Cellular Automata

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Cellular Automata
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Discrete Math

• A cellular automaton is a discrete model studied
  in computability theory and microstructure
  modeling.
• It consists of a regular grid of cells, each in
  one of a finite number of states.
• The grid can be in any finite number of
  dimensions.
• Time is also discrete, and the state of a cell at
  time t is a function of the states of a finite
  number of cells (called its neighborhood) at time
  t - 1.

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Dimensions

• A one-dimensional cellular automaton consists of
  two things a row of "cells" and a set of
  "rules".
• Two-dimensional cellular automata use rectangular
  grids of cells.
• "CA"means "one-dimensional cellular automaton.

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States of Automata

• Each of the cells can be in one of several
  "states". The number of possible states depends
  on the automaton. Think of the states as colors.
  In a two-state automaton, each of the cells can
  be either black or white. Of course, you might
  just as easily use purple and orange to represent
  the states, if you thing that's prettier. In a
  three-state automaton, the states might be black,
  red, and blue.

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Rules over Time

• A CA doesn't just sit there. Over time, the cells
  can change from state to state. The cellular
  automaton's rules determine how the states
  change. It works like this When the time comes
  for the cells to change state, each cell looks
  around and gathers information on its neighbors'
  states. You should think of the row of cells as a
  miniature "world" that runs through a sequence of
  "years" or "generations." At the end of each
  year, all the cells simultaneously decide on
  their state for the next year.

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• http//www.bitstorm.org/gameoflife/

The Rules For a space that is 'populated' Each
cell with one or no neighbors dies, as if by
loneliness. Each cell with four or more
neighbors dies, as if by overpopulation. Each
cell with two or three neighbors survives. For a
space that is 'empty' or 'unpopulated' Each cell
with three neighbors becomes populated.
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Examples

• The White Rabbit put on his spectacles.
• Where shall I begin, please your
• Majesty?" he asked.
• Begin at the beginning," the King said, very
• gravely, and go on till you come to the end then
  stop."
• Lewis Carroll, Alice's Adventures in Wonderland

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Self Replication

• http//en.wikipedia.org/wiki/Von_Neumann_cellular_
  automata
• Chou-Reggia loop
• Codd's cellular automaton
• Evoloop
• Langton's loops
• SDSR loop
• von Neumann's Universal constructor
• Asynchronous cellular automaton

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Continuous Automata

• There are continuous automata. These are like
  totalistic CA, but instead of the rule and states
  being discrete (e.g. a table, using states
  0,1,2), continuous functions are used, and the
  states become continuous. The state of a location
  is a finite number of real numbers. Certain CA
  can yield diffusion in liquid patterns in this
  way.

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Moore Neighborhood

• In two dimensions, with no threshold and the von
  Neumann neighborhood or Moore neighborhood, this
  cellular automaton generates three general types
  of patterns sequentially, from random initial
  conditions on sufficiently large grids,
  regardless of n.
• For larger neighborhoods, similar spiraling
  behavior occurs for low thresholds, but for
  sufficiently high thresholds the automaton
  stabilizes in the block of color stage without
  forming spirals.
• At first, the field is purely random. As cells
  consume their neighbors and get within range to
  be consumed by higher-ranking cells, the
  automaton goes to the consuming phase, where
  there are blocks of color advancing against
  remaining blocks of randomness.

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mathworld.wolfram.com/CellularAutomaton.html
http//www.cs.bgu.ac.il/sipper/papabs/epcm.pdf
http//en.wikipedia.org/wiki/Cellular_automatonS
elf-replication_in_cellular_automata
http//en.wikipedia.org/wiki/Belousov-Zhabotinsk
y_reaction http//math.hws.edu/xJava/CA/
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Patterns

• Patterns produced by the parity rule, starting
  from a 20 20 rectangular
• pattern. White squares represent cells in state
  0, black squares represent
• cells in state 1. (a) after 30 time steps (t
  30), (b) t 60, (c) t 90, (d)
• t 120.

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Schematic Diagram
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History

• Over the years they have been applied to the
  study of general phenomenological
• aspects of the world, including communication,
  computation, construction, growth, reproduction,
  competition, and evolution.

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By Katherine Kulik