Generalization of Boole-Shannon expansion, consistency of Boolean equations and elimination by orthonormal expansion

Abstract

The well known Boole-Shannon expansion of Boolean functions in several variables (with co-efficients in a Boolean algebra $B$) is also known in more general form in terms of expansion in a set $\Phi$ of orthonormal functions. However, unlike the one variable step of this expansion an analogous elimination theorem and consistency is not well known. This article proves such an elimination theorem for a special class of Boolean functions denoted $B(\Phi)$. When the orthonormal set $\Phi$ is of polynomial size in number $n$ of variables, the consistency of a Boolean equation $f=0$ can be determined in polynomial number of $B$-operations. A characterization of $B(\Phi)$ is also shown and an elimination based procedure for computing consistency of Boolean equations is proposed.