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here the
z
parameters cannot be analytically determined. Instead, a computationally
intensive global optimiation algorithm is used to find all gene values. Compared
to (3.), the realiation of (3.), which is also piecewise-linea r, is much more com-
pact in terms of absolute value functions (usually
) and therefore better
suited for physical implementations via circuits such as the one depicted in Fig. 3.3.
It is interesting to note that all (3.) (3.) and (3.) are defined around a certain
optimal number
m
of transitions points thus revealing a certain
local structural com-
m
k
|
log
2
plexity of the cell
, with simpler cells being the linearly separable ones (i.e.
m=
1).
ecent results
[3] indicate that at least
m=2
is required for computational univer-
sality in a cellular automata, although interesting computational effects are possible
even for
m=1
as seen for the gene ID76, performing the function of
character
segmentation
(see Chap. ).
The following atlab function returns the set of transition points
and the
t
,...
1
t
m
sign
s
for any gene given as the decimal ID of length
N
bits.
function [s, T]=id2t(ID,N)
% generate the s T parametrs for a given ID
% from the corresponding string of N bits
%- this is a preliminary step for representing any
% Boolean function using a canonical PWL "polynomial"
% Copyright Radu Dogaru, July 2006.
% radu_d@ieee.org
%-------------------------------------------------------------------
y=dec2bin(ID,N);
if y(N)=='0' s=-1; else s=1; end
ks=y(N); off=0.5;
T=[];
for i=1:(N-1)
if y(N-i)~=ks
T=[T off];
ks=y(N-i);
end
off=off+1;
end
3.4.2 Extended Families of Cells
One may consider expanded families of semitotalistic functions. For instance if
one replaces
E
with
1
(i.e. the negate version of the central cell is considered
instead of the original) another set of 1,024 cells is obtained, with different emer-
gent behaviors, among which certain computationally meaningful genes could be
found. y convention in the next we w ill denote with ID* the genes belonging
to such a family. Note that either ID and ID* are defined by the
same
unique set
E
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