CELLULAR AUTOMATAS

RULES AND APPLICATIONS

MARIA LUISA MARTINEZ · FI UPM · MASTER CACI

COMPLEX SYSTEMS

CONTENT

1.CELLULAR AUTOMATA

2.SPECIFIC RULES

3.NATURAL CA-LIKE PHENOMENA & APPLICATIONS

4.LET’S PLAY: How to use it?

5.REFERENCES

1.CELLULAR AUTOMATA

CELLULAR AUTOMATON (pl.), commonly

abbreviated CA.

A CA is a dynamical system that is discrete in space

and time.

A cellular automaton consists of a regular grid of

cells, each in one of a finite number of states, such

as on and off.

All interactions all local, with the next state of cell

being a FUNCTION of the current state of itself and

its neighbors.

1.CELLULAR AUTOMATA

Radius of a cell: r

Neighborhood size: consists of total of 2r+1 cells, 2r+1 = N

We will denote state of cell i at time t by c

i

(t)

Number of states: k

c

i

(t) = function (c

i-1

(t-1) , c

i

(t-1) , c

i+1

(t-1) ) left, right and

current cell.

ONE DIMENSIONAL CA

C

i-r

(t-1)

C

i-1

(t-1)

C

i

(t-1)

C

i+1

(t-

1)

C

i+r

(t-1)

C

i

(t)

…

SPACE

TIME

GENERAL CA PARAMETERS

1.CELLULAR AUTOMATA

ONE DIMENSIONAL CA

→ One possible rule table for a one-dimensional CA, with r=1 and k=2.

→ Last column contains the next state of the cell. Since there are 8 rows

in the table, there are 256 rule tables!!!!

1.CELLULAR AUTOMATA

TWO DIMENSIONAL CA

Cellular automata are mathematical models for systems in

which many simple components act together to produce

complicated patterns of behavior.

The extension to 2D is significant for comparisons with

many experimental results on pattern formation in physical

systems.

Immediate applications include dendritic crystal growth,

reaction-diffusion systems, and turbulent flow patterns.

(The Navier-Stokes equations for fluid flow appear to admit

turbulent solutions only in two or more dimensions.)

1.CELLULAR AUTOMATA

Wolfram´s CA Classification.

Conway’s Game of Life.

TWO DIMENSIONAL CA

NW

N

NE

W

CELL

E

SW

S

SE

→ Next cell state = function(states of the nearest neighbors)

→ You can specify rules, in the same way as in 1D CA.

S(t+1)=F(S

j

(t); j € N

S, state

F, rule function

1.CELLULAR AUTOMATA

N DIMENSIONAL CA

2.SPECIFIC RULES

INITIAL CONDITIONS

To start with the updating of the cells of the CA we must specify

the initial state of the cells, the initial conditions or seed.

2.SPECIFIC RULES

1D CA RULES

N=3 neighborhood size (2 neighbors)

k=2 binary value range (two states on/off)

2

N

number of neighborhoods

R= k

k

N

number of different rules

→ The rule number R, is its decimal equivalent, which will

range from 00000000b to 1111111b, and R from 0 to 255.

→ At each time step, the value of a site at position is updated

according to the rule

2.SPECIFIC RULES

RULE 30

1D CA

00011110b 30

Wolfram uses this rule in

Mathematica® for

creating random

integers.

2.SPECIFIC RULES

RULE 184

1D CA

THE TRAFIC RULE

Each 1 cell corresponds to a vehicle, and each vehicle moves forwards

only if it has open space in front of it.

RULE 184 is used as a simple model for traffic flow in a

single lane of a highway, and forms the basis for many CA

models of traffic flow more sophisticated.

2.SPECIFIC RULES

RULE 184

THE TRAFIC RULE

2.SPECIFIC RULES

RULE FROMS AND EQUIVALENCES

2.SPECIFIC RULES

RULE FROMS AND EQUIVALENCES

2.SPECIFIC RULES

RULE 90

01011010b = 90 RULE

Rule 90 is also called the Sierpiński automaton, due to

the characteristic Sierpiński triangle shape it generates.

Each cell's new value is the exclusive or of the two

neighboring values.

