Construction of smooth convex extensions of Boolean functions

Abstract

Systems of Boolean equations are widely used in mathematics, computer science, and applied sciences. In this regard, on the one hand, new research methods and algorithms are being developed for such systems, and on the other hand, existing methods and algorithms for solving such systems are being improved. One of these methods is that, firstly, the system of Boolean equations given over the ring of Boolean polynomials is transformed into a system of equations over the field of real numbers, and secondly, the transformed system is reduced either to the problem of numerical minimization of the corresponding objective function, to a MILP or QUBO problem, to a system of polynomial equations solved on the set of integers, or to an equivalent system of polynomial equations solved by symbolic methods. There are many ways to transform a system of Boolean equations into a continuous minimization problem, since the fundamental difference between such methods and “brute force” local search algorithms is that at each iteration of the algorithm, the shift along the antigradient is performed on all variables simultaneously. But one of the main problems that arise when applying these methods is that the objective function to be minimized in the desired area can have many local minima, which greatly complicates their practical use. In this paper, a non-negative convex and continuously differentiable extension of any Boolean function is constructed, which is applied to solving an arbitrary system of Boolean equations. It is argued that the problem of solving an arbitrary system of Boolean equations can be constructively reduced to the problem of minimizing a function, any local minimum of which in the desired domain is a global minimum.