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

TL;DR

This paper extends the Boole-Shannon expansion to a broader class of Boolean functions using orthonormal expansions, providing an elimination theorem and polynomial-time methods for checking equation consistency.

Contribution

It introduces a new elimination theorem for Boolean functions in the class B(Φ), generalizing classical expansion techniques and offering polynomial algorithms for consistency checking.

Findings

Elimination theorem proven for Boolean functions in class B(Φ)

Polynomial-time algorithm for checking Boolean equation consistency

Characterization of the class B(Φ) of Boolean functions

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.