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On the Approximate Non-Deterministic Degree of Total Boolean Functions

arXiv

Authors: Samruddhi Pednekar, Supartha Podder

Year

2026

Paper ID

68426

Status

Preprint

Abstract Read

~2 min

Abstract Words

239

Citations

N/A

Abstract

The approximate non-deterministic degree of a Boolean function f, denoted mathsf{ndeg}_ε(f) written $mathsf{N}_ε(ffor brevity), is the minimum degree of a real polynomialpsuch that0 \le |p(x)| \le εwheneverf(x) = 0, and|p(x)| \ge 1wheneverf(x) = 1. Unlike exact non-deterministic degree, which only requires the polynomial to be nonzero on1-inputs, this measure enforces a uniform gap: the polynomial must stay close to zero on all0-inputs and bounded away from zero on all1-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 functionf, \[ deg(f) le widetilde Oleft$operatorname{rdeg}(f$^3right). \] In their paper, they explicitly propose a stronger conjecture: that approximate degree is polynomially bounded by\mathsf{N}_ε(f)and\mathsf{N}_εoverline{f}jointly, i.e., for every total Boolean functionfand every constant0<ε<1$, \[ \widetilde{deg}(f) \le \operatorname{poly}mathsf N_ε(f, \mathsf N_εoverline f). \] 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.

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The approximate non-deterministic degree of a Boolean function f, denoted mathsfndeg_ε(f) written mathsfN_ε(ffor brevity), is the minimum degree of a real polynomialpsuch that0...

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References & Citation Signals

[1] DOI https://doi.org/arXiv:2605.23336 [2] arXiv https://arxiv.org/abs/2605.23336 [3] Publisher https://arxiv.org/abs/2605.23336

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