RULE FUNCTION: s(t+1) = s_leftCell(t) XOR s_right(t)

RULE 110

01101110b = 110 RULE

Class 4 Wolfram Classification

Each cell's new value is a function or of the two

neighboring values and the current cell state.

RULE FUNCTION:

s(t+1) = [NOT(s_leftCell(t)) AND s_currentCell(t)] OR

[s_currentCell(t) XOR s_right(t)]

2.SPECIFIC RULES

SOME 1D SPECIAL RULES

RULE 51 (00110011)

COMPLEMENTARY

s(t+1) = NOT(s_currentCell(t))

RULE 170 (10101010)

LEFT SHIFT

s(t+1) = s_leftCell(t)

RULE 204 (11001100)

IDENTITY

s(t+1) = s_currentCell(t)

RULE 240 (11110000)

RIGHT SHIFT

s(t+1) = s_rightCell(t)

1D RULES

EXPLORATIONS:

2.SPECIFIC RULES

2D CA RULES

5 neighbor and simple

central seed:

9 neighbor and simple

central seed:

2.SPECIFIC RULES

2D CA RULES

2CA parameters

k = 2

4 neighbors and 1 active cell

N = 5

Number of rules

R = 2

2

5

= 1024

R = [0..1023]

10 binary digits

SEED: central cell of the grid.

Initial Rule: 452 (111000100b)

2D RULES

EXPLORATIONS:

1.NATURAL CA-LIKE PHENOMENA

Lattice-gas CA

The lattice-gas automaton is a model of how of gas or plasma

interacts in a local space.

This model use local interactions (its nearest neighbors).

STATISTICAL MECHANICAL SYSTEMS

HPP simulation of gas flow. Source:

Wikipedia.org

1.NATURAL CA-LIKE PHENOMENA

The Belousov-Zhabotinsky reaction (or B-Z)

B-Z is a chemical reaction.

Which oscillates towards a single equilibrium state, back and

forth between two such states, as a "chemical oscillator“.

This model is a CA that produces spiral waves that resemble

those produced by the B-Z reaction. Similar spiral waves

observed in biochemical reactions.

AUTOCATALYTIC CHEMICAL SETS

Computer simulation of the

Belousov–Zhabotinsky reaction

occurring in a Petri dish. Source:

Wikipedia.org

1.NATURAL CA-LIKE PHENOMENA

GENE REGULATION

1 gene can inhibit or activate another gene, which activates

a third an so on.

Stuart Kauffman.

Random Boolean Networks (RBNs) as models of biological

processes.

Different rules for each cell, unlike the Wolfram CA, which had

cells that all have the same rule table.

1.NATURAL CA-LIKE PHENOMENA

MULTICELLULAR ORGANISMS

At the heart of the growth of a multi-cellular organism is

the process of cellular division…

1.NATURAL CA-LIKE PHENOMENA

COLONIES AND SUPER-ORGANISMS

FLOCKS AND HERDS

ECOSYSTEMS

ECONOMIES AND SOCIETY

IMAGE PROCESSING

PARALLEL COMPUTATION

Etc.

OTHER:

1.NATURAL CA-LIKE PHENOMENA

An SDSS, is computer system for solving spatial problems.

It is designed to assist the spatial planner in making land

use decisions.

An SDSS usually exists in the form of a computer model,

including a land use model.

Various techniques are available to simulate land use

dynamics,

Cellular Automata (CA) based models

Agent Based Models (ABM).

A SPATIAL DECISION SUPPORT SYSTEMS (SDSS)

MATRIX CA

The chess-board is the world; the pieces are the phenomena

of the universe; the rules of the game are what we call the

laws of Nature.

T. H. Huxley

4. LET’S PLAY

THE IDEA

①TO GENERATE THE MATRIX MOVIE SCREEN

4. LET’S PLAY

THE IDEA

Possible states of cells: k = 256

0 black | 1 random range of green color (based on R,G,B)

② THE COLOR

pixel = color (R,G,B) ; R,G,B = 0..255

green = color (0, 0..255,0)

4. LET’S PLAY

SIMPLEST CASE: 1D CELLULAR AUTOMATA

Possible states of cells: k = 256

Neighborhood size (2 neighbors, r=1) N = 3

Number of entries in the rule table: k

N

= 16777216

Number of different rules: R= 2

k

N

=2

16777216

③ SIMPLE CELLULAR AUTOMATA 1D

④ NEIGHBORHOOD GEOMETRY

4. LET’S PLAY

SIMPLEST CASE: 1D CELLULAR AUTOMATA

The initial state must be a random value for cells, which

some will be black an other will be in a random range of

green color.

