MAS 2011 Lecture 2

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Agent-Based Simulation


Cellular Automata

Federico Pecora

School of Science and Technologyrebro [email protected]

Federico Pecora Agent-Based Simulation Part 2 1 / 35
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Outline

1 Introduction to CA

2

Simple CA (apparently)

3 Other classes of CA

Reversible CA

Totalistic CA

4 Applications of CA

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Introduction to CA

Outline


2



Reversible CA

Totalistic CA



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Introduction to CA

What is a Cellular Automaton?

A discrete computational model

Relevant in a number of fields

theoretical computer science

mathematics

biology

physics

engineeringcryptography

. . .


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Introduction to CA


Regular n-dimensional grid of cells

Each cell has finte number of states

Each cell has a neighborhood consisting of other cells

Time advance function: the state of a cell at time t in function of

the state of the cells neighborhood at time t1

Every cell shares the same time advance function

At every tick (instant of discrete time), the rule is applied for all

cells in the grid, and a new generation is formed

Initial condition: all cells have the same state except for a given

set of cells, called configuration

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Introduction to CA


states: 2 (full/empty cell)

survival: 2 or 3

birth: 2 or 3


A t B d Si l ti

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Introduction to CA


states: 2 (full/empty cell)

survival: 2 or 3

birth: 2 or 3


Agent Based Sim lation
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Introduction to CA


Generations S23/B23/C8

states: 8 (colors in a

spectrum)

survival: 2 or 3

birth: 2 or 3

colors: only first state gives

birth, others decay in coloruntil they die


Agent Based Simulation
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Introduction to CA

Rules and periodic bondary conditions

Assume an infinite, 2-dimensional grid

Assuming 2-state cells, we have for each cell 29 = 512 possible

patterns

Since we can only represent a finite grid, the cells on the borders

are subject to periodic boundary conditions

cell coordinates are mod(N) (wrap around both vertically and

horizontally)


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Introduction to CA

Properties of CA

Parallelism: individual cell updates are performed independently of

each other all of the updates being done at once

Locality: when a cell is updated, its new value is based solely on

the values of its neighbors (and on the cells history in

n-order CA)

Homogeneity: each cell is updated according to the same rules

CA are good models for physical, biological and sociolog-

ical phenomena: each person/cell/small region of space up-dates itself independently (parallelism), based on its immedi-

ate surroundings (locality) and on some generally shared laws

of change (homogeneity)



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Introduction to CA

Properties of CA

Parallelism: individual cell updates are performed independently of

each other all of the updates being done at once

Locality: when a cell is updated, its new value is based solely on

the values of its neighbors (and on the cells history in

n-order CA)

Homogeneity: each cell is updated according to the same rules

CA are good models for physical, biological and sociolog-

ical phenomena: each person/cell/small region of space up-dates itself independently (parallelism), based on its immedi-

ate surroundings (locality) and on some generally shared laws

of change (homogeneity)


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Outline


2 Simple CA (apparently)


Reversible CA

Totalistic CA



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g


Conways Game of Life

Simple rules can lead to spectacular results! An example: Conways

Game of Life

A simple CA, with 2 states and the rule (aka S23/B3)

Any live cell with fewer than two live neighbours dies (loneliness)

Any live cell with more than three live neighbours dies

(overcrowding)

Any live cell with two or three live neighbours lives

Any dead cell with exactly three live neighbours comes to life

Due to John Horton Conway, british mathematician

This is interesting for two reasons. . .


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Properties of Conways Game of Life (1)

There are no initial patterns for which there is a simple proof that

the population can grow without limit

There exist initial patterns that apparently grow without limit

Ther exist simple initial patterns that grow and change for a long

time before (a) fading away completely, or (b) going into a stable

(static or oscillating) state

Boat (still)

Blinker (2-phase oscillator)

Glider (spaceship)

Pulsar (3-phase oscillator)


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In the 1940s, John von Neumann wanted to find a hypothetical

machine that could replicate itself

He built a mathematical abstraction of a self-replicating robot

(based on Stanisaw Ulams work on crystal growth models)

von Neumann proved that a particular pattern existed which could

make endless copies of itself the von Neumann universal

constructor

Conway simplified von Neumanns method into the GoL

The GoL has the power of a universal Turing machine, i.e., any-

thing that can be computed algorithmically can be com-

puted within Conways GoL


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In the 1940s, John von Neumann wanted to find a hypothetical

machine that could replicate itself

He built a mathematical abstraction of a self-replicating robot

(based on Stanisaw Ulams work on crystal growth models)

von Neumann proved that a particular pattern existed which could

make endless copies of itself the von Neumann universal

constructor

Conway simplified von Neumanns method into the GoL

The GoL has the power of a universal Turing machine, i.e., any-

thing that can be computed algorithmically can be com-

puted within Conways GoL


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Complex patterns from simple rules

