Emergence, Locating and Measuring It - Systematic Design for Emergence in Cellular Nonlinear Networks - page 59

represents a much smaller and manageable fraction. Since the optimal degree of
coupling (for a given CNN size) is not a crisp value we would expect to find

also some emergent phenomena for less coupled cells although we expect them

to be more sparse than from the set of totalistic functions. Therefore it would be
reasonable to consider a second, extended set of local rules, i.e the class of
semitotalistic functions (note that the set of totalistic functions is included in

the set of all semitotalistic functions). Here, one of the coupling coefficients

is allowed to have a different absolute value. It follows that in this case, the b

i
coupling degree ranges as 1

. Note than in his work, Kauffman [60]

2 d c d k

k

found that a degree of coupling of 2 as minimal for producing emergent

computation. It follows from the above that for a coupling

d log 2 ( N )

max

degree of 2. For cells (i.e. a 125 ×125 two-dimensional CNN) this N 15625

gives , i.e. at the most superior limit of the known natural systems.

d

max |

14

On these grounds we may postulate that most of the emergent behaviors
have to be found within the families of semitotalistic local Boolean func-
tions . It is thus no surprise that looking at most of the useful CNN tem-
plates [32] we find that they obey the above conjecture.

Second, the nonlinear representation of the Boolean function allows to define a
a local structural complexity, N defined [53] as the optimal (minimum) value

x

of p (the roots of the discriminant function w in (4.1) for the realization of a given

Boolean function. This coefficient is actually the same as m used in the piece-

wise linear definition (3.8) from the previous chapter. The linearly separable

N , while all the other functions, with different
levels of complexities correspond to the linearly not separable functions. The
notion of local structural complexity was first introduced in [53], where it was
conjectured that a necessary condition for emergence and complexity in an ar-
ray of Boolean cells is that they are linearly not separable . This conjecture was
then proved for one-dimensional CAs with 3 cells neighborhoods (the 1a3 fam-
ily). The tools for detecting emergence presented in the next sections confirm
this conjecture for many other types of 1D and 2D semitotalistic and totalistic
cellular systems with various neighborhoods.

functions correspond to 1

4.4 Clustering and Transients and as Measures
of Emergence

Emergence is observed as a global effect of the local interconnection of cells.
Wolfram employs a visually based classification of emergent phenomena using
four qualitative classes. Although his classification proved useful to derive certain
conclusions, it is quite unpractical to evaluate large numbers of rules in such a
way. In [61] we introduced several tools to detect emergence as a long transient