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CELLULAR AUTOMATA

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?

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CELLULAR AUTOMATA

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  1. CELLULAR AUTOMATA A Presentation By CSC

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

  3. 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

  4. ONE-DIMENSION CA • Binary, nearest-neighbor, one-dimensional • 256 rules, using Wolfram code

  5. 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

  6. TWO DIMENSION CA • Neighborhood definition: • von Neumann Neighborhood • Moore Neighborhood

  7. 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

  8. 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

  9. CONWAY’S GAME OF LIFE • Still Life, Ex: boat • Oscillator, Ex: Blinker • Spaceship Ex: Glider

  10. CONWAY’S GAME OF LIFE • Three phase oscillator • Guns, Ex:Glider Gun

  11. 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)

  12. 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

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