Solutions of Boolean equations by orthonormal expansion

Developments in solving equations over Boolean algebras and their applications are as old as George Boole’s monograph [1], while Shannon pioneered applications of Boolean logic to switching circuits. Boolean equations and their solutions are of central importance to many problems across Sciences such as Chemistry, Biology or Medicine while traditionally Boolean problems arise in domains such as verification and design of logic circuits, software verification and Artificial Intelligence in Computer Science as well as Decision Sciences such as Operational Research. Search for assignments of Boolean variables over the two element Boolean algebra B0 = {0, 1} for which given Boolean constraints hold true or function has specified value are known as satisfiability SAT problems and has lead to development of efficient algorithms such as DPLL and other approaches [4, 5] over past half century. Boolean equation solving methods of [3, 2] on the other hand are applicable for solving equations over general Boolean algebras. These two developments have largely evolved independently. The potential of Boolean equation solving methods is yet to be fully explored as is apparent from literature [4, 5]. Similarly development of efficient and scalable parallel computational algorithms for the problem of solving systems of Boolean equations in large number of unknowns over large number of processors is yet to mature fully. In this report we shall explore utilization of expansion in terms of orthonormal systems [2] and solutions of Boolean equations in orthonormal variables developed in [3]. Boolean equations in orthonormal variables are always linear in either the Boolean disjunction + (denoting OR) or the ring operation ⊕ (denoting XOR). Hence these are rich systems for algorithmic study. However it is not clear which applications involve orthonormal variables naturally. In this report we explore one indirect application that of studying consistency of Boolean equations.