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ure 11-c. It appears that some defects in
φ
c
move randomly which suggests a
stochastic model of dislocation motion like it happens for some chaotic CA [5,4].
Finally Figure 12 shows in a log-log plot the decaying of the density of
defects
ρ
d
, where for the chaotic rule
φ
c
only the sequence of symbols 010110
is counted and its average density decreases due to recombinations with
t
−
0
.
39
.
For rules
φ
a
,
φ
b
and
φ
d
the density of defects
ρ
d
decreases as the time grows
until a non-zero asymptotic value.
10
0
φ
b
10
−1
φ
d
φ
a
10
−2
φ
c
10
−3
10
0
10
1
10
2
10
3
10
4
Log ( t )
Fig.12.
Log-log plot of the density of defects versus time. Data points are averaged
over 10 different experiments in a lattice of 2000 cells. For rule
φ
c
the experiments were
done in a lattice of 10
5
cells and only defects of type “010110” were counted. The slope
of the dashed line is
−
0
.
39.
4 Conclusion
Non-trivial collective behavior in cellular automata (CA) is a striking phenom-
ena not well understood yet. In this work we presented results in which a genetic
algorithm (GA) is used to evolve one and two-dimensional binary-state CA to
perform a non-trivial collective task in which the concentration of activated
cells oscillates among three different values. We found that with an appropriate
fitness function the artificial evolutionary process is able to detect several CA
rules with the desired behavior.
In
d
= 2 the stability of the collective behavior is very sensitive to small
amounts of noise and as the lattice size increases the size of the attractor
diminishes. In
d
= 1 the GA detects some rules that organize the space-time
