  This paper explores the genotype-phenotype relationship. It outlines
conditions under which the dependence of a quantitative trait on the genome
might be predictable, based on measurement of a limited subset of genotypes. It
uses the theory of real-valued Boolean functions in a systematic way to
translate trait data into the Fourier domain. Important trait features, such as
the roughness of the trait landscape or the modularity of a trait have a simple
Fourier interpretation. Ruggedness at a gene location corresponds to high
sensitivity to mutation, while a modular organization of gene activity reduces
such sensitivity.
  Traits where rugged loci are rare will naturally compress gene data in the
Fourier domain, leading to a sparse representation of trait data, concentrated
in identifiable, low-level coefficients. This Fourier representation of a trait
organizes epistasis in a form which is isometric to the trait data. As Fourier
matrices are known to be maximally incoherent with the standard basis, this
permits employing compressive sensing techniques to work from data sets that
are relatively small -- sometimes even of polynomial size -- compared to the
exponentially large sets of possible genomes.
  This theory provides a theoretical underpinning for systematic use of Boolean
function machinery to dissect the dependency of a trait on the genome and
environment.
