A General Coarse-Graining Framework for Studying Simultaneous Inter-Population Constraints Induced by Evolutionary Operations: Extended Version

The use of genotypic populations is necessary for adaptation in Evolutionary Algorithms. We use a technique called form-invariant commutation to study the immediate effect of evolutionary operations on populations of genotypes. This technique allows us to understand compositional changes induced by evolutionary operations in terms of constraints between populations. Deep insight into the population-level effect of some evolutionary operation is possible when multiple constraints can be derived for all pairs of pre and post operative populations; for each such pair of populations the constraints between them are then said to hold simultaneously. When selection is fitness proportional we show that any coarse-graining of the genotype set can be used to systematically derive single constraints between between all pairs of pre and post selection populations. Matters are not so simple in the case of variation. We develop an abstract condition called ambivalence and show that when a coarse-graining and a variation operation satisfy this condition then a systematic derivation of single constraints between all pairs of pre and post variation populations is possible. Finally we show that it is possible to use schema partitions to systematically derive simultaneous constraints for any combination of variation operations that are commonly used in GAs.