![](bg6.png)
However, it is still unclear why the uniform
crossover is preferable for Decode and Multiply,
while Encode shows different trends. Both former
benchmarks are harder benchmarks compared to En-
code and Parity, though. It is possible that the
n−point crossovers are too destructive for harder
problems, as too many useful structures are dissi-
pated. This may indicate that the positional bias
plays only a minor role in CGP’s performance with
its crossover operators. Nevertheless, we can val-
idate the results of Husa and Kalkreuth (Husa and
Kalkreuth, 2018), that different recombination oper-
ators are needed for different settings.
6 CONCLUSION
The crossover operator has been an active research
topic for Cartesian Genetic Programming (CGP)
since its introduction. To this day, it is unclear why
CGP does not generally benefit from a recombination
algorithm and finding universally well-performing
operators might improve CGP’s performance greatly.
In the process to find answers to this question,
we argued that the influence of CGP’s positional bias
might be a possible issue. The uneven distribution of
active and inactive nodes might lead to problems for
untailored crossover operators, as simply swapping
genes might destroy useful structures. To mitigate the
effects of the positional bias, the reorder extension to
CGP was reintroduced.
Our hypothesis was tested empirically and prelim-
inary results were presented. By comparing differ-
ent CGP variants with different crossover operators, a
first conclusions can be given: The uniform crossover
generally benefits CGP with reorder.
However, no clear conclusion to our hypothesis
can be drawn yet. Further evaluations and configura-
tions must be tested before the influence of the posi-
tional bias can be truly assessed. For future work, a
more diverse set of benchmarks must be tested. Test-
ing only Boolean benchmarks are not sufficient and
evaluations must be extended to regression and real-
world benchmarks. Furthermore, we did not com-
pare our results to the standard CGP formula with
crossover operators, which is an important step to
assess REORDERs influence on the crossover opera-
tors. In addition, statistical analysis methods should
be added to the evaluations as well. By only compar-
ing mean values and their standard deviations, a first
assessment can be made. However, without proper
statistical analysis, it is not possible to truly judge the
results.
Along with these evaluations mentioned, the ideas
of Kalkreuth et al. should be included (Kalkreuth
et al., 2017). By only recombining active nodes in
CGP with reorder, some effects might be observed
which could help to accept or reject our hypothesis.
Additionally, new crossover operators should be
tested to evaluate our hypothesis. New algorithms
similar to those introduced by Cai et al. could recom-
bine genes differently or with different probabilities,
based on the genes location in the graph (Cai et al.,
2006). Other operators could be influenced by exist-
ing works and be based on the density of active nodes.
Furthermore, the counts of successful crossovers,
the fitness changes per crossover, distributions of ac-
tive and inactive nodes, etc. should be also accounted
for.
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