Representation invariant genetic operators.

A genetic algorithm is invariant with respect to a set of representations if it runs the same no matter which of the representations is used. We formalize this concept mathematically, showing that the representations generate a group that acts upon the search space. Invariant genetic operators are those that commute with this group action. We then consider the problem of characterizing crossover and mutation operators that have such invariance properties. In the case where the corresponding group action acts transitively on the search space, we provide a complete characterization, including high-level representation-independent algorithms implementing these operators.

Reinterpreting no free lunch.

Abstract Since its inception, the "No Free Lunch" theorem (NFL) has concerned the application of symmetry results rather than the symmetries themselves. In our view, the conflation of result and application obscures the simplicity, generality, and power of the symmetries involved. This paper separates result from application, focusing on and clarifying the nature of underlying symmetries. The result is a general set-theoretic version of NFL which speaks to symmetries when arbitrary domains and co-domains are involved. Although our framework is deterministic, we note situations where our deterministic set-theoretic results speak nevertheless to stochastic algorithms.

State aggregation and population dynamics in linear systems.

We consider complex systems that are composed of many interacting elements, evolving under some dynamics. We are interested in characterizing the ways in which these elements may be grouped into higher-level, macroscopic states in a way that is compatible with those dynamics. Such groupings may then be thought of as naturally emergent properties of the system. We formalize this idea and, in the case that the dynamics are linear, prove necessary and sufficient conditions for this to happen. In cases where there is an underlying symmetry among the components of the system, group theory may be used to provide a strong sufficient condition. These observations are illustrated with some artificial life examples.

Structural search spaces and genetic operators.

In a previous paper (Rowe et al., 2002), aspects of the theory of genetic algorithms were generalised to the case where the search space, omega, had an arbitrary group action defined on it. Conditions under which genetic operators respect certain subsets of omega were identified, leading to a generalisation of the term schema. In this paper, search space groups with more detailed structure are examined. We define the class of structural crossover operators that respect certain schemata in these groups, which leads to a generalised schema theorem. Recent results concerning the Fourier (or Walsh) transform are generalised. In particular, it is shown that the matrix group representing omega can be simultaneously diagonalised if and only if omega is Abelian. Some results concerning structural crossover and mutation are given for this case.

Efficient linkage discovery by limited probing.

This paper addresses the problem of discovering the structure of a fitness function from binary strings to the reals under the assumption of bounded epistasis. Two loci (string positions) are epistatically linked if the effect of changing the allele (value) at one locus depends on the allele at the other locus. Similarly, a group of loci are epistatically linked if the effect of changing the allele at one locus depends on the alleles at all other loci of the group. Under the assumption that the size of such groups of loci are bounded, and assuming that the function is given only as a "black box function", this paper presents and analyzes a randomized algorithm that finds the complete epistatic structure of the function in the form of the Walsh coefficients of the function.

Group properties of crossover and mutation.

It is supposed that the finite search space omega 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 omega are invariant under crossover are investigated, leading to a generalization of the term schema. Finally, it is sometimes possible for the group acting on omega to induce a group structure on omega itself.