On Restricting No-Junta Boolean Function and Degree Lower Bounds by Polynomial Method

TL;DR

This paper investigates the structure of Boolean functions depending on all variables, establishing a property of their subfunctions, and applies this to derive degree lower bounds for polynomial representations over finite rings, highlighting differences between symmetric and non-symmetric functions.

Contribution

It proves a new property of Boolean functions regarding their subfunctions and derives novel degree lower bounds for polynomial representations over finite rings, especially distinguishing symmetric and non-symmetric cases.

Findings

For any Boolean function depending on all variables, some restriction depends on fewer variables.

Degree bounds for polynomial representations over finite rings depend on symmetry and prime factorization.

Symmetric functions have degree bounds of o(√n), non-symmetric o(√log n) in finite fields.

Abstract

Let $\mathcal{F}_{n}^*$ be the set of Boolean functions depending on all $n$ variables. We prove that for any $f\in \mathcal{F}_{n}^*$, $f|_{x_i=0}$ or $f|_{x_i=1}$ depends on the remaining $n-1$ variables, for some variable $x_i$. This existent result suggests a possible way to deal with general Boolean functions via its subfunctions of some restrictions. As an application, we consider the degree lower bound of representing polynomials over finite rings. Let $f\in \mathcal{F}_{n}^*$ and denote the exact representing degree over the ring $\mathbb{Z}_m$ (with the integer $m>2$) as $d_m(f)$. Let $m=\Pi_{i=1}^{r}p_i^{e_i}$, where $p_i$'s are distinct primes, and $r$ and $e_i$'s are positive integers. If $f$ is symmetric, then $m\cdot d_{p_1^{e_1}}(f)... d_{p_r^{e_r}}(f) > n$. If $f$ is non-symmetric, by the second moment method we prove almost always $m\cdot d_{p_1^{e_1}}(f)... d_{p_r^{e_r}}(f) > \lg{n}-1$. In particular, as $m=pq$ where $p$ and $q$ are arbitrary distinct primes, we have $d_p(f)d_q(f)=\Omega(n)$ for symmetric $f$ and $d_p(f)d_q(f)=\Omega(\lg{n}-1)$ almost always for non-symmetric $f$. Hence any $n$-variate symmetric Boolean function can have exact representing degree $o(\sqrt{n})$ in at most one finite field, and for non-symmetric functions, with $o(\sqrt{\lg{n}})$-degree in at most one finite field.