Polynomial-Time Algorithms for Checking Some Properties of Boolean Functions Given by Polynomials

Abstract

In this paper, we show that checking some properties of Boolean functions which are given by the lists of monomials in their polynomial representations can be implemented in polynomial time. Multi-linear polynomials over GF(2) are often a convenient way to represent Boolean functions. There is a single polynomial for each Boolean function, and the length of the polynomial (i.e. the number of its monomials) which represents a function of n variables can be far less than 2ⁿ. Therefore, in some cases, polynomials are a compressed description of Boolean functions. Besides, polynomial representations of Boolean functions have applications in circuit lower bounds, computational learning, error-correcting codes, cryptography. We construct polynomial-time algorithms for checking some properties of Boolean functions which are given by the lists of monomials in their polynomial representations. The considered properties are self-anti-duality (evenness), self-duality, periodicity, 1-invariance (Möbius transform invariance, coincidence). Note that checking each of these properties directly by definition gives, in the general case, only exponential-time algorithms. The approach to construct our algorithms is the following. Firstly, we prove that if a Boolean function has a certain property then its polynomial has a special structure. And secondly, we check the property by its characterization, cutting negative cases by proven facts. More precisely, we analyze an exponential time algorithm and prove that if the number of steps of the algorithm exceeds a bound, which is polynomial in the input size, and may be computed in advance, then the input will be necessarily rejected.