Quotient evolutionary space: abstraction of evolutionary process w.r.t. macroscopic properties

Abstract

Darwinian evolution, which is characterized in terms of particular macroscopic behavior that emerges from microscopic organismic interaction, considers populations as units of evolutionary change. We formalize these concepts in evolutionary computation by developing notion of quotient evolutionary space (QES). We map set of all finite populations to a set of macroscopic properties of population those are chosen a priori; and we call this mapping as evolutionary criteria. On the 'quotient set of populations' that is induced by evolutionary criteria, we define mathematical structures to define evolutionary change with respect to chosen macroscopic parameters at populational level. This allows us to transform the objective defined on the search space that is imposed by the fitness function to an objective on the population space. We call quotient set of populations along with the mathematical structures the quotient evolutionary space. To demonstrate the abstraction we consider fitness distribution of population as evolutionary criteria and give a detailed analysis of resulting spaces and basic convergence results.