Elementary bit string mutation landscapes

Abstract

Genetic Programming parity with only XOR is not elementary. GP parity can be represented as the sum of k/2 + 1 elementary landscapes. Statistics, including fitness distance correlation (FDC), of Parity's fitness landscape are calculated. Using Walsh analysis the eigen values and eigenvectors of the Laplacian of the two bit, three bit, n-bit and mutation only Genetic Algorithm fitness landscapes are given. Indeed all elementary bit string landscapes are related to the discrete Fourier functions. However most are rough (λ/d ≈ 1). Also in many cases fitness autocorrelation falls rapidly with distance. GA runs support eigenvalue/graph degree (λ/d) as a measure of the ruggedness of elementary landscapes for predicting problem difficulty. The elementary needle in a haystack (NIH) landscape is described.