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value unchanged. It is worth mentioning, that these
genotypic changes are smaller than the ones observed
for positive mutations (according to WL-Subtree ker-
nel: < 0.2 for neutral and > 0.2 for positive muta-
tions). That is, these changes are small enough not
to escape local optima, as well as, they are not big
enough to enter valleys.
Finally, in the case of negative mutations, we see
that the most detrimental changes can be observed for
individuals with high original fitnesses (> 70). From
the behavioral perspective, all individuals experience
very high changes to their behaviors. This landscape
is stable in comparison to the behavior changes ob-
served for positive mutations. It is interesting to see
that for negative mutations, structural changes are
overall higher than the ones observed for neutral mu-
tations, but are lower than for positive mutations.
5 CONCLUSIONS
Two contributions of this paper are: (i) extending the
definition of GP by considering genotype to behav-
ior mapping; and (ii) proposed to use the family of
similarity measures, Graph Kernels, as a way to cal-
culate genotypic distance. Our proposals were tested
on two benchmark problems, artificial ant and even
parity problem.
First of all, traditionally used Tree Edit Distance
(TED) was compared to the Weisfeiler-Lehman Sub-
tree Kernel. This comparison showed that TED con-
siders trees to be significantly more different than
WL-Subtree kernel. This is due to the way, the
dissimilarity between the trees is calculated, that is
WL-Subtree kernel considers both local (node level)
and global (neighborhoods) properties of trees. This
means that the WL-Subtree kernel uses both syntactic
(structural) and semantic (behavioral) information, to
evaluate dissimilarity.
Investigations on the locality of three basic muta-
tion operators: subtree, structural, and one-point mu-
tations, showed that subtree mutation has the highest
locality. This result is counter-intuitive because sub-
tree mutation changes subtrees (and possibly whole
tree). This result is common for both genotype to fit-
ness and genotype to behavior mappings. However,
for the artificial ant problem, all three mutation oper-
ators have low locality in genotype to behavior map-
ping. That is, the frequency of a highly local change
to occur is much less common than a low locality
change to happen (frequencies at the level of 10
5
vs
10
2
, respectively). In the case of even-6-parity, dis-
tributions of fitness, and behavior distances were ap-
proximately the same.
Finally, our results showed that all mutations are
detrimental to the original behavior of an individual,
regardless of its original fitness and the size of in-
troduced genotypic change. Three considered muta-
tion operators are of low behavioral locality, that is,
they rarely preserve behaviors of programs. There-
fore, even small changes in the genotype, drastically
change an individual’s behavior, thus resulting in
slight improvements or considerable deterioration of
the fitness of an individual. Small behavioral changes
were also recorded, however, these did not change the
fitness of programs.
This work can be extended in several ways. First
of all, it could be applied to a wider variety of prob-
lems, e.g. symbolic regression (both benchmark and
real-world problems), maze navigation, image pro-
cessing, and scheduling. Furthermore, this work ap-
plied the Weisfeiler-Lehman Subtree kernel as a dis-
similarity measure, however, various methods could
also be used, such as, Shortest Path, Random Walk,
Graphlet Sampling, and Laplacian kernels. Finally,
it may be worthwhile to investigate how different
reperesentations of intermediate phenotypes (e.g. bi-
nary decision diagrams) affect the fitness landscape.
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