cellular automata : a simple introduction

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Presented by

ADEKUNLE ONAOPEPO HUSAMAT

CELLULAR AUTOMATA

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INTRODUCTION BACKGROUND SYNTAX COMPONENTS BEHAVIOUR VARIANTS APPLICATIONS CASE STUDIES LEVEL OF KNOWLEDGE ADVANTAGES DRAWBACKS REFERENCES

OVERVIEW

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“A CA is an array(Spatial Lattice) of identically programmed automata,

or cells,which interact with one another in a

neighborhood and havedefinite state”

What are Cellular Automata? CA are discrete dynamic systems.

CA's are said to be discrete because they operate in finite space and time and with properties that can have only a finite number of states.

CA's are said to be dynamic because they exhibit dynamic behaviours.

Basic Idea: Simulate complex systems by interaction of cells following easy rules.

“Not to describe a complex system with complex equations, but let the complexity emerge by interaction of simple individuals following simple rules.”

INTRODUCTION

From Another Perspectiveit is a Finite State Machine, with one transition function for all the cells, this transition function changes the current state of a cell depending on the previous state for that cell and its neighbors.

BACKGROUND

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Time Frame Major Players Contribution

Early 50’s J. Von Neuman, E.F. Codd, Henrie & Moore , H Yamada & S. Amoroso

Modeling biological systems - cellular models

‘60s & ‘70s A. R. Smith , Hillis, Toffoli Language recognizer, Image Processing

‘80 s S. Wolfram ,Crisp,Vichniac Discrete Lattice,statistical systems, Physical systems

‘87 - ‘96 IIT KGP, Group Additive CA, characterization,applications

‘97 - ‘99 B.E.C Group GF (2p) CA

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Cellular Automata: Lattice, Neighbourhood, Set of discrete states, Set of transition rules, Discrete time.

“CAs contain enough complexity to simulate surprising and novel change as reflected in emergent phenomena”(Mike Batty)

SYNTAX

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Cell Basic element of a CA. Cells can be thought of as memory

elements that store state information. All cells are updated synchronously

according to the transition rules. Lattice

Spatial web of cells. Simplest lattice is one dimensional. Others include 2,3… Dimensional

COMPONENTS

Initialcurrent

1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0Rule #126

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• 2 dimensional

• 3 dimensional•For 1D CA:

23 = 8 possible “neighborhoods” (for 3 cells)

28 = 256 possible rules

• For 2D CA:29 = 512 possible “neighborhoods”2512 possible rules (!!)

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•The cells on the end may (or may not) be treated as "touching" each other as if the line of cells were circular.

If we consider them as they touch each other, then the cell (A) is a neighbor of cell (C)

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• if #alive =< 2, then die• if #alive = 3, then live• if #alive >= 5, then die

• if #alive =< 2, then die• if #alive = 3, then live• if #alive >= 5, then die

• if #alive =< 2, then die• if #alive = 3, then live• if #alive >= 5, then die

“A CA is an array of identically programmed automata, or cells,which interact with one another in a neighbourhood and havedefinite state”

BEHAVIOUR

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“A CA is an array of identically programmed automata, or cells,which interact with one another in a neighborhood and havedefinite state”

BEHAVIOUR

Von Neumann Neighborhood

Moore Neighborhood

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“A CA is an array of identically programmed automata, or cells,which interact with one another in a neighborhood and havedefinite state”

2 possible states: ON OFF

O

W JA

R

I T

D

G M

X E

N Z

R

P

A

Z

26 possible states: A … Z

Never infinite!

BEHAVIOUR

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Rules Space and Time

t

t1

BEHAVIOUR

Initial Configuration

Initial Starting state of all cells in the lattice e.gthe initial configuration for all the cells is state 0, except for 4 cells in state 1.

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Asynchronous CA CA rules are typically applied simultaneously across all cells in the lattice.

This variant allows the state of the cells to be updated asynchronously. Probabilistic CA

The deterministic state-transitions are replaced with specifications of the probabilities of the cell-value assignments.

Non-homogenous CA State transition rules are allowed to vary from cell to cell.

Mobile CA Some or all lattice sites are free to move about the lattice. Essentially primitive models of mobile robots. Used to model some aspects of military engagements.

Structurally Dynamic CA The topology (the sites and connections among sites) are allowed to evolve.

VARIANTS

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Self-reproduction Diffusion equations Artificial Life Digital Physics Simulation of Cancer cells growth Predator – Prey Models Art Simulations of Social Movement Alternative to differential

equations CA based parallel processing

computers Image processing and pattern

recognition

SimulationsGas behaviorBiological

processesForest fire

propagationUrban

developmentTraffic flowAir flowCrystallization

process

APPLICATIONS

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Study of evolution of rules involving one dimensional cellular automata

CASE STUDY

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CASE STUDY

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CASE STUDY

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CASE STUDY

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I. Always reaches a state in which all cells are dead or alive

II. Periodic behavior

III. Everything occurs randomly

IV. Unstructured locally organized patterns and complex behavior

Results: Classifying Cellular Automata RulesCASE STUDY

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CASE STUDY

During each time step the system is updated according to the rules:

Forest Fire Model is a stochastic 3-state cellular automaton defined on a d-dimensional lattice with Ld sites.

