Predicting Emergence from Cell’s Structure - Systematic Design for Emergence in Cellular Nonlinear Networks - page 123

the active area is large (at most 1) there will be a slow implosion while if the un-
certainty within this area ua is 0 or close to 0 it is quite likely that a fast implo-
sion will take place. If ue is larger than 0, the CA may have an “explosive” dy-
namics and this is the more likely as ue approaches 1 (or

p U ). It is expected,
as in the case of the experimentally determined exponent of growth U , that com-
plex and interesting behaviors are likely for values of

2

p U .
Note that for a given cell ID and a specified interconnectivity the probabilistic
exponent of growth is determined only by the structure of the cell (the bits of the
cells' ID) without the need to simulate a large CA for many iterations. Still, nu-
merical simulations shows that it has a very good predictive capability with re-
gards to an arbitrarily sized CA using a specified cell.

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7.5.1 Predicting Behaviors from ID, an Example

Let us consider the 1s5 case. Using the above formulae one finds the following
condition for the existence of “explosive” behaviors, in the case of “0” quiescent
state:

if and only if (7.28) ue ! 0

222 y y y 2 y y 2 .

1 0 2 1 0

It is seen that only the less significant 3 bits of the ID are relevant for this fea-
ture. Moreover, since there only eight possible arrangements of these 3 bits it is
possible to check (7.28) for each of them, concluding that it holds in 75% cases,
i.e. in all cases except those when all bits are 1 or 0. The milder case (correspond-
ing to yy . This is the case

where slow growth and emergence (potentially emergence of gliders) is more
likely to occur. Expressed in decimal the above condition writes

) holds when or when ue 0 25 y y y 011 y 100

2 1 0 2 1 0

or

ID k 8 3

ID , where k is any arbitrary integer.
Simulations confirmed the prediction as seen in Fig. 7.7. where several exam-
ples of

k 8 4

are considered.

ID k 8 4

7.5.2 Other Properties of the Probabilistic Exponents of Growth

As seen from the above example, the predictive power of the probabilistic expo-
nents of growth is impressing. But many other interesting properties of the CA can
be now established by imposing conditions like (7.28). Calculating the output pro-
babilities while imposing (7.28) reveals that in the case of 1s5 semi-totalistic CAs
there are only three possible cases for the “exploding” behaviors; namely:
25 (with potential for complexity and gliders), and (mostly

ue 0 ue 0 75 ue 1

related to random number generators).