Evolving Transition Rules for Multi Dimensional Cellular Automata

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Evolving Transition Rules for Multi Dimensional
Cellular Automata
University of Leiden (LIACS)
Ron Breukelaarrbreukel_at_liacs.nl
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Evolving Transition Rules for Multi Dimensional
Cellular Automata
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Evolving Transition Rules for Multi Dimensional
Cellular Automata
Cellular Automata
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Evolving Transition Rules for Multi Dimensional
Cellular Automata
Cellular Automata Transition Rules for Cellular
Automata
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Evolving Transition Rules for Multi Dimensional
Cellular Automata
Cellular Automata Transition Rules for Cellular
Automata Evolving Transition Rules for Cellular
Automata
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Evolving Transition Rules for Multi Dimensional
Cellular Automata
Cellular Automata Transition Rules for Cellular
Automata Evolving Transition Rules for Cellular
Automata Multi Dimensional Cellular Automata
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Evolving Transition Rules for Multi Dimensional
Cellular Automata
Cellular Automata Transition Rules for Cellular
Automata Evolving Transition Rules for Cellular
Automata Multi Dimensional Cellular
Automata Transition Rules for Multi Dimensional
Cellular Automata
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Evolving Transition Rules for Multi Dimensional
Cellular Automata
Cellular Automata Transition Rules for Cellular
Automata Evolving Transition Rules for Cellular
Automata Multi Dimensional Cellular
Automata Transition Rules forMulti Dimensional
Cellular Automata Evolving Transition Rules
forMulti Dimensional Cellular Automata
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Cellular Automata
Evolving Transition Rules for Multi Dimensional
Cellular Automata
C a1, a2, , an
a2
a1
a7
a6
a5
a4
a3
a8
an
an is linked to a1
ai ? 0,1
2n different states of C
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Cellular Automata
Evolving Transition Rules for Multi Dimensional
Cellular Automata
si
si is the neighborhood of ai with r as radius
(here r 3)
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Cellular Automata
Evolving Transition Rules for Multi Dimensional
Cellular Automata
Ct state of CA at time t
C0
C1
C2
C3
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Cellular Automata
Evolving Transition Rules for Multi Dimensional
Cellular Automata
Ct state of CA at time t
C0
? 0,12r1 ? 0,1 ? (si) ? ai
0
0
0
1
1
0
1
1
1
C1
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Transition Rules
Evolving Transition Rules for Multi Dimensional
Cellular Automata
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0
1
0
0
1
0
0
1
1
0
1001011101101022r1 bits in rule
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Majority Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata

• relative number of ones in C0
• Task? 0.5 ? iterate to all ones state ?
  
• within maximum I iterations.

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Majority Problem (GKL)
Evolving Transition Rules for Multi Dimensional
Cellular Automata
(81,6 correct classifications)
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M. Mitchell, J.P. Crutchfield, P.T. Hraber
Evolving Transition Rules for Multi Dimensional
Cellular Automata

• A Genetic Algorithm to evolve the rules
• A pool of 100 transition rules
• Evaluation by iterating CA on 100 random
  initial states uniformly dist. over nr. of ones
• Selecting 10 to survive every generation
• Generating the other 90 using crossover on the
  selected 10 and then mutation
• r 3, therefore 27 128 bits in rule and
  2128 possible rules

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M. Mitchell, J.P. Crutchfield, P.T. Hraber
Evolving Transition Rules for Multi Dimensional
Cellular Automata
Fn,m the relative number of correct
classifications out of m initial states with a
width of n cells.
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M. Mitchell, J.P. Crutchfield, P.T. Hraber
Evolving Transition Rules for Multi Dimensional
Cellular Automata
(copied experiment)
(a) and (b) are block expanding rules(c) and (d)
are particle based rules
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M. Mitchell, J.P. Crutchfield, P.T. Hraber
Evolving Transition Rules for Multi Dimensional
Cellular Automata
(copied experiment)
Block expanding rules
Particle communication based rules
Very bad rules
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M. Mitchell, J.P. Crutchfield, P.T. Hraber
Evolving Transition Rules for Multi Dimensional
Cellular Automata
(copied experiment)
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Two Dimensional Cellular Automata
Evolving Transition Rules for Multi Dimensional
Cellular Automata
von Neumann neighborhood
Moore neighborhood
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Two Dimensional Cellular Automata
Evolving Transition Rules for Multi Dimensional
Cellular Automata
r 1
r 2
r 3
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Multi Dimensional Cellular Automata
Evolving Transition Rules for Multi Dimensional
Cellular Automata
In a CA with d dimensions e1, e2, , ed and
connected borders von Neumann neighborhood
Moore neighborhood
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Number of cells
Evolving Transition Rules for Multi Dimensional
Cellular Automata
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Transition Rules
Evolving Transition Rules for Multi Dimensional
Cellular Automata

