Predicting Emergence from Cell’s Structure - Systematic Design for Emergence in Cellular Nonlinear Networks - page 119

case but in a different topology as seen from Fig. 7.6. This situation takes in consi-
deration the fact that the cells are semi-totalistic and therefore permuting any input
except the one corresponding to the central cell, has no effect on the output. In the
case of arbitrary cells, the same method may be applied but then there is no sym-
metry around the central cell and therefore the uncertainty for each cell in the “next

state” shall be computed independently.

Fig. 7.6. CA cells involved in computing the probabilistic growth index, for the case of 2s5
CA family

7.4.3 Computing the Cells Function of Probability

In the above we used some special cases of the cells function of probability ( FP )
when the input probabilities were defined according with some of the possibilities
imposed by the initial state profile. In fact, an analytic function of FP can be es-
tablished for any type of cell, starting from the definitions and taxonomies intro-
duced in Chap. 3, Table 3.4. Here we will exemplify the procedure for the case of
“1a3” cells i.e. Boolean cells with three inputs. In this case, the excitation defined
¦

n

k 1

by e.g. (3.4) V is an integer between 0 and n (for the 1a3 cells, n = 3) 2 u

k

k 1

having the significance of an index for the ID bit y . There are eight possible
values for V in the case of Boolean functions with three inputs. Let assume that
such a cell has assigned a set of three input probabilities pp interpreted as:

The probability that input “ k ” is “1” is p . Then, the probability that the output is
“1” can be defined as “the probability that V is 0 and

, , p

1 2 3

”, or “the probability

y

1

that V is 1 and

y ”. Using prin-
ciples of probabilistic reasoning the above phrase translates into the following
formula:

”,…or “the probability that and

y

1 V 7 1