Sub-tree Swapping Crossover, Allele Diffusion and GP Convergence

We provide strong evidence that sub-tree swapping crossover when applied to tree-based representations will cause alleles (node labels) to diffuse within length classes. For a-ary trees we provide further confirmation that all programs are equally likely to be sampled within any length class when sub-tree swapping crossover is applied in the absence of selection and mutation. Therefore, we propose that this form of search is unbiased within length classes for a-ary trees. Unexpectedly, however, for mixed-arity trees this is not found and a more complicated form of search is taking place where certain tree shapes, hence programs, are more likely to be sampled than others within each class. We examine the reasons for such shape bias in mixed arity representations and provide the practitioner with a thorough examination of sub-tree swapping crossover bias. The results of this, when combined with crossover length bias research, explain Genetic Programming’s lack of structural convergence during later stages of an experimental run. Several operators are discussed where a broader form of convergence may be detected in a similar way to that found in Genetic Algorithm experimentation.