Mutation-Crossover Isomorphisms and the Construction of Discriminating Functions

We compare the search power of crossover and mutation in Genetic Algorithms. Our discussion is framed within a model of computation using search space structures induced by these operators. Iso-morphisms between the search spaces generated by these operators on small populations are identiied and explored. These are closely related to the binary reeected Gray code. Using these we generate discriminating functions which are hard for one operator but easy for the other and show how to transform from one case to the other. We use these functions to provide theoretical evidence that traditional GAs use mutation more eeectively than crossover, but dispute claims that mutation is a better search mechanism than crossover. To the contrary we show that methods that exploit crossover more effectively can be designed and give evidence that these are powerful search mechanisms. Experimental results using GIGA, the Gene Invariant Genetic Algorithm, and the well known GENESIS program support these theoretical claims. Finally, this paper provides the initial approach to a diierent method of analysis of GAs that does not depend on schema analysis or the notions of increased allocations of trials to hyperplanes of above average tness. Instead it focuses on the search space structure induced by the operators and the eeect of a population search using them.