Generalized cofactors and decomposition of Boolean satisfiability problems

We propose an approach for decomposing Boolean satisfiability problems while extending recent results of \cite{sul2} on solving Boolean systems of equations. Developments in \cite{sul2} 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 \emph{cofactors} as co-efficients with respect to a set of CNFs called a \emph{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.