On the weight and density bounds of polynomial threshold functions

TL;DR

This paper establishes new upper bounds on the weight and density of polynomial threshold functions representing Boolean functions, including special cases like Bent functions and sparse functions, improving understanding of their complexity.

Contribution

It provides the best known bounds on the number of monomials and coefficient magnitudes for polynomial threshold functions of Boolean functions.

Findings

All n-variable Boolean functions can be represented with at most 0.75 * 2^n non-zero integer coefficients.

Bent functions can be represented with coefficients less than 2^n while maintaining density bounds.

Sparse Boolean functions have small weight PTFs with density at most m + 2^{n-1}.

Abstract

In this report, we show that all n-variable Boolean function can be represented as polynomial threshold functions (PTF) with at most $0.75 \times 2^n$ non-zero integer coefficients and give an upper bound on the absolute value of these coefficients. To our knowledge this provides the best known bound on both the PTF density (number of monomials) and weight (sum of the coefficient magnitudes) of general Boolean functions. The special case of Bent functions is also analyzed and shown that any n-variable Bent function can be represented with integer coefficients less than $2^n$ while also obeying the aforementioned density bound. Finally, sparse Boolean functions, which are almost constant except for $m << 2^n$ number of variable assignments, are shown to have small weight PTFs with density at most $m+2^{n-1}$.