The Algebraic Theory of Recombination Spaces

Peter F. Stadler and Guenter P. Wagner

A new mathematical representation is proposed for the configuration space structure induced by recombination which we called ``P-structure''. It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental ``genotypes'' the set of all recombinant genotypes obtainable from the parental ones. It is shown that this construction allows a Fourier-decomposition of fitness landscapes into a superposition of ``elementary landscapes''. This decomposition is analogous to the Fourier decomposition of fitness landscapes on mutation spaces. The elementary landscapes are obtained as eigenfunctions of a Laplacian operator defined for P-structures. For binary string recombination the elementary landscapes are exactly the $p$-spin functions (Walsh functions), i.e. the same as the elementary landscapes of the string point mutation spaces (i.e. the hypercube). This supports the notion of a strong homomorphisms between string mutation and recombination spaces. However, the effective nearest neighbor correlations on these elementary landscapes differ between mutation and recombination and among different recombination operators. On average, the nearest neighbor correlation is higher for one-point recombination than for uniform recombination. For one-point recombination the correlations are higher for elementary landscapes with fewer interacting sites as well as for sites which have closer linkage, confirming the qualitative predictions of the Schema-Theorem. We conclude that the algebraic approach to fitness landscape analysis can be extended to recombination spaces and provides an effective way to analyze the relative hardness of a landscape for a given recombination operator.

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