Theoretical analysis of generalised recombination

In this paper we propose, model theoretically and study a general notion of recombination for fixed-length strings where homologous crossover, inversion, gene duplication, gene deletion, diploidy and more are just special cases. The analysis of the model reveals similarities and differences between genetic systems based on these operations. It also reveals that the notion of schema emerges naturally from the model's equations even for the strangest of recombination operations. The study provides a variety of fixed points for the case where recombination is used alone, which generalise Geiringer's manifold.

1 Introduction

An important objective in evolutionary computation (EC) is to exactly model classes of evolutionary algorithms (EAs) and, further, to be able to draw inferences from these models that enhance theoretical understanding and, hopefully, aid “practitioners” in finding more competent EAs. Early models for GAs, proposed by Holland, Goldberg, Whitley and others in the seventies and eighties were either approximate or not easily scalable [4], [3], [28], [29]. Exact probabilistic models have been developed, such as the dynamical systems model of Vose and collaborators [27], [20]. More recently, an alternative exact approach, based on a coarse graining of the dynamics and directly involving schemata, has been introduced, leading to a spate of both new theoretical results [26], [24], [25], [11], [13], [14] and practical recipes for implementation [7], [12].