The GoL (and CA in general) demonstrate how complex patters

are obtainiable from very simple rules

There exist oscillators of period 4, 8, 14, 15, 30, and a few others

have been seen on rare occasions

Methuselah patterns: less than 10 initial live cells that take

longer than 50 generations to repeat

Diehard

dies after 130 generations

Acorn

takes 5206 generations to generate

at least 25 gliders and stabilise as

many oscillators


http://videos/acorn.ogghttp://videos/diehard.ogghttp://find/http://goback/

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Even simpler CA

The simplest CA are one-dimensional, with two states per cell

Neghborhood = two adjacent cells on either side

23 = 8 possibile patterns for a neighborhood28 = 256 possible rules (256 possible CA)

Each possible CA is identified in Wolfram notation as the

decimal number which, in binary, gives the rule table

current pattern 111 110 101 100 011 010 001 000new state of middle

cell0 0 0 1 1 1 1 0


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Rule 30 (00011110)

Seed = one cell on

Generations

Generates apparent randomness, passes many randomtests...

. . . but there exist an infinite number of input patterns that result in

repeating patterns (e.g., 00001000111000, discovered by

Matthew Cook)



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Rule 110 (01101110)

Seed = one cell on

Generations

Very simple structure, but difficult to obtain desired behaviors

Of the 256 simple" CA, Rule 110 is the only one known to be

Turing complete


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Rule 110 (01101110)

Like the GoL, Rule 110 is capable of universal computation

This has been used to suggest that many of the properties of

many natural systems (e.g., patterns on some seashells) are

undecidable



Oth l f CA
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Other classes of CA

Outline




Reversible CA

Totalistic CA




Other classes of CA
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Other classes of CA

Reversible CA

Reversible CA

A CA is reversible if its update function is bijective

the current configuration is obtainable by exactly one previous

configuration

an irreversible CA is one for which there exist patterns for which

there are no previous states

these patterns are called Garden of Eden patterns

(Garden of Eden pattern for GoL)



Other classes of CA
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Other classes of CA

Reversible CA

Reversible CA

Any one-dimensional CA can be proved to be either

reversible or irreversible

For n-dimensional CA (n 2), reversability is undecidable

i.e., there exist n-dimensional rules for which the complexity of

describing its inverse vastly exceeds the complexity of the rule

itself

i.e., the only way of proving it is to simulate!

There exist, however, methods to build a reversible CA from an

existing rule

second-order technique

vaious partitioning techniques, e.g., Margolus neighborhood

. . . although the properties of the original rule are lost



Other classes of CA
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Other classes of CA

Reversible CA

Reversible CA

Wait a minute. . . doesnt the opposite rule give the inverse CA?

Rule 30111 110 101 100 011 010 001 000

0 0 0 1 1 1 1 0

Rule 225111 110 101 100 011 010 001 000

1 1 1 0 0 0 0 1



Other classes of CA
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Other classes of CA

Reversible CA

Reversible CA

Wait a minute. . . doesnt the opposite rule give the inverse CA?

. . . the reverse step is not deterministic!



Other classes of CA
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Other classes of CA

Reversible CA

Second order technique

State at time t+1 is function of neighborhood at time tand stateof cell at time t1

s(t+1) = f(n(t)) s(t1)

s(t1) = f(n(t)) s(t+1)

Rule 30 111 110 101 100 011 010 001 0000 0 0 1 1 1 1 0

Rule 30R

1 1 1 1 1 1 1 1 (t1)111 110 101 100 011 010 001 000 t

1 1 1 0 0 0 0 1 (t+1)

0 0 0 0 0 0 0 0 (t1)111 110 101 100 011 010 001 000 t

0 0 0 1 1 1 1 0 (t+1)



Other classes of CA
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Reversible CA

Second order technique

State at time t+1 is function of neighborhood at time tand stateof cell at time t1

s(t+1) = f(n(t)) s(t1)

s(t1) = f(n(t)) s(t+1)

Rule 30 111 110 101 100 011 010 001 0000 0 0 1 1 1 1 0

Rule 30R

1 1 1 1 1 1 1 1 (t1)

111 110 101 100 011 010 001 000 t

1 1 1 0 0 0 0 1 (t+1)

