Cellular Automata

Simulation of discrete

spatio-temporal

systems

Systems with many variables

Iterative function systems describe systems with a

single variable

A iterative system with two variableswas given by

the Julia set

Systems in economy, meteorology, ecology,

sociology, etc., consists often of a large number of

variables, interacting with each other

Besides chaos many new phenomena occur in such

many variable systems

These are for example evolution, self-organization,

emergence, self-reproduction, phase-transitions

Complexity, Chaos, and Anti-Chaos

The study of spatio-temporal systems will

reveal that complexity and chaos are not the

same

Chaotic processes can produce simple

patterns called anti-chaos

The emergence of order out of chaos can be

observed (Cohen, Stewart 1994)

S.A. Kauffman (1991) Antichaosand adaptation, Scientific American,

265 (2), 64-70

J. Cohen and I. Stewart 1994: The collapse of chaos, Penguin, NY

Simple models of complex systems

A realistic treatment of complex systems is computationally

expensive and often intractable

Scientists are looking for simple models of complex systems

Often the predictions made with these simple models are

surprisingly realistic or provide deep insights in the dynamics of

real world systems (e.g. explaining phase-transitions).

However, the science of complexity is in its very beginning and

a new frontier in nonlinear systems dynamics; definitions are

evolving and scientists have not yet discovered unifying

theories;

T. Bohr et al. (1998): Dynamical systems approach to turbulence,Cambridge

University Press, New York

Locally interacting cell arrays

One of the simplest models involve a spatial

array of cells

The cells interact with nearby cells by simple

rules

These systems often exhibit spatio-temporal

chaos

–Spatial patterns in time are aperiodicand difficult

to predict

–Complex, often self-similar,patternsevolve in time

Cellular automata

Von Neumann introduced cellular automata

1966, Wolfram studied them extensively and

classified them (“A new kind of science”)

CA are perhaps the most simple models of

spatio-temporal systems, but their behavioral

spectrum is wide and interesting to study

Wolfram, S. (1986) Theory and application of cellular automata, World

Scientific, Singapore

Von Neumann (1966) Theory of self-reproducing cellular automata, Univ.

Of Illinois Press,Urbana, Il.

Wolfram S. (2002) A new kind of science, Wolfram Press

A motivating example –the XOR 1D

Automaton

Consider a ring of people

Each one is wearing a cap with the bill forwards,

except one who is wearing the bill to the back

Now, each one is looking at his/her left and right

neighbor, and adapts using these rules:

–Left and right neighbor have bill forwards wear bill

forwards

–Left and right neighbor have bill backwards wear bill

forwards

–If only one neighbor has bill forwards wear bill backwards

Evolve system over a number of generations for a

large ring of people

Simulation for 15 people, 7 time-steps

BBBB

BBBB

BB

BBBB

BB

BB

B

time

space

Periodic boundary condition

BBBB

BBBB

BBBB

BBBB

BB

BB

BBBBBBBB

Sierpinskicones and maximal-speed of

information

Long term behavior shows self-

similar cone-like structures

They resemble Sierpinski

triangles

The maximal speed-of-

information gives rise to the

boundary of the cones, similar

to the speed-of-light giving rise

to Minkowski’sspace-time

cones

Indeed, in CA literature the set

of possible states that

influenced the system in some

past are called ‘light cones’

Complex organization, but no

chaos is evident

SierpinskiTriangle in Nature

Formal expression

The given game is an example of a

1-D cellular automaton

Several ways to express a cellular

automata rule

–

X(t+1,i)=(X(t,i-1)+X(t,i+1)) modulo 2

–

X(t+1,i)=X(t,i-1) XOR X(t,i+1)

The XOR statement is a logical

function

24

= 16 logical functions could be

tried instead of XOR

x411

x301

x210

x100

t+1,it,i+1t,i-1

Rule of a cellular automaton

Initial state

The evolution of a cellular

automaton is defined also by its

initial state

The left figure shows the

evolution of a cellular

automaton with random initial

conditionusing the XOR

function

The behavior is not chaotic, but

propagate the initial conditions

in an ordered way forward in

time

Other rules may give rise to

chaotic behavior from ordered

starting conditions

The size of the rule space

The discussed XOR automaton is an example of an

1-D Cellular automaton

The size of the neighborhood and the number of

possible states determine the number of possible

rules for a cellular automaton

If we consider N nearest neighbors to each side, the

number of possible rules would grow to:

Why?

