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*++//2a0aa22a33/0a231213122123*2aa1-103aa3a31-[72,11] = 1000 (17)
-*//-3-a2a02123-+/-1-3320331a0*3a-1*1a2a3201a-[81,45] = 1000 (27)
*a2a--aa30a333a*+//1a/a2aa032a/*a00*aa3a0331a-[84,38] = 1000 (19)
*a+3/a*2130a1200-2212100101aa0/*31*-0a310a2aa-[96,33] = 1000 (19)
And a remarkable thing can be straightaway observed just by computing
the average program size of these perfect solutions in both experiments: in
the first experiment, we get an average size of 25.8 against 18.4 in the sec-
ond, which corresponds to a decrease of 28.7%. And this means that the
solutions created in the second experiment are much more compact and,
therefore, have much less room for neutral or redundant motifs to appear,
which is indeed a serious handicap in evolutionary terms (see a discussion of
The Role of Neutrality in Evolution in chapter 12). Consequently, in the
second experiment, the algorithm has much more difficulties in creating and
manipulating small building blocks and combining them in different arrange-
ments to test how they work together. In other words, the GEP-NC system is
much less flexible and consequently performs considerably worse in this
kind of situation.
However, when the constants required to solve a problem are small float-
ing-point constants, then the inclusion of numerical constants in the terminal
set is advantageous and the algorithm performs considerably better when
numerical constants are explicitly available. To demonstrate this, a simple
polynomial function with rational coefficients was chosen:
2
y
2
718
x
3
141636
x
(5.1)
For this study, a set of 10 random fitness cases chosen from the real inter-
val [-10, 10] was used (Table 5.2). For both experiments, the same function
set was used and consisted of F = {+, -, *, /}. For the basic GEA without
numerical constants, the terminal set consisted obviously of the independent
variable alone, giving T = {x}, whereas for the GEP-NC algorithm, a small
set of five numerical constants chosen randomly from the real interval [0, 2]
was used, giving T = {x, a, b, c, d, e}, where
x
corresponds to the independ-
ent variable,
a
= 0.298,
b
= 1.083,
c
= 1.466,
d
= 0.912, and
e
= 1.782. The
fitness function was evaluated by equation (3.3b), with a selection range of
100% and maximum precision (0% error), giving
f
max
= 1000. This experi-
ment, with its two approaches, is summarized in Table 5.3.
And as you can see in Table 5.3, the GEP-NC system performs consider-
ably better than the simpler approach, with an average best-of-run fitness of
987.384 and an average best-of-run R-square of 0.999980229 (as opposed to
