  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.
