cellular automata - plone sitebabaoglu/courses/cas12-13/slides/ca.pdf · 2013-09-02 · alma mater...

ALMA MATER STUDIORUM – UNIVERSITA’ DI BOLOGNA

Cellular Automata

Ozalp Babaoglu

© Babaoglu Complex Systems

History

■ Developed by John von Neumann as a formal tool to study mechanical self replication

■ Studied extensively by Stephen Wolfram

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1-Dimensional CA

■ An (infinite) array of “cells”■ Each cell has a value from a k-ary state (assume binary)■ Each cell has has a position in the array and has r left and r

right neighbors (assume r =1)

3

......


1-Dimensional CA

4

1

2

3

4

5

0

t


State Transitions (Look-up Table)

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Xt Xt+1


Wolfram Canonical Enumeration

■ With a binary state and radius r =1, there are 223=256 possible CAs

■ Read off the final state column of the look-up table as a binary number

■ Each possible CA identified through an integer 0-255

6



7

2

4

#8

#16

Rule = 30



8

2

4

#8

#32

Rule = 110

#64


Wolfram’s Classification

■ Class 1: Nearly all initial patterns evolve quickly into a stable, homogeneous state (fixed point)

■ Class 2: Nearly all initial patterns evolve quickly into stable or oscillating structures (periodic)

■ Class 3: Nearly all initial patterns evolve in a pseudo-random or chaotic manner (chaotic)

■ Class 4: Nearly all initial patterns evolve into structures that interact in complex and interesting ways. This class is capable of universal computation

9 © Babaoglu Complex Systems

Wolfram’s Classification: Class 1

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Rule 40 Rule 172 Rule 234

Source: http://liinwww.ira.uka.de/~rahn/ca.pics/





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Rule 30 Rule 101 Rule 146



13

Rule 110


NetLogo

■ CA 1D Elementary

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Wolfram’s Classification

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Periodic Chaotic

Complex

Fixed

1.00.0 λ


Langdon’s λ Metric

■ Seek a compact characterization of the CA behavior class■ Count the number of “ones” in the look-up table final state

column

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0 1 1 8 4


Langdon’s λ Metric

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λ ALL#Rules Class#III Class#IV

0 1 0 0

1 8 0 0

2 28 2 0

3 56 4 1

4 70 20 4

5 56 4 1

6 38 2 0

7 8 0 0

8 1 0 0


Conway’s Game of Life

■ 2-Dimensional Cellular Automata■ Developed by British mathematician John Conway■ Similar to Schelling’s model■ Each cell has eight neighbors■ Each cell can be “alive” or “dead”■ Instead of moving or staying, cells come alive, die or survive

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19

1 2 3

4 X 5

6 7 8

“Dead” (off)

“Alive” (on)



■ Each cell has eight neighbors■ Each cell can be “alive” or “dead”■ Rules:■ a live cell with fewer than 2 live neighbors dies (loneliness)■ a live cell with 2 or 3 live neighbors survives (stasis)■ a live cell with more than 3 live neighbors dies (over crowding)■ a dead cell with exactly 3 live neighbors come alive (reproduction)

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Conway’s Game of Life Behaviors



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Fixed point





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26





28



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Periodic


Glider

30


Fish

31

t=0 t=1 t=2

t=3 t=4


NetLogo

■ Game of Life

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Universal Computation

■ Both Conway’s game of life and CA rule 110 are capable of universal computation

■ Prove by showing that the game of life is equivalent to a Turing Machine


Logical Operators from Game of Life

■ Building blocks:

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Logical Operators from Game of Life

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NOT AND OR

cellular automata - plone sitebabaoglu/courses/cas12-13/slides/ca.pdf · 2013-09-02 · alma mater...

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