Study of Evolution in Genetic Algorithms by Eigen's Theory Including Crossover Operator

A theory representing the dynamics of in nite population Genetic Algorithms is given by using Eigen's evolution model. The effect of crossover is included to study the evolution of in nite population GAs. We present a theory of GA dynamics, in which we give discrete time equations for GA evolution including the mutation and crossover processes. The Walsh analysis of allele frequencies provides a very powerful tool for studying the evolutionary processes by mutation and crossover. In previous papers [1, 2], we gave a model for describing the evolution of allele frequencies in in nite population GAs. In the rst paper [1], we applied Eigen's model for describing evolutionary processes of GAs with selection and mutation. In this system, the combined e ect of selection and mutation is represented by the selection-mutation matrix. In the second paper [2], we modi ed di erential equations to a system of di erence equations for simulating generational GA processes and gave a procedure for solving equations. We consider the in nite population model of the Simple Genetic Algorithms with selection, mutation and crossover. We treat binary strings of length l as chromosomes having one gene with n = 2 alleles. Let xi(t) be the relative frequency of the i-th allele at generation t. The change of the distribution in the mutation process is expressed by the mutation matrix M