CELLULAR AUTOMATA A Presentation By CSC

OUTLINE History One Dimension CA Two Dimension CA Totalistic CA & Conway’s Game of Life Classification of CA

HISTORY First CA: Ulam & von Neumann, 1940 Simulation of crystal growth Study of Self-replicating systems What is CA? Mathematical idealizations of natural systems Consist 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-dimensional 256 rules, using Wolfram code

ONE-DIMENSION CA Rule 30: Chaotic, random number generator in Mathematica Black cells b(n), closely fit by the line b(n) = n Rule 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, 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: Class 3: 0.4 < λ < 1