In this paper, we study the Boolean function parameters sensitivity ($\mathsf{s}$), block sensitivity ($\mathsf{bs}$), and alternation ($\mathsf{alt}$) under specially designed affine transforms and show several applications. For a function $f:{\mathbb{F}}_2^n\to \{-1,1\}$, and $A=Mx+b$ for $M\in {\mathbb{F}}_2^{n\times n}$ and $b\in {\mathbb{F}}_2^n$, the result of the transformation g is defined as $\forall x\in {\mathbb{F}}_2^n,g(x)=f(Mx+b)$.

As a warm up, we study alternation under linear shifts (when M is restricted to be the identity matrix) called the shift invariant alternation (the smallest alternation that can be achieved for the Boolean function f by shifts, denoted by $\mathsf{salt}(f)$). By a result of Lin and Zhang (2017) [7], it follows that $\mathsf{bs}(f)\le O(\mathsf{salt}{(f)}^2\mathsf{s}(f))$. Thus, to settle the Sensitivity Conjecture ($\forall f,\mathsf{bs}(f)\le \mathsf{poly}(\mathsf{s}(f))$), it suffices to argue that $\forall f,\mathsf{salt}(f)\le \mathsf{poly}(\mathsf{s}(f))$. However, we exhibit an explicit family of Boolean functions for which $\mathsf{salt}(f)$ is $2^{\mathrm{\Omega }(\mathsf{s}(f))}$.

Going further, we use an affine transform A, such that the corresponding function g satisfies $\mathsf{bs}(f,0^n)\le \mathsf{s}(g)$. We apply this in the setting of quantum communication complexity to prove that for $F(x,y)\stackrel{\text{def}}{=}f(x\wedge y)$, the bounded error quantum communication complexity of F with prior entanglement, $Q_{1/3}^{⁎}(F)$ is $\mathrm{\Omega }(\sqrt{\mathsf{bs}(f,0^n)})$. Our proof builds on ideas from Sherstov (2010) [17] where we use specific properties of the above affine transformation. Using this, we show the following.

(a)

For a fixed prime p and an ϵ, $0<ϵ<1$, any Boolean function f that depends on all its inputs with ${\mathsf{deg}}_p(f)\le (1-ϵ)\log n$ must satisfy $Q_{1/3}^{⁎}(F)=\mathrm{\Omega }\left(\frac{n^{ϵ/2}}{\log n}\right)$. Here, ${\mathsf{deg}}_p(f)$ denotes the degree of the multilinear polynomial over ${\mathbb{F}}_p$ which agrees with f on Boolean inputs.

(b)

For Boolean function f such that there exists primes p and q with ${\mathsf{deg}}_q(f)\ge \mathrm{\Omega }({\mathsf{deg}}_p{(f)}^{\delta })$ for $\delta >2$, the deterministic communication complexity - $\mathsf{D}(F)$ and $Q_{1/3}^{⁎}(F)$ are polynomially related. In particular, this holds when ${\mathsf{deg}}_p(f)=O(1)$. Thus, for this class of functions, this answers an open question (see Buhrman and de Wolf (2001) [15]) about the relation between the two measures.

在本文中，我们研究布尔函数参数的敏感性（$\mathsf{s}$），块敏感度（$\mathsf{s}$）和交替（$\mathsf{alt}$）经过特殊设计的仿射变换并展示了几种应用。对于功能$F：{\mathbb{F}}_2^{ñ}\to \{-\mathrm{1个}，\mathrm{1个}\}$和 $\mathrm{一种}=\mathrm{中号}X+b$ 对于 $\mathrm{中号}\in {\mathbb{F}}_2^{ñ\times ñ}$ 和 $b\in {\mathbb{F}}_2^{ñ}$，转换g的结果定义为$\forall X\in {\mathbb{F}}_2^{ñ}，G（X）=F（\mathrm{中号}X+b）$。

作为热身，我们研究线性位移（当M被限制为单位矩阵时）下的交替，称为位移不变交替（布尔函数f可以通过位移实现的最小交替，表示为$\mathsf{盐}（F）$）。根据Lin和Zhang（2017）[7]的结果，$\mathsf{s}（F）\le Ø（\mathsf{盐}{（F）}^2\mathsf{s}（F））$。因此，要解决敏感性猜想（$\forall F，\mathsf{s}（F）\le \mathsf{聚}（\mathsf{s}（F））$），足以证明 $\forall F，\mathsf{盐}（F）\le \mathsf{聚}（\mathsf{s}（F））$。但是，我们展示了一个明确的布尔函数族，$\mathsf{盐}（F）$ 是 $2^{\mathrm{\Omega }（\mathsf{s}（F））}$。

更进一步，我们使用仿射变换A，使得对应的函数g满足$\mathsf{s}（F，0^{ñ}）\le \mathsf{s}（G）$。我们将其应用在量子通信复杂性的设置中，以证明$F（X，ÿ）\stackrel{\text{定义}}{=}F（X\wedge ÿ）$，先验纠缠F的有界误差量子通信复杂度，${问}_{\mathrm{1个}/3}^{⁎}（F）$ 是 $\mathrm{\Omega }（\sqrt{\mathsf{s}（F，0^{ñ}）}）$。我们的证明基于Sherstov（2010）[17]的思想，其中我们使用了上述仿射变换的特定属性。使用此，我们显示以下内容。

（一种）

对于固定的素数p和ε，$0<ϵ<\mathrm{1个}$，任何依赖于其所有输入的布尔函数f${\mathsf{度}}_p（F）\le （\mathrm{1个}-ϵ）\mathrm{日志}ñ$ 必须满足 ${问}_{\mathrm{1个}/3}^{⁎}（F）=\mathrm{\Omega }（\frac{{ñ}^{ϵ/2}}{\mathrm{日志}ñ}）$。这里，${\mathsf{度}}_p（F）$ 表示上的多元线性多项式的次数 ${\mathbb{F}}_p$与布尔输入上的f一致。

（b）

对于布尔函数f，使得存在质数p和q为${\mathsf{度}}_q（F）\ge \mathrm{\Omega }（{\mathsf{度}}_p{（F）}^{\delta }）$ 对于 $\delta >2$，确定性的通信复杂度- $\mathsf{d}（F）$ 和 ${问}_{\mathrm{1个}/3}^{⁎}（F）$与多项式相关。特别是，当${\mathsf{度}}_p（F）=Ø（\mathrm{1个}）$。因此，对于此类功能，这回答了有关两个度量之间关系的公开问题（参见Buhrman和de Wolf（2001）[15]）。