S(N)=2

22N

Four dynamical classes of Cellular

automata

Cellular automata were classified by Wolfram (2002), into four

classes based on their dynamics

1.Class 1 reach a homogeneous state with all cells the same for

all initial conditions

2.Class 2 reach a non-uniform state that is either constant or

periodic in time, with a pattern depending on initial conditions

3.Class 3 have somewhat random patterns, are sensitive to

initial conditions, and small scale local structure

4.Class 4 have relatively simple localized structures that

propagate and interact in very complicated ways

The four classes correspond roughly to fixed points,

periodicity, and chaos in dynamical systems examples will

follow

Langton’s λ

λ λ λquantity

Langton’squantity λis the

number of state

configurations that map to 1

divided by the total number

of state configurations

For instance in the left figure

λ=3/8

As the numbers of 0 equals

1-λ, only the range from 0

to 0.5 is of particular interest

0111

0011

0101

1001

0110

0010

1100

1000

t+1,it,i+1t,it,i-1

Langton, C. (1986) Studying artificial

life with cellular automata, PhysicaD 22, 120-49

Langton’s λ λ λ λquantity and dynamic

behaviour

Solid at zero temperature

Melting Fluid

Solid at finite temperature

Turbulent Fluid

Melting fluid

Solid at finite temperature

λ

By increasing λfrom 0 to 0.5 (1 downto0.5) roughly

the system goes through the same states than the

logistic map for different values of the constant a

Assignment

Higher dimensional Cellular automata

Cellular automata can be defined not only for

1-D arrays but also on higher dimensional

arrays

Some mathematical notation:

–1-D arrays are called chains

–2-D arrays are called grids

–Arrays of any dimension d are called d-

dimensional lattices

Cellular automata in 2-D

A classical cellular automaton

was defined by Conway –

Conway’s game of life

Consider a ‘game’played on a

rectangular grid, each grid cell

can have two states –dead or

alive

The neighbors of a center cell

are the nearest neighbors to the

north, south, east, west, north-

west, south-west, south-east,

north-east

This is termed the Moore

Neighborhood

SE

S

SW

E

C

W

NE

N

NW

Cellular automata in 2-D

Rule

–A cell that is alive, stays

alive, if it has two or three

living neighbors

–A dead cell becomes alive,

when it has exactly three

living neighbors

–For all other cases a cell

dies or remains dead

Example of outer totalistic

rule, i.e. a rule that involves

only the sum of neighbor

states

SE

S

SW

E

C

W

NE

N

NW

Evolution of the game of life

Starting from

an initially random

Configuration

Colonies of cells

emerge, some

of them periodic

some of them fixed

or moving through

space, shooting

pixels (glider guns*)

etc.

*Berlekampet al. (1982) Winning ways for your mathematical plays, Academic

Press, New York

The glider gun

Conway offered 50$ for everyone, who

could find an endlessly growing

configuration or prove that none exists

William Gosper and 5 other MIT students

discovered the glider gun and won the price

–The glider gun shoots a copy of itself

–On an infinite grid it would grow and evolve

without limit

Other possible configuration spaces

Regular tilingsof the 2-

D plane (there are three

possibilities)

More than 3-

dimensional

configuration spaces

Most generally:

–Configuration spaces

represented by Caley

graphs of some group

All possible regular tilingsof

the 2-D plane,i.e. tilingsconsisting

only of the same objects

hexagonal

grid

General CA definition via Caleygraph

Groupsdescribe symmetric

structures

(M,+)is a group, iff

∀

a,b,c

∈

M:

–

a+b

∈

M and a+(b+c)=(a+b)+c

–There exists

e

∈

M

with

e+a=a

–each a

∈

Mhas an inverse called

(-a),

such that

a+(-a)=e

.