0 0 0 0 0 0 0 0 (t1)111 110 101 100 011 010 001 000 t

0 0 0 1 1 1 1 0 (t+1)



Other classes of CA
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Reversible CA

Rule 30R



Other classes of CA
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Totalistic CA

Totalistic CA

The state of each cell is represented by a numberState at time tdepends only on the sum of the values of thecells in its neighborhood at time t1

outer totalistic CA: neighborhood does not contain the cell itself

totalistic CA: neighborhood does contains the cell itself

GoL is an outer totalistic CA with values 0 and 1

Generations S0235678/B3468/C9



Applications of CA
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Outline




Reversible CA

Totalistic CA




Applications of CA

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CA-based encryption

Low speed of encryption and/or decryption causes big problems

for practical implementations

CA are inherently paralleleasy to implement on distributed processing platforms

Some CA exhibit strong irregularities

e.g., Wolframs rule 30

Reversible CA can be used as backdoor functions

computing f() is easy, computing f1() is difficult



Applications of CA
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Stream ciphers with Rule 30

Stream cipher: a symmetric key cipher

where plain text is combined with PRNG

bitstream

Wolframs Rule 30 does not repeat for any

short period and has no obvious structure

The central column of Rule 30 has been

subject to many randomness tests, and

has passed every one so farCryptotext: combination (e.g., XOR) of

clear stream with a column from Rule 30

Cleartext

Cryptotext

XOR

Rule30

column



Applications of CA
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Encryption with reversible CA

Asymmetric cipher: the key used to encrypt a message differs

from the key used to decrypt

e.g., public key cryptography

It is undecidable if a 2D CA is invertible it can be very

difficult to invert a 2D CA

Use the message to encrypt as initial configuration for a

reversible CA A

Cryptotext: configuration of A at generation n (n fixed or

depending on size of message)Keys: A is public key, A1 is private key

it seems possible to obtain such difficult to invert CA by

combining several simply invertible CA



Applications of CA
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Reversible CA for gasses

Reversible CA can be employed to simulate gasses

Homogeneity: every molecule follows one rule

Locality: each molecule affects its neighbors only, not

others far away from itReversibility: all information the motion of molecules is

backward traceable

In gasses, heat entails the motion of molecules

As in CA, the randomness (entropy) of the system increases. . . gasses look like reversible CA!



Applications of CA
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Hardy, de Pozzis, Pomeau (HPP) gas model

Rules are based on Margulos Neighbourhood: 22

partitioning of the lattice which alternates at every time step

Rules Step 1



Applications of CA
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Rules Step 1



Applications of CA

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Rules Step 2



Applications of CA
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Rules Step 2

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Applications of CA

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Rules Step 3



Applications of CA
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Rules Step 4



Applications of CA
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Rules Step 4


Agent-Based SimulationApplications of CA
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Rules Step 5


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Rules Step 5


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Rules Step 6


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Rules Step 6


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Rules Step 7


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Rules Step 7


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HPP gas model already comes close to Navier-Stokes equation

realistic simulation of gasses

More precise models allow movements in more than four

directions

e.g., FHP (Fritsch, Hasslacher, Pomeau) model, allowsmovements of particles in six directions used to simulate

aerodynamics

Solute around obstacles

Poiseiulle flow


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CA simulation of epidemics

Cells have 3 states: susceptible,

infected, immune

A susceptible cell with neighboring

infected cell

becomes infected with probabilityp

becomes immune with probability

1p

Rate of spread depends on

susceptibility to virus pdefinition of neighborhood

Equivalent to forest fire model

SIR Epidemic model


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CA simulation of epidemics

Cells have 3 states: susceptible,

infected, immune

A susceptible cell with neighboring

infected cell

becomes infected with probabilityp

becomes immune with probability

1p

Rate of spread depends on

susceptibility to virus pdefinition of neighborhood

Equivalent to forest fire model

SIR Epidemic model



Applications of CA
http://videos/epidemic.ogghttp://find/

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CA simulation of traffic flow

Dynamics of vehicles represented

in coarse-grained way

Space and time are discretized

One dimensional toroid grid

Emergence of stop-and-go

behavior from injection of simple

non-determinism

slowdown probability

slow-to-start rules

An interesting example of

agent-like CA

Nagel-Schreckenberg model



Applications of CA
http://home/fpa/Documents/teaching/MAS-course/trafficCA/trafficCA.htmlhttp://goforward/http://find/http://goback/

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Cellular Automata

Thank you!



References
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References

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MAS 2011 Lecture 2

Documents

Transcript of MAS 2011 Lecture 2