Generalized cofactors and decomposition of Boolean satisfiability problems

We propose an approach for decomposing Boolean satisfiability problems while extending recent results of [12] on solving Boolean systems of equations. Developments in [12] were aimed at the expansion of functions f in orthonormal (ON) sets of base functions as a generalization of the Boole-Shannon expansion and the derivation of the consistency condition for the equation f = 0 in terms of the expansion co-efficients. In this paper, we further extend the Boole-Shannon expansion over an arbitrary set of base functions and derive the consistency condition for f = 1. The generalization of the Boole-Shannon formula presented in this paper is in terms of cofactors as co-efficients with respect to a set of CNFs called a base which appear in a given Boolean CNF formula itself. This approach results in a novel parallel algorithm for decomposition of a CNF formula and computation of all satisfying assignments when they exist by using the given data set of CNFs itself as the base.