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emergence”, the topic of this topic. Note also that since CA genes are usually
described by a small number of bits, the CA “program length” is usually quite
small, at least compared to traditional programs. Though, even for a “small” pro-
gram length of 12 bits, the number of possible programs is huge and therefore
adequate tools and methods have to be defined for an effective way to locate those
programs.
In one discrete time step a synchronous update of all cells takes place, based on
the status of their neighborhoods at the previous time step. For a semitotalistic
C, the update depends
only on the values of
and
D
y
t
y
t
y
t
y
t
2
3
4
5
E
. Therefore the transition local oolean function (gene) can be defined us-
ing Table 3.1 where the gene is a 10-dimensional binary vector
y
1
t
>
@
:
G
g
,
g
,...,
g
9
8
0
Table 3.1.
A cell definition using a table
G
g
9
g
8
g
7
g
6
g
5
g
4
g
3
g
2
g
1
g
0
4
3
2
1
0
4
3
2
1
0
D
E
1
1
1
1
1
0
0
0
0
0
The state of the neighborhood is entirely described by two integers:
E
D
,
hen each of the binary components
g
of the gene
G
are defined, the above
table precisely indicates
a local rule
, i.e. how to update the state (output) of the
central cell for any possible configurations of the neighborhood given by
E
D,
(3.3)
y
t
1
G
D ,
t
E
t
1
The gene
>
@
G
is a binary string. In practice is often more conven-
ient to use its decimal representation, denoted ID (cell identifier). For a given
constraint of the cellular model (e.g. specifying that semitotalistic cells with nine
g
,
g
,...,
g
9
8
0
inputs are used, or the label of a taxonomy such as “2s” - see Sect. 3.7) an ID
clearly individualies a cell within a fami ly composed of all possible cells given
the constraints.
3.4.1 Piecewise-Linear Representation and Implementation
The function in (3.3) can be implemented directly using a look-up table (a 1,024
bits M or O) or via compact mixed-
signal cells [37]. This later method is
the most convenient in terms of density of cells.
In order to define a circuit as in [37], but also for other reasons, a nonlinear rep-
resentation of (3.3) is more convenient that a table. For instance, as seen later, the
use of such a nonlinear description in atlab allows the use of the intrinsic matrix
operations, therefore leading to an important speedup compared to the case when
tables would be used.
In order to understand the principles of nonlinear representations [2] let us
consider the case of implementing the function ID103 (or 0001100111 binary).
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