Download - 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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