Covers: A Theory of Boolean Function Decomposition

In this paper, we develop a theory of covers for funct ions defined over boolean str ings. Given a function , a cover is a decomposition , though not necessarily a partition, of the bits into subproblems that can be solved in parallel. In the paper , we formulat e th e notion of cover size , which equals the size of the largest subproblem in a decomposition for a parti cular function. Cover size is defined relative to a function 's upper contour sets . An implicat ion of cover theory is th at a problem's difficulty, as measured by its cover size, necessarily decreases as the function value improves. We also show how cover theory lends insight to the performance of hillclimbing algorithms and genet ic algorithms, and present data from simulation s that support a covers-based explanation for genet ic algorithm performance.