We define a group Mvia a set of

generators

X

⊆

M

, such that for

every element

a

∈

M

and generator

x

,

both

a+x

∈

M

and

a+(-x)∈M

; Moreover,

all elements belong to the group that

can be obtained by concatenated

application of generators.

Given a generator

{x

1,…,x

m}

we can

define a Caleygraph

C=(V,E)

of the

group:

–

vertex set: V=M

–

edges Eare given by (v1,v2)

∈

E, iffv1

= v2+a, or v1=v2+(-a)for some ain X.

The fundamental neighborhoodN(C)

of the Caleygraph is defined by the

union of the set of generators and the

set of its inverse elements.

For each element in the graph we can

get its neighbors by using the generator

elements in N(C).

A cellular automaton

(C, N(C),A, T)

is

defined as a tupleof a Caleygraph with

labeled vertices, its fundamental

neighborhood, an finite alphabet, and a

transition function T

Node labels are chosen from a finite

local state spaceA

Transition rule

T: A

|N(C)+1|

A

assigns

each element of a cell a new value

based on the neighbors in the Caley

graph, obtained by applying the

generator.

Fd

=x1

,...,x

m

|Freegroup

Zd

=x1

,...,x

d

|∀m,n:xm

+xn

=xn

+xm

Zm

×Zn

=a,...,b|ab=ba,ma=nbTorus

Neighborhood types and sizes

Von Neumann neighborhood and Moore

neighborhood are most commonly used in 2-D grids

The radius of these neighborhoods can be

increased, e.g. by applying group generators twice

Examples of groups and their Caley

graphs

Fd

=x1

,...,x

m

|Freegroup

Zd

=x1

,...,x

d

|∀m,n:xm

+xn

=xn

+xm

Questions

How many transition rules can we define on

a cellular automaton (C, N(C), A, T)?

What could be the set of generators for the -

2D integer lattice with Moore neigborhood?

Extensions of Cellular automata

A multidimensional state space

–In Lattice gas models each cell is assigned a

vector (velocity of the fluid flow)

Memory of states in t-1, t-2, etc.

Dynamic rule sets, dynamic neighborhoods,

etc.

Finite cellular automata and chaos

Finite CA cannot be truly chaotic because the

number of states is finite, and thus the

system will eventually return to some

previous state and be trapped in a circle from

then on

To obtain maximal periods, prime numbers

are chosen as cell array sizes

Self-organization

Simple rules such as the game of lifecan cause an

initially chaotic state to evolve into a highly ordered

one Self-organization

This somehow contradicts the third law of

thermodynamics (3LT), that the entropy is always

increasing

Hakenattributed the self-organizing behavior to

cooperative effects of the systems components

(synergetics)

The 3LT is motivated by deterministic systems, but in

fact also stochastic systems can self-organize

Forest simulation model by Sprott

Consider a forest with trees

placed on grid cells, 0=fur,

1=oak

We choose a random tree

that dies

We replace this tree with a

new tree

Five trees in the

neighborhood are chosen

randomly

If the vast majority (s=4,5) is

oak, the new tree gets an

oak

If the vast majority is fur

(s=0,1), the new tree gets a

fur

Otherwise (s=2,3), the same

tree than before will grow

Connected patterns emerge

from a random starting set

Broken symmetry

It is surprising, that despite the highly symmetrical starting

conditonthe emerging system does not converge to a

symmetrical object

This phenomenon is called spontaneous symmetry breaking

and can be observed in highly ordered systems, deterministic

systems (Wolfram 2002)

