关于布尔函数在仿射子空间上的代数次数与非线性度的最大最小值

布尔函数的近一致闭minions

Near-unanimity-closed minions of Boolean functions

本文完整描述了布尔函数的近一致闭minions（目标代数含近一致函数的clonoids），关键借助小项-小项偏序及其序理想这一核心概念实现该结果。

Irreducible Boolean Functions

This paper is a contribution to the study of a quasi-order on the set $Ω$ of Boolean functions, the \emph{simple minor} quasi-order. We look at the join-irreducible members of the resulting poset $\tildeΩ$. Using a two-way correspondence between Boolean functions and hypergraphs, join-irreducibility translates into a combinatorial property of hypergraphs. We observe that among Steiner systems, those which yield join-irreducible members of $\tildeΩ$ are the -2-monomorphic Steiner systems. We also describe the graphs which correspond to join-irreducible members of $\tildeΩ$.

Bounding threshold dimension: realizing graphic Boolean functions as the AND of majority gates

A graph $G$ on $n$ vertices is a \emph{threshold graph} if there exist real numbers $a_1,a_2, \ldots, a_n$ and $b$ such that the zero-one solutions of the linear inequality $\sum \limits_{i=1}^n a_i x_i \leq b$ are the characteristic vectors of the cliques of $G$. Introduced in [Chv{á}tal and Hammer, Annals of Discrete Mathematics, 1977], the \emph{threshold dimension} of a graph $G$, denoted by $\dimth(G)$, is the minimum number of threshold graphs whose intersection yields $G$. Given a graph $G$ on $n$ vertices, in line with Chv{á}tal and Hammer, $f_G\colon \{0,1\}^n \rightarrow \{0,1\}$ is …

Join-irreducible Boolean functions

This paper is a contribution to the study of a quasi-order on the set $Ω$ of Boolean functions, the \emph{simple minor} quasi-order. We look at the join-irreducible members of the resulting poset $\tildeΩ$. Using a two-way correspondence between Boolean functions and hypergraphs, join-irreducibility translates into a combinatorial property of hypergraphs. We observe that among Steiner systems, those which yield join-irreducible members of $\tildeΩ$ are the -2-monomorphic Steiner systems. We also describe the graphs which correspond to join-irreducible members of $\tildeΩ$.

具有最小值的简单博弈

Simple games with minimum

本文研究具有唯一最小获胜向量的简单博弈（单调布尔函数），推导出此类博弈数量的闭式公式及其以玩家数和等价类表示的维度。

Alon-Füredi覆盖中的小型布尔截面

Small Boolean Sections in Alon-Füredi Covers

该研究针对Alon-Füredi覆盖问题，构造出显式的n个仿射超平面族，证明其最大布尔交集的最小值渐近达到平均下界，解决了覆盖的超平面规模与布尔交集大小的相关问题。

关于向量布尔函数与仿射函数之间的最小Hamming距离

On the minimum Hamming distance between vectorial Boolean and affine functions

本文研究向量布尔函数与仿射函数的Hamming距离，证明了2017年关于该度量的猜想上界对两类向量bent函数成立，改进了单项式函数的已知界，并计算了维数不超过9的APN函数到仿射函数的精确距离。

Generalized Maiorana-McFarland Constructions for Almost Optimal Resilient Functions

In a recent paper \cite{Zhang-Xiao}, Zhang and Xiao describe a technique on constructing almost optimal resilient functions on even number of variables. In this paper, we will present an extensive study of the constructions of almost optimal resilient functions by using the generalized Maiorana-McFarland (GMM) construction technique. It is shown that for any given $m$, it is possible to construct infinitely many $n$-variable ($n$ even), $m$-resilient Boolean functions with nonlinearity equal to $2^{n-1}-2^{n/2-1}-2^{k-1}$ where $k<n/2$. A generalized version of GMM construction is further desc…

Quadratization of Symmetric Pseudo-Boolean Functions

A pseudo-Boolean function is a real-valued function $f(x)=f(x_1,x_2,\ldots,x_n)$ of $n$ binary variables; that is, a mapping from $\{0,1\}^n$ to $\mathbb{R}$. For a pseudo-Boolean function $f(x)$ on $\{0,1\}^n$, we say that $g(x,y)$ is a quadratization of $f$ if $g(x,y)$ is a quadratic polynomial depending on $x$ and on $m$ auxiliary binary variables $y_1,y_2,\ldots,y_m$ such that $f(x)= \min \{g(x,y) : y \in \{0,1\}^m \}$ for all $x \in \{0,1\}^n$. By means of quadratizations, minimization of $f$ is reduced to minimization (over its extended set of variables) of the quadratic function $g(x,y)…

Self-Predicting Boolean Functions

A Boolean function $g$ is said to be an optimal predictor for another Boolean function $f$, if it minimizes the probability that $f(X^{n})\neq g(Y^{n})$ among all functions, where $X^{n}$ is uniform over the Hamming cube and $Y^{n}$ is obtained from $X^{n}$ by independently flipping each coordinate with probability $δ$. This paper is about self-predicting functions, which are those that coincide with their optimal predictor.

反极布尔函数的黑林格猜想

The Hellinger Conjecture for Antipodal Boolean Functions

该研究证明反极布尔函数的黑林格猜想，通过格拉姆矩阵核范数等方法将重叠度最小界扩展到任意独立二元输入信道，为信息论与测试提供相关界。