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Cellular Automata

CS6800 Summer I 2009

Thap Panitanarak

Cellular Automata

Definition

� Discrete dynamic system

� Discrete in space, time and state

� “Cells” in space

� “States” in time

� Current states of all its neighborhoods (may including itself) defines next state of that cell

� State changed according to “Local rule”

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Cellular Automata

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Formal Definition� Cellular Automaton C = <S, s0, G, d, f>� S is finite set of states� s0 is initial state, s0 in S� G is cellular neighborhood

� G = {i, i+r1, i+r2, …, i+rn} where n is neighborhood size

� d – dimension� f: Sn � S is local rule� C(t) is configuration at time t

� C(t) = (s0(t), s1(t), …, sN(t)) where N is finite size of CA & si(t) is state of cell i at time t

� Global mapping, F� F: C(t) � C(t+1)

Cellular Automata

Varieties of Cellular Automata

� Dimension

� Shape

� Color

� Neighborhood

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Cellular Automata

Dimension

one-dimensional two-dimensional

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Cellular Automata

Shape

square hexagon

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Cellular Automata

Color

2 colors 3 colors

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Cellular Automata

Neighborhood

Moore von Neumann

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Cellular Automata

� Theoretically, space is defined infinitely

� Practically, space is bounded

� How to handle cells along the edges?

� Keep them static

� Differently defined neighborhoods

� Wrap-around: torus

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Cellular Automata

Elementary Cellular Automata (ECA)

� Simplest

� Dimension: one

� Shape: square

� Color: two (binary)

� Neighborhood: nearest

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Cellular Automata

ECA Properties

� Next state can be derived by current states of itself & its two neighborhoods (left & right)

� Neighborhood: 3 cells � 2 x 2 x 2 = 8 possible patterns of states

� Total rules = 28 = 256 rules

� Rule’s number represented in binary

� Rule 30 = 111102

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Cellular Automata

ECA with rule 30

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Cellular Automata

ECA more examples

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Cellular Automata

ECA more examples

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Cellular Automata

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Totalistic Cellular Automata (TCA)

� Dimension: one

� Shape: square

� Color: k

� Neighborhood: nearest

� Use average of cells in neighborhood to derive next state

Cellular Automata

TCA Properties

� Next state can be derived by “average” of current states of itself & its two neighborhoods (left & right)

� Neighborhood consists of three cells, itself, left & right cells

� With k = 3, all possible averages (sums) are0, 1, 2, 3, 4, 5, 6 � 7 possible sets of states

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Cellular Automata

TCA Properties

� Indexed with (3k – 2)-digit k-ary number called “code”

� With k = 3 � 7-digit 3-ary number

� Code 777 = 10012103

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Cellular Automata

TCA with code 777

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Cellular Automata

TCA with code 777

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Cellular Automata

TCA more examples

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Cellular Automata

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Conway's Game of Life (Life)

� Dimension: two

� Shape: square

� Color: two (binary)

� Neighborhood: Moore/nearest (without itself)

� Use average of cells in neighborhood to derive next state

� It can view as 2-dimensional, binary totalistic cellular automata

Cellular Automata

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Life’s Rule

� Count – summation of its neighborhoods

� Use 3 rules; death, survival, birth

� Death – on cell changed to off if count < 2 or count > 3

� Survival – on cell left unchanged if count = 2 or count = 3

� Birth – off cell changed to on if count = 3

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Cellular Automata

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Life – Simple Example

Cellular Automata

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Life – Still Life

� Special Patterns that their states do not change on time

� No death & no birth, only survival

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Cellular Automata

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� Life – More Examples

� http://www.math.com/students/wonders/life/life.html

� 2D Cellular Automata

� http://psoup.math.wisc.edu/mcell/mjcell/mjcell.html

� 3D Cellular Automata

� Visions of Chaos

Cellular Automata

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� Universal Cellular Automata

� Capability of simulating any cellular automata or Turing machine

� It had been proved that ECA (with rule 110) [Cook 2004] & Life [Berlekamp, Conway & Guy1982][Gardner 1983] are universal

� In 2002, Wolfram had proved that one-dimensional, two-color cellular automata with nearest neighbor rules is sufficient to exhibit universality

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Cellular Automata

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Conclusion

� Three fundamental properties of CA

� Parallelism : each cell updates independently

� Locality : new state derived from previous states of neighborhoods

� Homogeneity : use local rule, common for all cells

� Simple systems � more complex, dynamic systems

Cellular Automata

6/10/2009CS6800 Summer I 2009 Thap Panitanarak

References

� http://mathworld.wolfram.com/CellularAutomaton.html

� http://en.wikipedia.org/wiki/Cellular_automata

� L. N. de Castro, “fundamentals of natural computing: an overview”, Physics of Life Reviews, 4 (2007), 1–36.

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Cellular Automata

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Question

� Show the first five generations of elementary cellular automata using rule 30 with a start state that has only middle cell black