Group Properties of Crossover and Mutation

Abstract

It is supposed that the finite search space Ω has certain symmetries that can be described in terms of a group of permutations acting upon it. If crossover and mutation respect these symmetries, then these operators can be described in terms of a mixing matrix and a group of permutation matrices. Conditions under which certain subsets of Ω are invariant under crossover are investigated, leading to a generalization of the term schema. Finally, it is sometimes possible for the group acting on Ω to induce a group structure on Ω itself.

Exact Schema Theory and Markov Chain Models for Genetic Programming and Variable-length Genetic Algorithms with Homologous Crossover

“…This group action describes the symmetries that are inherent in the definition of crossover for fixed-length strings [42]. This idea can be generalised to other finite search spaces (see [31] for the detailed theory). However, in the case of GP, where the search space is a set of trees (up to some depth), the symmetry is more complex and does not seem to give rise to a single mixing matrix.…”

Structure and metaheuristics

“…Rowe, Vose and Wright [19,20] investigate this scenario for GA. They formally define a notion of structure in the search space, and study the condi-tions for which mutation and crossover respect the symmetries induced by the structure.…”

“…Respecting the symmetries of the search space is an important property of an efficient search-operators -in fact, designing such operators is, perhaps, the main motivation for [19,20,13]. In section 5 we define a generic model of random search heuristics which explicitly assumes that the search operators respect the structure of the search space.…”

NP-Completeness of Deciding Binary Genetic Encodability

“…In order to apply an evolutionary algorithm to attack a specific optimization problem, one needs to model the algorithm in a suitable manner. The importance of finding appropriate models is emphasized in much of the research literature: see, for instance, the introduction to chapter 17 of [13], [14], [12] and [11]. The general methodology for how to construct the search space and the appropriate recombination operators with the aim of applying the classical genetic algorithm first appeared in [9].…”