Evolving Complexity is Hard

Understanding the evolution of complexity is an important topic in a wide variety of academic fields. Implications of better understanding complexity include increased knowledge of major evolutionary transitions and the properties of living and technological systems. Genotype-phenotype (G-P) maps are fundamental to evolution, and biologically-oriented G-P maps have been shown to have interesting and often-universal properties that enable evolution by following phenotype-preserving walks in genotype space. Here we use a digital logic gate circuit G-P map where genotypes are represented by circuits and phenotypes by the functions that the circuits compute. We compare two mathematical definitions of circuit and phenotype complexity and show how these definitions relate to other well-known properties of evolution such as redundancy, robustness, and evolvability. Using both Cartesian and Linear genetic programming implementations, we demonstrate that the logic gate circuit shares many universal properties of biologically derived G-P maps, with the exception of the relationship between one method of computing phenotypic evolvability, robustness, and complexity. Due to the inherent structure of the G-P map, including the predominance of rare phenotypes, large interconnected neutral networks, and the high mutational load of low robustness, complex phenotypes are difficult to discover using evolution. We suggest, based on this evidence, that evolving complexity is hard and we discuss computational strategies for genetic-programming-based evolution to successfully find genotypes that map to complex phenotypes in the search space.