Quotient geometric crossovers and redundant encodings

We extend a geometric framework for the interpretation of search operators to encompass the genotype-phenotype mapping derived from an equivalence relation defined by an isometry group. We show that this mapping can be naturally interpreted using the concept of quotient space, in which the original space corresponds to the genotype space and the quotient space corresponds to the phenotype space. Using this characterization, it is possible to define induced geometric crossovers on the phenotype space (called quotient geometric crossovers). These crossovers have very appealing properties for non-synonymously redundant encodings, such as reducing the size of the search space actually searched, removing the low locality from the encodings, and allowing a more informed search by utilizing distances better tailored to the specific solution interpretation. Interestingly, quotient geometric crossovers act on genotypes but have an effect equivalent to geometric crossovers acting directly on the phenotype space. This property allows us to actually implement them even when phenotypes cannot be represented directly. We give four example applications of quotient geometric crossovers for non-synonymously redundant encodings and demonstrate their superiority experimentally.