionality
=
2
as
iterations
go
to
infinity
even
though it is a single line

Fractals: measure of object increases as the measuring
device
decreases
2-D
substitution
systems
L-Systems
for
plant
growth
Substitution system
Continuous
State
Cellular
Automata

Each cell’s state is based on a numeric function of
neighbors

Diffusion = each

cell’s state is average of itself and its 2 neighbors

Space,
state,
and
time
can
all
be
continuous

Partial differential equations: Specify the rate at which gray levels change with time
at every point in space. Depends on gray level at each point in space, and on the
rate at which gray levels change with position

Partial Differential Equation for Diffusion


t
u
[
t
,
x
]

1
4

xx
u
[
t
,
x
]
1 1 1 1 5 5 5 5 1 1 1
0 0 0 4 0 0 0 -4 0 0
0
0
4
-4
0
0
-4
4
0
0
0
1
-1
0
0
-1
1
0
1 1 2 4 5 4 2 1 1

x

xx
U
[1,6]
U
[2,6]
+
Continuous
States
Diffusion
=
every
cell
takes
on
the
average
of
itself
and
its
two
neighbors
Continuous
States
and
Space
Discrete
transitions
from
continuous

systems
Order from random configurations
Apparent randomness from orderly configurations
Crystal
Formation
When ice added to snowflake, heat is released, which
inhibits the addition of further ice nearby
Cellular automata: cell becomes black if they have exactly
one black neighbor, but stay white if they have more than
one
black
neighbor
Crystal
Formation
Shell
formation
(following
Raup)
Model-world
comparison
Plant
Formation
Pine
Cone
Spirals
The
numbers
of
clockwise
and
counter-clockwise
spirals
are
successive
numbers
in
the
Fibonacci
sequence:
1
1
2
3
5
8
13
21
34
55
The
angle
between
successive
leaves
on
the
pine
cone
is
137.5
degrees
Golden
Mean
A
C=1
A
C-A
C
A
=
A
C - A
C
2
-AC=A
2
A
2
+A-1=0
The
Golden
Section
A
C=1
C-A
C
2
-AC=A
2
A
2
+A-1=0
C
A
=
A
C - A
Find the A such that
1

Golden Rectangle
The
Golden
Section
The
angle
between
successive
sunflower
seeds
is
the
golden section of a circle
The
ratio
of
successive
numbers
of
a
fibonacci
sequence
approximate


=.6180… 3/5=.6 8/13=.615 34/55=.6182
The
Golden
Section
in
Plants
So,
are
sunflowers
good
mathematicians?
No,
137.5
degrees
emerges
from
simple
interactions
among
plant
leaves/seeds
Sunflower
Seed
Interactions
1
Sunflower
Seed
Interactions
1
2
New seed is positioned maximally as far away from
existing seeds as possible.
Sunflower
Seed
Interactions
1
2
3
Seeds 1 and 2 both push Seed 3 away, but Seed 2 pushes more
because it is closer to Seed 3.
Find location on circle for seed that minimizes the sum of the
“push”
exerted by other seeds, where push is an inverse square
function of distance
Sunflower
Seed
Interactions
1
2
3
4

137.5
o
A simple model based on these interactions can
provide an account of many plant forms that are
found by varying only a few parameters.
Goodwin
-
evolutionary
pressures
as
overrated?
Cellular
Automata
in
Shell
Patterns
Pattern
Formation
Pattern
Formation
(Morphogenesis)

Spots and Stripe formation

Activator-inhibitor systems

Cells activate and inhibit neighboring cells

Close neighbors activate each other

Further neighbors inhibit each other

Mexican hat function in vision
Distance from cell
Influence on cell
Turing’s
Reaction-Diffusion
Model

Show how patterns can emerge through a self-organized
process from random origins

Each cell has two chemicals

Chemical A is an autocatalyst - it produces more of itself

Chemical B inhibits production of A

Diffusion: each chemical spreads out

Reaction: each chemical reacts to the presence of the
other chemical and to itself

Activator chemical diffuses more slowly than inhibitor
chemical

If there is local variation in chemicals and chemical
amounts do not increase without bound, then stable
states of inhibitor and activator chemicals are found
Turing’s
(1952)
Reaction-Diffusion
Model
Diffusion
Reaction
reaction
diffusion
ion
A
B
-
+
A
difference
equation
account
of
diffusion
a=f(x)
a
i
-1
a
i
a
i
+1
x
i
x
i+1
x
i-1
(a
i-1
+a
i+1
)-2a
i


2
a

x
2

a
i

1

a
i

1

2
a
i
a
i
-
a
i
-1
a
i
+1
-
a
i

a/

x


a/

x
2

a
x
,
y
t

1

a
x
,
y
t


tD
a
a
x

1
,
y
t

a
x

1
,
y
t

a
x
,
y

1
t

a
x
,
y

1
t

4
a
x
,
y
t


Pattern
Formation
with
activator-inhibitor
system
Stripe
formation
Greater
diffusion
in
one
direction
than
the
other
Cellular
Automata
for
Animal
Pigmentation
Patterns
Murray
(1993)
Cellular
Automata
for
Animal
Pigmentation
Patterns
Diffusion Limited Aggregation for Population Growth