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the case of small local connectivity. Without any structural restriction, i.e. accepting
all local rules as potentially emergent we have to investigate
2
k
2
32
9
2
|
4
10
local rules, for
k
. It is quite impractical, if not impossible, to test all of them
for emergence. The situation is getting even worse for larger neighborhoods.
But are all these functions equally important? In the following I will propose a
framework which differentiates among these Boolean functions and reveals how
to address only a smaller fraction of them, while maximizing the chance to get
emergent behaviors.
As shown in [52] and in the previous chapter any Boolean function can be rep-
resented as a piecewise linear (but nonlinear) function:
5
ys
sgn
w
V
,
(4.1)
k
(4.2)
b
u
,
¦
V
ii
i
1
where
(
w
is a nonlinear function with up to
p
roots, and
V
is the one-
dimensional projection of the
k
-dimensional input vector
>
)
@
through
u
u
,
2
u
,...
u
1
k
the projection
3
(or coupling) vector
>
@
. In the CA framework, the bi-
b
b
,
b
,..,
b
1
2
k
nary output
y
of any cell in the network is updated according to (4.1) and (4.2)
with an input vector
>
@
formed by the binary outputs of the cells in the
u
neighborhood at the previous discrete time moment.
u
,
2
u
,...
u
1
k
The importance of describing an arbitrary Boolean function as (4.1) and (4.2) is
twofold:
x First, the coupling equation (4.2) indicates the
coupling degree
of a cell with its
k
¦
b
i
neighbors. The
coupling degree
can be defined as
i
1
and
it is not
c
max
b
i
revealed unless a Boolean cell is described as a nonlinear function. The maxi-
mum degree of coupling
c
is obtained for
totalistic
cells (defined in pre-
vious works [56] as cells where all
b
are equal in their absolute value). As
shown above, in a network of cells there is a critical range for the cell coupling
so that emergence may occur (postulating that emergence is a characteristic of a
natural system characterized by
k
d
). For any other values of
b
, the
degree of coupling will decrease resulting in a departure from the optimal
critical coupling
k
responsible for emergence (as shown in Table 4.1). Thus
many of the Boolean functions characterized by a low degree of coupling may
be discarded. Correspondingly, it would be more reasonable to focus first on
the group of
3
8
k
2
totalistic functions characterized by the maximal degree of
coupling
k
for a given neighborhood. Compared to all
k
k
2
Boolean cells it
2
3
Here we will assume
optimal projections
i.e.
b
is optimized such that using the represen-
tation (4.1) and (4.2) of a Boolean function it is achieved with a minimum of
p
roots of
the
w
() function. The linearly separable function correspond to
p=1
.
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