# Jan30-Feb13: Cellular Automata - UNM Computer Science

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Cellular Automata

CS 591

Spring 2008

Professor Melanie Moses

2/4/08

Review of concepts of complexity

Measures of complexity

Computational measures

Computational complexity:
Measures program running time, e.g., P, NP, NC

Language complexity:

Classes of languages that can be computed (recognized) by different
abstract machines
, e.g.
regular, context free, context sensitive languages

Information
-
theoretic approaches:

Algorithmic Complexity

(Komogorov, AIC): Length of (# of bits to represent) the
shortest program that can produce the phenomenon; grows with unpredictability

Problems: Uncomputable, machine dependent, high for random strings

Shannon entropy
: Number of bits to represent a string

*
Mutual information
:amount of info that one random variable contains about another

Logical depth (Bennett):
running time of the shortest program, combines
computational complexity and AIC

Effective complexity (Gell
-
Mann):
AIC of the
regularities;
Effective
complexity is highest of non
-
random, non
-
regular strings

)
(
log
)
(
)
(
2
x
p
x
p
X
H

Mutual Information

Measures the amount of information that one random variable contains

Mutual information is a measure of reduction of uncertainty due to another random
variable.

That is, mutual information measures the dependence between two random variables.

It is symmetric in X and Y, and is always non
-
negative.

Recall: Entropy of a random variable X is
H(X).

Conditional entropy of a random variable X given another random
variable
Y = H(X | Y).

The

mutual information

of two random variables
X

and
Y is:

Gell
-
Mann: If a string is divided into two parts the mutual AIC is the sum
of the AIC's of the parts minus the AIC of the whole.

y
x
y
p
x
p
y
x
p
y
x
p
Y
X
H
X
H
Y
X
I
,
)
(
)
(
)
,
(
log
)
,
(
)
|
(
)
(
)
,
(
Kauffman’s Boolean Networks

from Lansing 2003

What is the relationship between the average connectedness of genes

to the ability of organisms to evolve?

Given N bulbs and K connections behavior is

1) Chaotic: If K is large, the bulbs keep twinkling chaotically

2) Frozen or periodic: If K is small (K = 1), some flip on and off, most soon stop

3) Complex: If K is around 2, complex patterns appear, in which twinkling

islands of stability develop, changing shape at their borders.

A network that is either frozen solid or chaotic cannot transmit information

A Boolean Net has 2
N

possible states, but many fewer
basins of attraction

Introduction to Cellular Automata (CA)

Invented by John von Neumann (circa~1950).

A cellular automata consists of:

A regular arrangement (lattice) of cells.

Cells can be in one of a finite number of states at each time step.

All cells have the same synchronous update rule.

Cells have a local interaction neighborhood.

Example: The game of life.

Many cellular automata applications:

Hydrodynamics, fluid dynamics

Forest fire simulation

Models of ecosystems and epidemics

Ferro
-
magnetic modeling (ISING models)

One
-
Dimensional CA Illustration

t

i
-

r

...

i
-

1

i

i
+
1

...

i
+
r

t
+
1

i

h

f

i

r

One
-
Dimensional Cellular Automaton (CA)

Consists of a linear array of identical
cells
(called a
lattice
),

each
of which can be in a finite number of
k

states.

The (local)
state

of cell
i

at time
t

is denoted:

The (global) state s
t

at time
t
is the
configuration

of the entire
array,

where
N

is the (possibly infinite) size of the array.

}
1
,...,
1
,
0
{

k
s
i
t
N
N
t
t
t
t
s
s
s

)
,...,
,
(
s
1
1
0
One
-
Dimensional CA cont.

At each time step, all cells in the array update their state
simultaneously, according to a
local update

rule .

This update rule takes as input the
local neighborhood
configuration
of a cell.

A local neighborhood configuration consists of s
i
and its 2
r

nearest neighbors (
r

cells on either side):

r

is called the

of the CA.

The local update rule , which is the same for every cell in the
array, can be represented as a lookup table, which lists all
possible neighborhood configurations

s

:

h
f
h
h
)
,...,
,...,
(
r
i
r
i
i
s
s
s

h
)

(

1
i
t
i
t
s
h
f

f
)

|

(|
1
2

r
k
h
From Flake ch 15, rule 6

C
i
-
1
(t)

C
i
(t)

C
i+1
(t)

C
i
(t+1)

0

0

0

0

0

0

1

1

0

1

0

1

0

1

1

1

1

0

0

1

1

0

1

1

1

1

0

1

1

1

1

0

Wolfram rule

Sum to

0
--
> 0

1
--
> 1

2
--
> 1

3
--
> 0

0110 = ‘rule 6’

Flake: k(2r+1) rules (??)

(k
-
1)(2r+1) +1 works

Example One
-
Dimensional CA

Rule 110

The number of states,
k
=2.

