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Evolving Transition Rules for Multi Dimensional Cellular Automata

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3/22/09. Evolving Transition Rules for Multi Dimensional Cellular Automata. Ron Breukelaar ... M. Mitchell, J.P. Crutchfield, P.T. Hraber ... – PowerPoint PPT presentation

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Title: Evolving Transition Rules for Multi Dimensional Cellular Automata


1
Evolving Transition Rules for Multi Dimensional
Cellular Automata
University of Leiden (LIACS)
Ron Breukelaarrbreukel_at_liacs.nl
2
Evolving Transition Rules for Multi Dimensional
Cellular Automata
3
Evolving Transition Rules for Multi Dimensional
Cellular Automata
Cellular Automata
4
Evolving Transition Rules for Multi Dimensional
Cellular Automata
Cellular Automata Transition Rules for Cellular
Automata
5
Evolving Transition Rules for Multi Dimensional
Cellular Automata
Cellular Automata Transition Rules for Cellular
Automata Evolving Transition Rules for Cellular
Automata
6
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
7
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
8
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
9
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
10
Cellular Automata
Evolving Transition Rules for Multi Dimensional
Cellular Automata
si
si is the neighborhood of ai with r as radius
(here r 3)
11
Cellular Automata
Evolving Transition Rules for Multi Dimensional
Cellular Automata
Ct state of CA at time t
C0
C1
C2
C3
12
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
13
Transition Rules
Evolving Transition Rules for Multi Dimensional
Cellular Automata
1
0
1
0
0
1
0
0
1
1
0
1001011101101022r1 bits in rule
14
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.

15
Majority Problem (GKL)
Evolving Transition Rules for Multi Dimensional
Cellular Automata
(81,6 correct classifications)
16
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

17
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.
18
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
19
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
20
M. Mitchell, J.P. Crutchfield, P.T. Hraber
Evolving Transition Rules for Multi Dimensional
Cellular Automata
(copied experiment)
21
Two Dimensional Cellular Automata
Evolving Transition Rules for Multi Dimensional
Cellular Automata
von Neumann neighborhood
Moore neighborhood
22
Two Dimensional Cellular Automata
Evolving Transition Rules for Multi Dimensional
Cellular Automata
r 1
r 2
r 3
23
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
24
Number of cells
Evolving Transition Rules for Multi Dimensional
Cellular Automata
25
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.

26
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.

27
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.

28
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.)

29
Multi Dimensional Majority Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata
1D
3D
30
Multi Dimensional Majority Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata
31
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
32
Checkerboard Problem 1D
Evolving Transition Rules for Multi Dimensional
Cellular Automata
33
Checkerboard Problem 2D
Evolving Transition Rules for Multi Dimensional
Cellular Automata
34
Checkerboard Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata
35
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.
36
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.

37
AND Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata
(a) von Neumann, (b) Moore
38
XOR Problem
Evolving Transition Rules for Multi Dimensional
Cellular Automata
(a) von Neumann, (b) Moore
39
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)
40
Bitmap generation
Evolving Transition Rules for Multi Dimensional
Cellular Automata
41
Bitmap generation
Evolving Transition Rules for Multi Dimensional
Cellular Automata
42
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.
43
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.
44
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.

45
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.

46
Questions?
Any questions / ideas?
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