On the Approximate Non-Deterministic Degree of Total Boolean Functions

2026-05-22

AI Summary

This work investigates whether the approximate degree of a total Boolean function can be polynomially upper-bounded by the product of its approximate nondeterministic degrees for the function and its complement, thereby advancing the understanding of de Wolf's conjecture. For the first time, we systematically verify this stronger conjecture across several natural classes of Boolean functions—including monotone, unate, symmetric functions, k-uniform hypergraph properties, and read-k DNF formulas. By integrating tools from polynomial approximation theory, analysis of Boolean functions, and complexity measures, and through a refined characterization of function structure, we establish that all these classes satisfy the proposed upper bound, offering significant progress on the long-standing open question concerning the relationship between approximate degree and approximate nondeterministic degree.

Abstract

The approximate non-deterministic degree of a Boolean function $f$, denoted $\mathsf{ndeg}_ε(f)$ (written $\mathsf{N}_ε(f)$ for brevity), is the minimum degree of a real polynomial $p$ such that $0 \le |p(x)| \le ε$ whenever $f(x) = 0$, and $|p(x)| \ge 1$ whenever $f(x) = 1$. Unlike exact non-deterministic degree, which only requires the polynomial to be nonzero on $1$-inputs, this measure enforces a uniform gap: the polynomial must stay close to zero on all $0$-inputs and bounded away from zero on all $1$-inputs. The rational degree conjecture, open for over three decades, was recently resolved by Kothari, Kovacs-Deak, Wang, and Yang, who showed that for every total Boolean function $f$, \[ deg(f) \le \widetilde O\!\left(\operatorname{rdeg}(f)^3\right). \] In their paper, they explicitly propose a stronger conjecture: that approximate degree is polynomially bounded by $\mathsf{N}_ε(f)$ and $\mathsf{N}_ε(\overline{f})$ jointly, i.e., for every total Boolean function $f$ and every constant $0<ε<1$, \[ \widetilde{deg}(f) \le \operatorname{poly}(\mathsf N_ε(f), \mathsf N_ε(\overline f)). \] This conjecture, if true, would imply a polynomial version of the rational degree result and bring us closer to resolving de Wolf's longstanding non-deterministic degree conjecture. In this work, we make the first systematic progress on this problem, establishing the conjecture for several broad and natural function classes: monotone and unate functions, functions of bounded alternation number, symmetric functions, $k$-uniform hypergraph properties, and read-$k$ Disjunctive Normal Form (DNF) formulas.

Problem

• approximate non-deterministic degree
• approximate degree
• Boolean functions
• polynomial bound
• rational degree conjecture

Innovation

• approximate non-deterministic degree
• approximate degree
• Boolean functions
• polynomial upper bound
• rational degree conjecture