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