Matrix and combinatorics solutions of Boolean differential equations

Abstract

Boolean differential equation is a logic equation containing Boolean differences of Boolean functions. We formulate the solution in terms of matrix notations and consider two methods. The focus of the first method is that the initial Boolean differential equation is represented by a system of logic equations in Reed–Muller canonical form, then it is solved by discrete orthogonal transform. The computational complexity of the first method (2 2 n +3 n in terms of logic operations) is reduced to sorting of 2 n elements and combinatorics forming of 2 2 n −1 solutions, where n is the number of variables in the equation.