Self-organized critically

So-called dissipative structures

will emerge

Connected regions with a

strange but not necessarily

fractal boundary (fat fractals)

The size distribution of the

clusters follows a power laws

Dissipative patterns are

observed in many spatio-

temporal processes

–Animal migration

–Spread of diseases

–Vegetation patterns

–Clouds and mud

Prigogine, I. (1997) The end of certainty: time, chaos, and the new laws of

nature, the free press, new york

Self-organized critically

Systems like the forest

converge to a pattern for which

there is no characteristic scale

size

Size distributions of objects

often obey power laws (this

they share with the fractals), i.e.

the distribution can be fitted to a

function

Recently, power laws are

applied in all kind of

applications

–Gene regulatory networks

–DNA pattern

–Stock prices

–City distributions

–Letter frequency in human/ape

generated random strings

(Zipf)

Not always SOC is the

explanation for the Power law

In case of city size distribution it

related to a least effort principle

(Zipf).

d

∼

1

/

f

α

Diffusion

Diffusion can be modelledvia:

Note, that there is a conservation rule fulfilled

Task: Implement diffusion system in 2-D in

MATLAB and visualize its behaviourover

time

a

j

(

t

+

1

)−

a

j

(

t

)=

a

j

+

1

(

t

)−

2

a

j

(

t

)+

a

j

−

1

(

t

)

Sand Pile –the prototype of a SOC

system

Consider a pile of sand to which we add

sand continuously

The sand-pile steepensuntil it reaches an

angle of repose, whereupon avalanches

keep the sandpileclose to this angle

The avalanches obey a power law scaling in

their size distribution and in their duration

A pile of cheese

Bak’sCA simulation of a pile of sand

Baksimulated a pile of sand using the

following CA model

The pile is represented by a N ×N matrix

of integers

Initially all cells are chose between 1 and

3

At each time step choose a random cell

i,jand set Z(t+1,i,j)=Z(t,i,j)

Cells outside the boundary are kept as 0

All other cells which exceed Z=3, and

their von-Neumann neighbors are

updated with:

Z(t+1,i,j)=Z(t,i,j)-4

Z(t+1,i±1,j)=Z(t,i±1,j)+1

Z(t+1,I,j±1)=Z(t,I,j±1)+1

Strictly speaking, this is not a cellular

automaton, as it evolves not

autonomously

d=2

Power spectrum of sum(Z(i,j)) over t

Dropping sand on the central point

Emergence vs. Reductionism

Reductionism assumes simple laws that govern

natural processes and that these simple laws help to

understand/explain global behaviour

Emergence holds that high level structure is

generally unpredictable from low level processes,

and does not even depend very much on its

properties

Due to Sprott(2006), systems are complex, if they

exhibit emergent behaviour

How to measure degree of

(self-)organization

The term of self-organization

is used since 1947, but up to

know there is no standard

definition except “I know it

when I see it”

Thermodynamic entropy

measures the degree of a

system’s “mixedupedness”

(to use Gibbs’s word), or

how far it departs from being

in a pure state

Organisms are essentially

never in pure states, and are

highly mixed up at the

molecular level, but are the

paradigmatic examples of

organization.

Furthermore, there are many

different kinds of

organization, and entropy

ignores all the distinctions

and gradations between

them

W. R. Ashby, “Principles of the self-organizing dynamic system,”Journal of

General Psychology 37, pp. 125–128, 1947.

How to measure a degree of self-

organization and complexity?

Another school of thought has

been put forward by

Kolmogorovand Solomonoff

“A complex phenomena is one

which does not admit of

descriptions which are both

short and accurate”

Problem exactness: Coin

tossing, produces sequences of

maximal Kolmogorov

complexity, though dynamics

are simple to describe.

