Design of Neural Networks - Gene Expression Programming - page 393

a.

012345678901234567890123456789012
TQaTaaababbbabaaa6305728327795806

W = {1.175, 0.315, -0.738, 1.694, -1.215, 1.956, -0.342, 1.088, -1.694, 1.288}

b.

T

Q a

T

a a a a

b b b

Figure 10.5. A perfect, slightly complicated solution to the exclusive-or problem
designed with the GEP-NN algorithm. a) Its chromosome and respective weights. b)
The fully expressed neural network encoded in the chromosome.

parsimonious solutions to the XOR function were found in our second experi-
ment (they were also found among the perfect solutions of the first experi-
ment, which was the reason why the second experiment was designed).
The parameters used per run in this experiment are summarized on the
second column of Table 10.1 and, as you can see, despite keeping the func-
tions “T” and “Q” with high connectivities, the smallest workable head size
for this problem was chosen (that is, h = 2) in order to increase the odds of
finding the most parsimonious solutions. One such solution is shown below:

01234567890123456
TDbabaabb73899388

W = {0.713, -0.774, -0.221, 0.773, -0.789, 1.792, -1.77, 0.443, -1.924, 1.161}

which, as you can see in Figure 10.6, is a perfect, extremely parsimonious
solution to the XOR problem.
Curiously, in this experiment, several other solutions to the XOR function
were found that use exactly the same kind of structure of this parsimonious
solution. Indeed, the algorithm discovered not only one but several Boolean
functions of three arguments and invented new, unexpected solutions to the
XOR problem by using them as building blocks. This clearly shows that
GEP is an astonishing invention machine, totally devoid of preconceptions.