Some geometric and algebraic results on crossover

Abstract

In light of recent “No Free Lunch” theorems, crossover is neither inferior nor superior, in general, to other search operators. The present work formally establishes some properties of crossover that may be useful in analysis of when crossover performs well and when it does not. All forms of crossover which preserve the ordering of alleles are included. Alleles are restricted, however, to binary values. In this context, crossover is naturally treated as a bitwise Boolean operator parameterized by a mask indicating which alleles' values to exchange. Interesting properties of the Hamming distances between parents and offspring are derived. Several identities and inequalities of the distances are summarized in a diagram referred to as the crossover rectangle. Various algebraic properties, some counterintuitive, of crossover are also established. The algebraic definition of crossover gives rise to a simple style of formal reasoning about crossover that may prove to be generally useful.