Each site is occupied by a tree, a burning tree, or is empty.

1. empty site tree with the growth rate probability p 2. tree burning tree with the lightning rate probability f, if no nearest

neighbour is burning 3. tree burning tree with the probability 1-g, if at least one nearest

neighbour is burning, where g defines immunity. 4. burning tree empty site

Forest Fire Model (FFM)

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CASE STUDY

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The application

CASE STUDY

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Eventually

After some time forest reaches the steady state in which the mean number of growing trees equals the mean number of burned trees.

CASE STUDY

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Model predator/prey relationship by CA Begins with a randomly distributed population of fish, sharks, and empty

cells in a 1000x2000 cell grid (2 million cells) Initially,

50% of the cells are occupied by fish 25% are occupied by sharks 25% are empty

CASE STUDY

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Breeding rule: if the current cell is empty If there are >= 4 neighbors of one species, and >= 3 of them are of breeding

age, Fish breeding age >= 2, Shark breeding age >=3,

and there are <4 of the other species:then create a species of that type

+1= baby fish (age = 1 at birth) -1 = baby shark (age = |-1| at birth)

CASE STUDY

Initially cells contain fish, sharks or are emptyEmpty cells = 0 (black pixel)Fish = 1 (red pixel)Sharks = –1 (yellow pixel)

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EMPTY

CASE STUDY

Breeding Rule: Before

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CASE STUDY

Breeding Rule: After

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Shark rule: DetailsIf the current cell contains a shark: Sharks live for 20 generations If >=6 neighbors are sharks and fish neighbors =0, the shark dies (starvation) A shark has a 1/32 (.031) chance of dying due to random causes If a shark does not die, increment age

CASE STUDY

Fish rule: DetailsIf the current cell contains a fish: Fish live for 10 generations If >=5 neighbors are sharks, fish dies (shark food) If all 8 neighbors are fish, fish dies (overpopulation) If a fish does not die, increment age

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Results

Next several screens show behavior over a span of 10,000+ generations

CASE STUDY

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Generation: 0

CASE STUDY

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Generation: 500

CASE STUDY

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Generation: 100

CASE STUDY

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Generation: 1,000

CASE STUDY

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Generation: 2,000

CASE STUDY

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Generation: 4,000

CASE STUDY

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Generation: 8,000

CASE STUDY

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Generation: 10,500

CASE STUDY

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Borders tended to ‘harden’ along vertical, horizontal and diagonal lines

Borders of empty cells form between like speciesClumps of fish tend to coalesce and form convex shapes or

‘communities’

Long-term trendsCASE STUDY

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Generation 100 20001000

4000 8000

Medium-sized population (1/16 of grid)

Random placement of very small populations can favor one species over another

Fish favored: sharks die out Sharks favored: sharks predominate, but fish survive in

stable small numbers

CASE STUDY

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Cellular automata provides structural knowledge level through the initial configuration of the system that evolved

Generative knowledge level is also provided by the transition rule to generate next data set of the system

LEVEL OF KNOWLEDGE

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Powerful computation engines. Allow very efficient parallel computation

Discrete dynamical system simulator. Allow for a systematic investigation of complex phenomena.

Original models of fundamental physics. Instead of looking at the equations of fundamental physics, consider

modelling them with CA. Emergent behaviour of complex group from simple individual

behaviour can be studied. Simulation results are much more intuitive as it is well visually

represented Simple to Implement

ADVANTAGES

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Not suitable for systems that require synthesis. Since CA rules cannot be easily predict results

Results may contain redundant information. Patterns which seem complex can be generated but are un-important data

as concerned with emergent behaviour of the actual system. It is not sometimes easy to obtain perfect rules governing

evolution of the system It is difficult to understand whether a CA model captures the dynamics of

the system being modelled fully or adds superfluous dynamics

DISADVANTAGES

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Wolfram, S.: A new kind of science. Wolfram Media, Inc. (2002) Adamatzky, A., Alonso-Sanz, R., Lawniczak, A., Juarez Martinez, G.,

Morita, K., Worsch,T. (eds.): AUTOMATA-2008 Theory and Application of Cellular Automata (2008) http://cell-auto.com http://

www.brainyencyclopedia.com/encyclopedia/c/ce/cellular_automaton.html

Debasis Das: A Survey on Cellular Automata and Its Applications

REFERENCES

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