⑤ INITIAL STATE OR SEED

4. LET’S PLAY

SIMPLEST CASE: 1D CELLULAR AUTOMATA

Window size: 200 x 200 pixels

Size of each cell: int scl = 20 pixels (this value can change)

Number of cells in the window: 100

For a different values of window and “scl” we obtain

different number of cells.

⑥ THE CA GEOMETRY

⑦ THE BOUNDARY CONDITIONS

We are going to ignore edges that only have one neighbor

4. LET’S PLAY

SIMPLEST CASE: 1D CELLULAR AUTOMATA

Is possible to use different rules.

RULE 240 (11110000b) RIGHT SHIFT

Current_cell(t+1) = left_cell(t)

⑧ THE RULE CONDITIONS

⑨ STOPPING CONDITION

Repeatedly update all the cells of the CA, until some stopping

condition is met (for example, a pre-assigned number of steps

is attained, or the CA is in a quiescent state, or cycles in a

loop,...).

When the height of the Window is achieved.

4. LET’S PLAY

SIMPLEST CASE: 1D CELLULAR AUTOMATA

⑩ RUN YOUR CA!!

RULE 240

scl = 20

RULE 30

scl = 2

RULE 240

scl = 4

RULE 90

scl = 10

SEED = CENTER

4. LET’S PLAY

2D CELLULAR AUTOMATA

Possible states of cells: k = 256

0 black | 1 random range of green color (based on R,G,B)

② THE COLOR

Neighborhood size: N = 3

Number of entries in the rule table: k

N

= 16777216

Number of different rules: R= k

k

N

=k

16777216

③ SIMPLE CELLULAR AUTOMATA 1D

④ NEIGHBORHOOD GEOMETRY

4. LET’S PLAY

2D CELLULAR AUTOMATA

The initial state must be a random value for all the cells.

⑤ INITIAL STATE OR SEED

Window size: 400 x 400 pixels

Size of each cell: int scl = 2 pixels

Number of cells in the window: 40000!!

⑥ THE CA GEOMETRY

⑦ THE BOUNDARY CONDITIONS

We are going to consider all edges.

4. LET’S PLAY

2D CELLULAR AUTOMATA

for (int r=0; r<R; r++) {

for (int c=0; c<C; c++) {

int state = curr[r][c];

int lr = prev[r][(c-1+C)%C] + prev[r][(c+1)%C];

int ab = prev[(r-1+R)%R][c] + prev[(r+1)%R][c];

if (ab == 0) { //&0xff, bit mask to select the green color

curr[r][c] = (state+lr)&0xff; // adds new "life" to the ca

} else { //% es el resto, para no exceder nunca el valor de 255

curr[r][c] = (lr%tb)&0xff; // gives the "matrix"-y effect

}

⑧ THE RULE CONDITIONS

⑨ STOPPING CONDITION

No stopping condition.

4. LET’S PLAY

2D CELLULAR AUTOMATA

⑩ RUN YOUR CA!!

4. REFERENCES

Gary William Flake (1998). The Computational Beauty of Nature.

Computer explorations of Fractals, Chaos, Complex Systems and

Adaptation. The MIT Press. 231-259.

Ganguly, Niloy; Sikdar, Biplab K.; Deutsch, Andreas; Canright, Geoffrey;

Chaudhuri, P. Pal (December 2003), A survey on cellular automata, Tech.

Report, Centre for High Performance Computing, Dresden University of

Technology.

Stephen Wolfram web page. available at:

http://www.stephenwolfram.com/

Processing.org Wolfram tutorial. Available at:

http://processing.org/learning/topics/wolfram.html

Processing 1.5.1. Java Reference.

THANKS FOR YOUR ATTENTION!

REMEMBER …

THE MATRIX HAS YOU

Any question?

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