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speed of light” in some CA papers. The speed of light is expressed in cells per
iterations and is solely determined by the neighborhood and the CA topology.
For our examples, the speed of light is 2 cells per iteration and for a spatial
cluster located in the middle a good choice is
T
=(77-11)/2=33 iterations.
The
exponent of growth
is defined as the ratio between the area of the terminal
state spatial cluster
A(T)
and
A(0).
Therefore:
A
(
A
T
)
(5.1)
U
(
0
Several qualitatively different situations may occur:
(a)
(U=0, passive).
The
spatial cluster
“implodes rapidly”, i.e. its
area
as defined
above is rapidly shrinking until it becomes 0 (there will be no terminal spatial
cluster at all). This situation is similar to
passivity
as defined by the local activity
theory in the context of Reaction-Diffusion cellular systems [8]. Such an example
is ID = 924, as exemplified in Fig. 5.2.
In principle there is no computational meaning (or practical use) for a passive
behavior since the final state of the CA contains no information at all. However in
some cases stopping the CA array earlier (see iterations 2 or 4 in Fig. 5.2) may re-
sult in a form nonlinear filtering which may be of use for certain applications.
(b)
(0<U<1, stable but active)
. In this case
A(T)<A(0),
i.e. the initial state
spa-
tial cluster
shrinks (slower when
U
approaches 1) but it does not disappear. See
for instance, the example in Fig. 5.3, obtained for ID = 890. In this case U = 0.07.
In terms of
local activity theory
[8] this behavior corresponds to a
stable but ac-
tive
behavior. Its relevance for computation is that of
feature extraction
and
nonlinear filtering
. Also within this category, particularly for U approaching 1,
glider formation may occur
. The well-known “Game of Life” cellular automata, which
Fig. 5.2.
Time evolution for ID = 924. It is a typical evolution for
passive
behaviors. The
exponent of growth in such cases is always
U=0.
The upper row represents the time evolu-
tion of the cellular array, while the lower row indicates the evolution of the rectangular
boundary enclosing the evolving spatial cluster
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