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CELLULAR AUTOMATA
A Presentation By CSC
OUTLINE
HistoryOne Dimension CATwo Dimension CATotalistic CA & Conway’s Game of LifeClassification of CA
HISTORY
First CA: Ulam & von Neumann, 1940Simulation of crystal growthStudy of Self-replicating systems
What is CA?Mathematical idealizations of natural systemsConsist of a lattice of discrete identical sites,
each site taking on a finite set of, say, integer values.
The values evolve in discrete times, according to some rules depend on the state of neighboring sites
ONE-DIMENSION CA
Binary, nearest-neighbor, one-dimensional256 rules, using Wolfram code
ONE-DIMENSION CA
Rule 30:Chaotic, random number generator in
MathematicaBlack cells b(n), closely fit by the line b(n)
= nRule 110:
Class IV behavior, Turing-complete
TWO DIMENSION CA
Neighborhood definition:von Neumann Neighborhood Moore Neighborhood
TOTALISTIC CA
The state of each cell in a totalistic CA is represented by a number
The value of a cell at time t depends only on the sum of the values of the cells in its neighborhood
CONWAY’S GAME OF LIFE
Invented by J.H.Conway, 1970. Became famous since an article in Scientific American 223, by Martin Gardner.
States of each cell are {0,1} Survive if neighbor’s sum is 2 or 3
Birth if sum is 3
Representation: S23/B3 or 23/3
CONWAY’S GAME OF LIFE
Still Life, Ex: boat
Oscillator, Ex: Blinker
Spaceship Ex: Glider
CONWAY’S GAME OF LIFE
Three phase oscillator
Guns, Ex:Glider Gun
CLASSIFICATION OF CA
Class 1 : evolves to a homogeneous state. Class 2 : evolves to simple separated
periodic structures. Class 3 yields chaotic aperiodic patterns. Class 4 yields complex patterns of localized
structures, including propagating structures. (Wolfram, 1984)
CLASSIFICATION OF CA
λ = number of neighborhood states that map to a non-quiescent state/total number of neighborhood states. (Langton, 1986)
Class 1: λ < 0.2
Class 2,4: 0.2 < λ < 0.4 Game of Life: 0.2734
Class 3: 0.4 < λ < 1