The alphabet

The size of the array, N=11.

The configuration space

r

= 1.

The rule table :

neighborhood : 000 001 010 011 100 101 110 111

: 0 1 1 1 0 1 1 0

This is rule 110 (base 10) because the output states are: 01101110
(base 2). Read right to left. Known as Wolfram notation.

Rule 110 supports universal computation.

),...}
11
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
(
),
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
{(

N
f
h
)

(

1
i
t
i
t
s
h
f

k

}
1
,
0
{
1

1

0

0

1

1

0

0

0

1

0

1

1

0

1

1

1

0

1

1

1

1

Neighborhood

h

t
=
1

t
=
0

Periodic
boundary
conditions

Neighborhood:

Output bit:

Lattice:

000
001
010
011
100
101
110
111
0
1
1
1
0
1
1
0
Rule Table
f

:

Rule 110 Space
-
Time Plot

Rule 30

current pattern

111 110 101 100 011 010 001 000

new state for center cell

0 0 0 1 1 1 1 0

Generates apparent randomness, despite being finite

Wolfram proposed using the central column as a pseudo
-
random number generator

Passes many tests for randomness, but many inputs produce
regular patterns:

All zeroes

00001000111000 repeated infinitely (try separating by 6 1s)

Used in Mathematica for creating random integers (Wikipedia)

Wolfram’s CA Classification

Class I: Eventually every cell in the array settles into one state,
never to change again.

Analogous to computer programs that halt after a few steps and to dynamical
systems that have fixed
-
point attractors.

Class II: Eventually the array settles into a periodic cycle of states
(called a limit cycle).

Analogous to computer programs that execute infinite loops and to dynamical
systems that fall into limit cycles.

Class III: The array forms “aperiodic” random
-
like patterns.

Analogous to computer programs that are pseudo
-
random number generators
(pass most tests for randomness, highly sensitive to seed, or initial condition).

Analogous to chaotic dynamical systems. Almost never repeat themselves,
sensitive to initial conditions, embedded unstable limit cycles.

Wolfram’s Classification cont.

Class IV: The array forms
complex

patterns with localized
structure that move through space and time:

Difficult to describe. Not regular, not periodic, not random.

Speculate that it is interesting computation.

Hypothesis: The most interesting and complex behavior occurs
in Class IV CA
---
the edge of chaos.

Example: Rule 110

Wolfram Class I

Wolfram’s Class II

Wolfram’s Class III

Wolfram’s Class IV

More Cellular Automata

Wed. Feb 6

Langston’s lambda

Example calculations

Exercise: Finding complex CAs

Flake’s CA applet

Universal computation

Rule 110

Game of Life

Langton’s Lambda Parameter

Wolfram’s classification scheme is
phenomenological (
argument
by visual inspection of space
-
time diagrams).

Chris Langton (1986) quantified the classification scheme by
introducing the parameter .

Lambda is a statistic of the output states in the CA lookup table,
defined as
the fraction of non
-
quiescent states in this table.

The quiescent state is an arbitrarily chosen state

Example: For a 2
-
state CA ( ), and quiescent state s=0,
lambda is the fraction of 1s in the output states of .

s
}
1
,
0
{

f
Lambda Parameter cont.

Where n
q
= the number of rules that map to the quiescent state.

Using the lambda parameter to study CA s:

through the range:

Random table: Interpret lambda as a bias on the random selection of states
to fill up the table. (Get a new table for every new value of lambda).

Use various statistics to measure CA “average” behavior, as a
function of lambda:

Single
-
site entropy.

Same site mutual information across time steps.

Two
-
site mutual information.

|

|
|

|

h
h

q
n

k
1
0
.
1

0

From Flake ch 15, rule 6

C
i
-
1
(t)

C
i
(t)

C
i+1
(t)

C
i
(t+1)

0

0

0

0

0

0

1

1

0

1

0

1

0

1

1

1

1

0

0

1

1

0

1

1

1

1

0

1

1

1

1

0

Wolfram ‘sum’ rule

Sum to

0
--
> 0

1
--
> 1

2
--
> 1

3
--
> 0

0110 = ‘rule 6’

Flake: k(2r+1) rules (??)

(k
-
1)(2r+1) +1 works

Sum rule: 0110

To calculate

convert sum notation to full
rule table

Wolfram rule: 01111110 = rule 116

= representation of all rules:

= 1
-
1/k = 1/2

‘Interesting cases’: 0 <

< 1/2

For sum rule 0110,

= 3/4

|

|
|

|

h
h

q
n

Lambda Space and Wolfram Classes

Lambda Parameter cont.

Table walkthrough illustrates interesting change in dynamics as
function of lambda:

Transient lengths increase dramatically in middle of the range (see next
slide).

However, different traversals of lambda space using the walkthrough
method make the transition at different lambda values, although there is a
well defined distribution around a mean value.

