Information Technology Reference
In-Depth Information
The simple problem we are going to solve with the cellular system of gene
expression programming is the sextic polynomial
a
6
-2
a
4
+
a
2
, the same func-
tion chosen by Koza to show that ADFs allow the evolution of modular solu-
tions (Koza 1994). Indeed, its regularity and modularity can be easily guessed
by its factorization:
a
6
-2
a
4
+
a
2
=
a
2
(
a
-1)
2
(
a
+1)
2
(6.1)
For this problem, a very simple function set composed of the basic arith-
metical operators was chosen for both the ADFs and the main programs, that
is, F = F
H
= {+, -, *, /}. The set of terminals used in the conventional genes
consists obviously of the independent variable
a
, thus giving T = {a}; for the
homeotic genes, the terminal set consists obviously of the ADFs encoded in
the conventional genes (two, in this case) and will be represented by “0” and
“1”, thus giving T
H
= { 0, 1}. For this simple problem of just one variable, a
set of 10 random fitness cases chosen from the interval [-1, 1] will be used
(Table 6.1). The fitness will be evaluated by equation (3.3b), using a relative
error of 100% for the selection range and a precision for the error equal to
0.01%. Thus, for the 10 fitness cases used in this problem,
f
max
= 1000. The
complete list of the parameters used per run is shown in Table 6.2.
The evolutionary dynamics of the successful run we are going to analyze
is shown in Figure 6.2. And as you can see, in this run, a perfect solution to
the sextic polynomial was found in generation 8.
Table 6.1
Set of 10 random computer generated fitness cases used to solve
the simple modular function using automatically defined functions.
a
f(a)
-0.133270
0.0171355969283171
0.901673
0.0284259762561082
-0.179748
0.0302552836680942
0.221649
0.0444196789826592
-0.920441
0.0197775495439223
-0.633942
0.1437712766244980
0.797516
0.0842569554563209
-0.098480
0.0095111081469302
-0.808197
0.0785663809054614
-0.926209
0.0173313183876865
