On the Approximate Non-Deterministic Degree of Total Boolean Functions

• URL: http://arxiv.org/abs/2605.23336v1
• Date: Fri, 22 May 2026 07:52:09 GMT
• Title: On the Approximate Non-Deterministic Degree of Total Boolean Functions
• Abstract summary: approximate non-deterministic degree of a function $f$, $mathsfndeg_(f)$ (written $mathsfN_(f)$ for brevity)<n>The rational degree conjecture, open for over three decades, was recently resolved by Kothari, KovacsDeak, Wang, and Yang.
• Score: 0.2864713389096699
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Quantum and classical query complexities of functions of matrices [0.0]
  We show that for any continuous function $f(x):[-1,1]rightarrow [-1,1]$, the quantum query complexity of computing $brai f(A) ketjpm varepsilon/4$ is lower bounded by $Omega(widetildedeg_varepsilon(f))$.
  arXiv  Detail & Related papers  (2023-11-13T00:45:41Z)
• On the Rational Degree of Boolean Functions and Applications [2.929575660518211]
  We study a natural complexity measure of Boolean functions known as the rational degree.<n>We show that for quantum computers, post-selection and bounded-error are resources in the black-box model.
  arXiv  Detail & Related papers  (2023-10-12T03:14:44Z)
• Dimension-free discretizations of the uniform norm by small product sets [45.85600902330814]
  A classical inequality of Bernstein compares the supremum norm of $f$ over the unit circle to its supremum norm over the sampling set of the $K$-th roots of unity.<n>We show that dimension-free discretizations are possible with sampling sets whose cardinality is independent of $deg(f)$ and is instead governed by the maximum individual degree of $f$.
  arXiv  Detail & Related papers  (2023-10-11T22:46:09Z)
• A polynomial quantum computing algorithm for solving the dualization problem [75.38606213726906]
  Given two monotone prime functions $f:0,1n to 0,1$ and $g:0,1n to 0,1$ the dualization problem consists in determining if $g$ is the dual of $f$. We present a quantum computing algorithm that solves the decision version of the dualization problem in time.
  arXiv  Detail & Related papers  (2023-08-28T18:12:54Z)
• The Approximate Degree of DNF and CNF Formulas [95.94432031144716]
  For every $delta>0,$ we construct CNF and formulas of size with approximate degree $Omega(n1-delta),$ essentially matching the trivial upper bound of $n. We show that for every $delta>0$, these models require $Omega(n1-delta)$, $Omega(n/4kk2)1-delta$, and $Omega(n/4kk2)1-delta$, respectively.
  arXiv  Detail & Related papers  (2022-09-04T10:01:39Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Learning low-degree functions from a logarithmic number of random queries [77.34726150561087]
  We prove that for any integer $ninmathbbN$, $din1,ldots,n$ and any $varepsilon,deltain(0,1)$, a bounded function $f:-1,1nto[-1,1]$ of degree at most $d$ can be learned.
  arXiv  Detail & Related papers  (2021-09-21T13:19:04Z)
• Degree vs. Approximate Degree and Quantum Implications of Huang's Sensitivity Theorem [4.549831511476248]
  We show that for any total Boolean function $f$, $bullet quad mathrmdeg(f) = O(widetildemathrmdeg(f)2)$: The degree of $f$ is at mosttrivial quadratic in the approximate degree of $f$. We show that if $f$ is a non monotone graph property of an $n$-vertex graph specified by its adjacency matrix, then $mathrmQ(f)=Omega(n)$, which is also optimal.
  arXiv  Detail & Related papers  (2020-10-23T19:21:28Z)
• An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
  We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
  arXiv  Detail & Related papers  (2020-08-24T06:50:57Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.