• Rules are defined the similar way as in one dim.
  CA
• Cells are numbered
• ? 0,1n ? 0,1 , n S(d, r)
• Bitstring length explodes in high
  dimensions 22S(d, r) bits in a rule. Only small
  number of dimensions or small radius look seem
  doable.
• Experiments will focus on two dimensional CA
  with r 1 to simplify them.

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Genetic Algorithm
Evolving Transition Rules for Multi Dimensional
Cellular Automata

• Improved Genetic Algorithm
• Using tournament selection.
• Using a gliding distribution instead of uniform.
• Using crossover for only 60 of the population.

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Experiments
Evolving Transition Rules for Multi Dimensional
Cellular Automata

• Four multi-dimensional experiments
• Majority Problem with von Neumann
  neighborhood to compare two and three dim. with
  one dim.
• AND and XOR problem to better show
  communication in 2D CA.
• Checkerboard Problem to test robustness
  of algorithm and again compare dimensionality.
• Bitmap generation to show the potential of
  multi dimensional CA and work towards a
  real-time application.

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Multi Dimensional Majority Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata

• The same pool size, evaluation method,
  selection method, crossover and mutation for
  d1, 2 or 3.
• CA with linked borders.
• A von Neumann neighborhood with r 1. For d2
  25 32 bits in rule and 232 possible rules for
  d2. (This is 296 times smaller then one dim.)

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Multi Dimensional Majority Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata
1D
3D
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Multi Dimensional Majority Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata
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Checkerboard Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata
Given a random initial state Generate a
checkerboard pattern. (alterning blank and white
in every direction) Note that the CA must have
even dimensions
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Checkerboard Problem 1D
Evolving Transition Rules for Multi Dimensional
Cellular Automata
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Checkerboard Problem 2D
Evolving Transition Rules for Multi Dimensional
Cellular Automata
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Checkerboard Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata
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AND and XOR Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata
Given two special input cells v1 and v2 AND
problem If (v1 AND v2 TRUE) ? iterate to an
all ones stateelse ? iterate to an all zeros
state. XOR problem If (v1 XOR v2 TRUE) ?
iterate to an all ones stateelse ? iterate to
an all zeros state.
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AND and XOR Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata
v2
v1

• Borders of the CA are unlinked to increase
  the distance between v1 and v2.
• I was set to 10 to increase challenge.

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AND Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata
(a) von Neumann, (b) Moore
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XOR Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata
(a) von Neumann, (b) Moore
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Bitmap generation
Evolving Transition Rules for Multi Dimensional
Cellular Automata
Given a target bitmap and an initial state Find
a transition rule that generates the target
bitmap from the initial state.
Initial state
Five target bitmaps(5 x 5)
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Bitmap generation
Evolving Transition Rules for Multi Dimensional
Cellular Automata
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Bitmap generation
Evolving Transition Rules for Multi Dimensional
Cellular Automata
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Bitmap generation
Evolving Transition Rules for Multi Dimensional
Cellular Automata
A von neumann neighborhood trying to spell the
name RON in a 11x5 CA and ending up having only
3 errors.
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Bitmap generation
Evolving Transition Rules for Multi Dimensional
Cellular Automata
A CA using a Moore neighborhood generating a 9 x
9 image that looks like a gecco.
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Conclusions

• From the results can be concluded
• Multi dimensional CA are able to solve the
  Majority Problem with results similar to the
  one dimensional CA, but with shorter
  duration times and with d2 smaller
  neighborhood.
• Multi dimensional CA can be used to
  evolve transition rules that exhibit
  communicational behaviour.
• Multi dimensional CA can be trained to exhibit
  very diverse behaviour and might well have
  real-world applications in Parallel Computing
  and modelling social / biological behaviour.

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Further Work

• Some ideas on continuation of this project
• Explore how far Bitmap Generation is
  possible. (is compression an option?)
• Try this approach on real-world applications.
• Investigate how other forms of crossover
  influence the results.

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Questions?
Any questions / ideas?