Grassbergergave a more

general definition: ‘The

complexity of a process as the

minimal amount of information

about its state needed for

maximally accurate prediction’

Crutchfield and Young gave

operational definitions of

“maximally accurate prediction”

and “state”

The Crutchfield-Young

“statistical complexity”, C

µ, of a

dynamical process is the

Shannon entropy (information

content) of the minimal

sufficient statistic for predicting

the process’s future.

Shaliziand Shaliziused this

measure recently to quantify

self-organization in CA

practically

They used cyclic CA to assess

their method

Cyclic CA

Cyclic cellular automata

(CCA) are simple models of

chemical oscillators.

Started from random initial

conditions, they produce

several kinds of spatial

structure, depending on their

control parameters.

They were introduced by

David Griffeath, and

extensively studied by Fisch

Transition rule

–

Each site in a square

two-

dimensional lattice is in one

of κcolors.

–

A cell of colork will change

its colorto k + 1 mod κif

there are already at least T

cells of that colorin its

neighborhood

–

Otherwise, the cell retains

its current color

Fisch, R. (1990a). "The one-dimensional cyclic cellular automaton: A system

with deterministic dynamics that emulates an interacting particle system with

stochastic dynamics". Journal of Theoretical Probability3(2): 311–338.

Cyclic CA

The CCA has three generic

forms of long-term behavior,

depending on the size of the

threshold relative to the range.

At high thresholds, the CCA

forms homogeneous blocks of

solid colors, which are

completely static —so-called

fixation behavior.

At very low thresholds, the

entire lattice eventually

oscillates periodically;

–sometimes the oscillation takes

the form of large spiral waves

which grow to engulf the entire

lattice.

There is an intermediate range

of thresholds where incoherent

travelingwaves form,

propagate for a while, and then

disperse;

–this is called “turbulence”, but

whether it has any connection

to actual fluid turbulence is

unknown.

Spiralling

waves

Turbulent behavior

Spirals engulfing the space for Moore

neighborhood

Cyclic CA for a Moore

neighborhood and T=2

For Moore neighborhood the

following transitions can be

found:

–T=1: local oscillations

–T=2: spiraling waves

–T=3: turbulence, often

metastablein very long run

(then spirals can take over)

Cellular Automata and beyond

Statistical complexity of cyclic CA over time

CosmaRohillaShaliziand Kristina Lisa Shalizi: Quantifying Self-Organization

in Cyclic Cellular Automata, http://arxiv.org/abs/nlin/0507067v1

Cellular automata and beyond

Partial Differential

equations

ContinousContinuousContinuous

DiscreteContinuousContinuous

ContinuousDiscreteContinuous

DiscreteDiscreteContinuous

Coupled Flow

Lattices

ContinuousContinuousDiscrete

DiscreteContinuousDiscrete

Coupled Map

Lattices

ContinuousDiscreteDiscrete

Cellular AutomatonDiscreteDiscreteDiscrete

ModelStateTimeSpace

Summary (1)

Cellular automata are defined on a Caleygraph (with

state labels) with a neighbourhoodand transition

rule mapping the state of a center cell to a new state

based on its neighbor states.

The number of possible transition rules grows

exponentially with the size of the local state space

and neighborhood

Common neighborhood types are von Neumann and

Moore neighborhood, and the k-neighbors in 1-D

arrays (with periodic boundary conditions)

Summary (2)

CA are simple models of natural systems

Despite their simplicity the behavior of CA can be

extremely complex and difficult to predict

CA serve as models for studying emergent

phenomena and self-organization

Self organized systems are often at the boundary of

chaotic and ordered states; many open questions

remain, and definitions are not yet clarified

An interesting question if the type of global behavior

can be predicted from properties of the rules (e.g.

Langtonslambda)

Summary (3)

As simulators CA models are easy to

implemented (also in parallel) and can be

used to model phenomena such as diffusion,

cell systems, flow, pattern formation, etc.

CA can be seen as discretecounterparts of

partial differential equations

As such they belong to the class of spatio-

temporal models

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