Size of the array has an effect on dynamics only for intermediate values of
lambda.

Transient length depends exponentially on array size at lambda = 0.5.

Overall evolutionary pattern in time is more random as lambda increases
past the transition region (use entropy and mutual information).

Transition region supports both static and propagating structures.

Claims:

There is a phase transition between periodic and chaotic behavior. Most complex
behavior is in the vicinity of the transition: The “edge of chaos.”

CA s near the transition point correspond to Wolfram’s Class IV.

CA s capable of performing complex computations will be found near the transition
point (long transients).

E.g., the game of life has lambda=0.273 (in the transition region for K=2, N=9 2D CA
s).

Criticisms of the Lambda parameter:

CA s with high lambda
-
value can still have simple behavior. Lambda describes
“average” behavior.

Lambda does not take the initial state of the computation into account (see Assignment
3).

Exercise

Find a complex rule set with 3 states and r = 1 (neighborhood size = 3)

Represent in Flake (sum) rule notation

To convert from sum notation

Binomial coefficient (Number of combinations without repetition)

Number of sets with exactly k of n states = n!/(k!(n
-
k!))

In what range should

c

be?

How sensitive is the rule to initial conditions?

Entropy and Lambda

Critical Slowing Down,

Length of Transients Depends on Cell Size

Phase Transitions and Lambda

Computation in Cellular Automata

CA s as computers:

Initial configuration constitutes the data that the physical computer is processing.

Transition function implements the algorithm which is applied to the data.

Examples: Majority calculations, synchronization (Mitchell and Crutchfield).

CA s as logical universes within which computers can be embedded:

Initial configuration constitutes a computer.

Transition function is the “physics” obeyed by the parts of the embedded computer.

The algorithm and data are functions of the precise state of the initial configuration of
the embedded computer.

In the most general case, the initial configuration is a universal computer.

Examples: Rule 110, Game of life

CA s can implement storage, counting, logical operators

CA s are networks of FSAs

A Turing machine is equivalent to an FSA with an infinite storage tape

Example One
-
Dimensional CA

Rule 110

The number of states,
k
=2.

The alphabet

The size of the array, N=11.

The configuration space

r

= 1.

The rule table :

neighborhood : 000 001 010 011 100 101 110 111

: 0 1 1 1 0 1 1 0

This is rule 110 (base 10) because the output states are: 01101110
(base 2). Read right to left. Known as Wolfram notation.

Rule 110 supports universal computation.

),...}
11
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
(
),
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
{(

N
f
h
)

(

1
i
t
i
t
s
h
f

k

}
1
,
0
{
Rule 110 Space
-
Time Plot

Two
-
Dimensional Cellular Automata

Source: Wikipedia

Wire World

Conus textile

Cyclic Cellular Automaton

The Game of Life

The number of states, k=2.

The alphabet,

Call 0 “dead” and 1 “alive.”

The Moore neighborhood:

Transition rules:

Loneliness: If a live cell has less than 2 live neighbors, then it dies.

Overcrowding: If a live cell has more than 3 live neighbors, then it dies.

Birth: If an empty (dead) cell has 3 live neighbors, then it becomes alive.

Otherwise: If a cell has 2 or 3 live neighbors, then it remains unchanged.

}
1
,
0
{

NE

SW

S

SE

E

W

N

NW

Moore vs. Von Neumann Neighborhoods

NE

SW

S

SE

E

W

N

NW

S

E

W

N

Moore

Von Neumann

Example Transition

Center square changes to one (birth).

“Just 3 for birth, 2 or 3 for survival.”

0

0

1

0

1

0

1

0

0

1

Possible Life Histories

(Dynamical behaviors that enable universal
computation)

Static structures:

Beehive, Loaf, Pond, etc. (Figure 15.11)

Periodic structures:

Moving structures:

Gliders (next slide)

Glider guns

Logical gates (and, or, not)

Self
-
reproducing structures.

Static Objects

Periodic Objects

Gliders (Moving Objects)

Logical Operators

Can we predict the dynamics of a given
initial condition?

Consider a straight line of n live cells:

n = 1,2 Dies out immediately.

n = 4 Becomes a beehive

n = 5 Traffic lights

n = 6 Dies out at t = 12

n = 7 Interesting behavior, terminating in the honey farm.

N = 8 4 blocks and 4 beehives.

N = 9 2 sets of traffic lights.

Etc.

Wolfram A new kind of science, chapter 11, p 674
-
691

Rule 110

online book at
http://www.wolframscience.com/nksonline/toc.html

pdf will be posted on course site

http://cs.unm.edu/~melaniem/courses/CAS08.html

Background on tag systems (not required)

in Wolfram Chapter 3 and

http://mathworld.wolfram.com/CyclicTagSystem.html

http://mathworld.wolfram.com/TagSystem.html