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Table 4.2
Set of fitness cases used in the polynomial function problem.
a
f(a)
6.9408
44.909752
-7.8664
7.3409245
-2.7861
-4.4771234
-5.0944
-2.3067443
9.4895
73.493805
-9.6197
17.410214
-9.4145
16.072905
-0.1432
-0.41934688
0.9107
3.1467872
2.1762
8.8965232
First of all, the set of functions and the set of terminals must be chosen.
For the sake of simplicity, in this case we will choose just the basic arithme-
tic operators to design this model, thus giving F = {+, -, *, /}; and the termi-
nal set will include just the independent variable, giving T = {a}. Next, the
structural organization of the chromosomes, namely, the length of the head h
and the number of genes, must also be chosen. It is wise to start with short,
single-gene chromosomes and then gradually increase h . Figure 4.1 shows
such an analysis for this problem. Note how the success rate increases abruptly
in the beginning, from a small success rate of 29% for the most compact
organization (a gene length g of 13) to a high success rate of 86% obtained
with a moderately redundant organization ( g = 29). Note also that, from this
point onward, the success rate starts to decrease progressively. As you can
see, for each problem, it is possible to guess more or less accurately the ideal
head length in order to create a search landscape not only easy to navigate
through but also full of riches.
It is worth pointing out that GEP can be used to search for the most parsi-
monious solution to a problem by choosing smaller and smaller head sizes
(we will see in section 4.3, Logic Synthesis and Parsimonious Solutions,
how to do this more efficiently by using special fitness functions with parsi-
mony pressure). As shown in Figure 4.1, it was not possible to find a correct
solution to this problem using a head length of five. Only with h = 6 was it
possible to evolve a perfect solution. In this case, these perfect solutions are
 
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