Evolutionary dynamics on multiple scales: a quantitative analysis of the interplay between genotype, phenotype, and fitness in linear genetic programming

Abstract

Redundancy is a ubiquitous feature of genetic programming (GP), with many-to-one mappings commonly observed between genotype and phenotype, and between phenotype and fitness. If a representation is redundant, then neutral mutations are possible. A mutation is phenotypically-neutral if its application to a genotype does not lead to a change in phenotype. A mutation is fitness-neutral if its application to a genotype does not lead to a change in fitness. Whether such neutrality has any benefit for GP remains a contentious topic, with reported experimental results supporting both sides of the debate. Most existing studies use performance statistics, such as success rate or search efficiency, to investigate the utility of neutrality in GP. Here, we take a different tack and use a measure of robustness to quantify the neutrality associated with each genotype, phenotype, and fitness value. We argue that understanding the influence of neutrality on GP requires an understanding of the distributions of robustness at these three levels, and of the interplay between robustness, evolvability, and accessibility amongst genotypes, phenotypes, and fitness values. As a concrete example, we consider a simple linear genetic programming system that is amenable to exhaustive enumeration and allows for the full characterization of these quantities, which we then relate to the dynamical properties of simple mutation-based evolutionary processes. Our results demonstrate that it is not only the distribution of robustness amongst phenotypes that affects evolutionary search, but also (1) the distributions of robustness at the genotypic and fitness levels and (2) the mutational biases that exist amongst genotypes, phenotypes, and fitness values. Of crucial importance is the relationship between the robustness of a genotype and its mutational bias toward other phenotypes.