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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 variables was 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

• bserved (Cohen, Stewart 1994)

S.A. Kauffman (1991) Antichaos and 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 aperiodic and difficult

to predict

– Complex, often self-similar,patterns evolve 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

B B B B B B B B B B B B B B B B B B B

time space

Periodic boundary condition

B B B B B B B B B B B B B B B B B B B B B B B B B B B B

Sierpinski cones 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’s space-time cones

• Indeed, in CA literature the set
• f possible states that

influenced the system in some past are called ‘light cones’

• Complex organization, but no

chaos is evident

Sierpinski Triangle 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

x4 1 1 x3 1 x2 1 x1 t+1,i t,i+1 t,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 condition using 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?

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 λ quantity

Langton’s quantity λ is the

number of state configurations that map to 1 divided by the total number

• f 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

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

Langton, C. (1986) Studying artificial life with cellular automata, Physica D 22, 120-49

Langton’s λ λ λ λ quantity and dynamic behaviour

Solid at zero temperature Melting Fluid Solid at finite temperature Turbulent Fluid Melting fluid Solid at finite temperature

λ By increasing λ from 0 to 0.5 (1 downto 0.5) 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

• nly 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

• f them periodic

some of them fixed

• r moving through

space, shooting pixels (glider guns*) etc.

*Berlekamp et 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 tilings of 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 tilings of the 2-D plane,i.e. tilings consisting

• nly of the same objects

hexagonal grid

General CA definition via Caley graph

• Groups describe 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∈M has an inverse called (-a), such that a+(-a)=e.

• We define a group M via 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 {x1,… ,xm} we can

define a Caley graph C=(V,E) of the group:

–

vertex set: V=M

–

edges E are given by (v1,v2)∈E, iff v1 = v2+a, or v1=v2+(-a) for some a in X.

• The fundamental neighborhood N(C)
• f the Caley graph 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 tuple of a Caley graph with labeled vertices, its fundamental neighborhood, an finite alphabet, and a transition function T

• Node labels are chosen from a finite

local state space A

• 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.

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

a b

• Free

group

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 life can cause an

initially chaotic state to evolve into a highly ordered

• ne Self-organization

This somehow contradicts the third law of

thermodynamics (3LT), that the entropy is always increasing

Haken attributed 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

• ak, the new tree gets an
• ak

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

conditon the 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
• bserved 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
• ften 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).

Diffusion

Diffusion can be modelled via: Note, that there is a conservation rule fulfilled Task: Implement diffusion system in 2-D in

MATLAB and visualize its behaviour over time

Sand Pile – the prototype of a SOC system

Consider a pile of sand to which we add

sand continuously

The sand-pile steepens until it reaches an

angle of repose, whereupon avalanches keep the sandpile close to this angle

The avalanches obey a power law scaling in

their size distribution and in their duration

A pile of cheese

Bak’s CA simulation of a pile of sand

• Bak simulated a pile of sand using the

following CA model

• The pile is represented by a N ×N matrix
• f integers
• Initially all cells are chose between 1 and

3

• At each time step choose a random cell

i,j and 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

• rganization.

Furthermore, there are many

different kinds of

• rganization, 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-

• rganization and complexity?
• Another school of thought has

been put forward by Kolmogorov and 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.

• Grassberger gave 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
• perational definitions of

“maximally accurate prediction” and “state”

• The Crutchfield-Young

“statistical complexity”, C, of a dynamical process is the Shannon entropy (information content) of the minimal sufficient statistic for predicting the process’s future.

• Shalizi and Shalizi used 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

• f colors.

–

A cell of color k will change its color to k + 1 mod if there are already at least T cells of that color in 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 Probability 3 (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

• scillates periodically;

–

sometimes the oscillation takes the form of large spiral waves which grow to engulf the entire lattice.

• There is an intermediate range
• f thresholds where incoherent

traveling waves 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 metastable in very long run (then spirals can take over)

Cellular Automata and beyond

Statistical complexity of cyclic CA over time Cosma Rohilla Shalizi and Kristina Lisa Shalizi: Quantifying Self-Organization in Cyclic Cellular Automata, http://arxiv.org/abs/nlin/0507067v1

Cellular automata and beyond

Partial Differential equations Continous Continuous Continuous Discrete Continuous Continuous Continuous Discrete Continuous Discrete Discrete Continuous Coupled Flow Lattices Continuous Continuous Discrete Discrete Continuous Discrete Coupled Map Lattices Continuous Discrete Discrete Cellular Automaton Discrete Discrete Discrete Model State Time Space

Summary (1)

Cellular automata are defined on a Caley graph (with

state labels) with a neighbourhood and 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. Langtons lambda)

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 discrete counterparts of

partial differential equations

As such they belong to the class of spatio-

temporal models