New Upper Bounds on the Average PTF Density of Boolean Functions

Abstract

A Boolean function f:{1, − 1}ⁿ →{1, − 1} is said to be sign-represented by a real polynomial $p:{\mathbb R}^n \rightarrow {\mathbb R}$ if sgn(p(x)) = f(x) for all x ∈ {1, − 1}ⁿ. The PTF density of f is the minimum number of monomials in a polynomial that sign-represents f. It is well known that every n-variable Boolean function has PTF density at most 2ⁿ. However, in general, less monomials are enough. In this paper, we present a method that reduces the problem of upper bounding the average PTF density of n-variable Boolean functions to the computation of (some modified version of) average PTF density of k-variable Boolean functions for small k. By using this method, we show that almost all n-variable Boolean functions have PTF density at most (0.617) 2ⁿ, which is the best upper bound so far.