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It is worth pointing out how complex these polynomials might become: as
you can see, a simple organization like the one above with just two functions
in the head already gives rise to a very complex expression.
The two-parameter function âFâ described above is the function used in
the STROGANOFF system that composes complete bivariate basis polyno-
mial functions (Ivakhnenko 1971). In Table 7.1, this function is represented
by
F
9
. The implementation of similar functions in gene expression program-
ming, such as the incomplete bivariate basis polynomial functions used in
the enhanced STROGANOFF (Nikolaev and Iba 2001), is also very simple
and is done exactly as illustrated above for function
F
9
. The definition of all
these functions is given in Table 7.1.
Another function easily implemented is the following:
2
1
2
2
y
x
x
x
x
x
x
(7.4)
1
2
1
2
which, except for the coefficients, is identical to the function (7.1) or func-
tion
F
9
in Table 7.1. This and similar functions based on the 16 bivariate
functions of Table 7.1 are going to be included in the function kit of GEP in
order to analyze the importance of random numerical constants in the evolu-
tion of polynomials. Their description is given in Table 7.2.
Table 7.2
Bivariate basis polynomials with unit coefficients.
F
x
x
x
x
1
1
2
1
2
F
x
x
2
1
2
F
x
x
x
2
1
x
2
2
3
1
2
F
x
x
x
x
2
1
4
1
1
2
2
2
F
x
x
5
1
2
1
F
x
x
x
6
1
2
2
1
2
2
F
x
x
x
7
1
2
1
2
2
F
x
x
8
2
1
2
2
F
x
x
x
x
x
x
9
1
2
1
2
2
1
F
x
x
x
x
x
10
1
2
1
2
2
1
2
2
F
x
x
x
x
x
11
1
1
2
2
1
2
2
F
x
x
x
x
12
1
2
2
2
F
x
x
x
x
13
1
1
2
F
x
x
x
14
1
1
2
F
x
x
15
1
2
F
x
x
x
2
1
16
1
2
