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Fig. 5.3. Time evolution for ID = 890. It is a typical evolution for stable but active behaviors.
Here the exponent of growth is larger than 0, but not reaching 1 . The upper row represents
the time evolution of the cellular array, while the lower row indicates the evolution of the
rectangular boundary enclosing the evolving spatial cluster
has been proved to be an universal Turing Machine belongs to this category and its
exponent of growth was determined as U = 0.2 . Although values of U closer to 1
may be associated with long transients and complexity, there is no direct corre-
lation between the U measure and the transient length. Sometimes very long tran-
sients are observed even for relatively low values of U, and this is actually the
case with the “Game of Life”, (ID = 6,152 from “2s9”) with an experimentally
computed U=0.2 . Note however that a more accurate method described in Chap. 7
×
gives U1 (with U > 1 ) for the
Game of Life
”.
(c) (U=1, edge). This is an interesting case indicating behaviors that are located
at the boundary between stable (shrinking) and unstable (growing) behaviors. In
these cases the area of the initial state spatial cluster is not changing in time or it
is slowly and alternatively increasing and decreasing. Though, the CA dynamics
exhibit relevant information processing of the initial state spatial cluster. The gene
ID = 768 discussed above (which draws an opaque rectangle perfectly framing the
spatial cluster) belongs to this class. Other example is gene ID = 1011 with a
dynamics exemplified in Fig. 5.4. Among such genes one may find interesting
computational applications if a meaning of the transformation between the initial
and the terminal state can be established. For instance, gene ID = 768 is used suc-
cessfully for character segmentation, as shown later in Chap. 8.
(d) (1<U<4, unstable near edge) . This is actually the most interesting case, which
may have an equivalent in the unstable near the edge of chaos behaviors introduced in
cellular systems analyzed with the local activity theory [7,8]. The value of 4 in the
above boundary is not critical and may depend on the CA family under investigation.
The CAs within this category are characterized by a “slow explosion”, i.e. a slow
growing of the initial state spatial cluster. Sometimes (depending on the initial state)
they evolve to a stable cluster with a larger area while sometimes the growth is indefi-
nite but very slow, until the entire cellular array is filled. This process, in my opinion
reflects better living systems than imploding dynamics as observed for the “Game of
Life”. Another complex phenomena observed for the CAs in this category is the emer-
gence of self-making patterns, living for many iterations. Such behaviors were referred
as autopoiesis [64].
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