Predicting Emergence from Cell’s Structure - Systematic Design for Emergence in Cellular Nonlinear Networks - page 111

ship function indicating the degree of truth corresponding to a given sieve (e.g. how

good a random number generator is the CA using that cell). If a method to mea-
sure emergence based solely on cells structure is available, that method will pro-

duce a number assigned to the same

P , element of a parametrization. Therefore it is
convenient to compare parametrizations based on two different methods for measur-
ing emergence (experimental, based on running the CA, and analytic based on cells
structure and its neighborhood). By doing such a comparison it is possible to validate
or not a method for measuring emergence and complexity.
The line i and column j indexes are integers representing certain non-over-
lapping groups of bits in the cell ID definition. For instance, in the case displayed
in Fig. 7.1. five least significant bits from the 10 bits ID are assigned to i while the
remaining 5 bits are assigned to j according to the following rule: The upper left
pixel corresponds to ID = 0, then ID is increased by one on the vertical column
until ID = 31 (lower leftmost pixel) then the process of counting the genes contin-
ues with the next column and goes on until the last pixel (lower rightmost) corre-
sponding to ID = 1,023 is reached.
Each pixel in the image is represented as a gray level associated with the value

j

of

P , , here assigned to U (the exponent of growth).
Observing Fig. 7.1 leads to a positive answer to our question. Indeed, observe
that except several isolated cases, all pixels corresponding to the three main cate-
gories “unstable”, “edge” and “stable” behaviors are regularly distributed and
some simple rules for predicting the type of behavior may be inferred from the
string of bits defining the cells ID. Still we have to deal with some exceptions.
They may be related either to the precision of the measurement method (where
only a specific initial state is used instead of averaging over many, to reduce com-
putational effort) or with a more subtle relationship between structure and behavior,
particularly in the interesting border regions (

j

U ).
The reader may check that indeed a such a “good” rule for predicting “unsta-
ble” behaviors (including the “near edge” ones) is the following:
ID = 10xxxxxxxx OR ID = 01xxxxx1x OR ID = 0xxxxxxx01
And a good rule for predicting “edge” behaviors (i.e. U = 1) is the following:
ID = 110xxxxxxx

1

OR ID = 111xxx0xxx OR ID = 01xxxxxx00 OR

ID = 00xxxxxx1x,
where an x bit is either a 0 or a 1 bit.
The above suggests that a theory similar to that of local activity but developed
for Boolean discrete-time cellular systems is possible and it has to be further de-
veloped. It will eventually give the analytic formulation of how to deduce rules as
those mentioned above for a larger category of systems (e.g. with different number

of cell inputs and with different topologies). Observe that although the above rules
apply for a large majority of cells, there are still some irregular boundaries to be
predicted, where the relationship between ID (cell's structure) and the shape of the
boundaries is not so obvious. Such refinements shall be pursued for a comp-
lete theory of predicting emergence based on cells' structure.