$\newcommand{\F}{\mathbb{F}}$We study the Boolean function parameters sensitivity ($s$), block sensitivity ($bs$), and alternation ($alt$) under specially designed affine transforms. For a function $f:\F_2^n\to \{0,1\}$, and $A=Mx+b$ for $M \in \F_2^{n\times n}$ and $b\in \F_2^n$, the result of the transformation $g$ is defined as $\forall x\in\F_2^n, g(x)=f(Mx+b)$. We study alternation under linear shifts ($M$ is the identity matrix) called the shift invariant alternation (denoted by $salt(f)$). We exhibit an explicit family of functions for which $salt(f)$ is $2^{\Omega(s(f))}$. We show an affine transform $A$, such that the corresponding function $g$ satisfies $bs(f,0^n) \le s(g)$, using which we proving that for $F(x,y)=f(x\land y)$, the bounded error quantum communication complexity of $F$ with prior entanglement, $Q^*_{1/3}(F)=\Omega(\sqrt{bs(f,0^n)})$. Our proof builds on ideas from Sherstov (2010) where we use specific properties of the above affine transformation. We show, * For a prime $p$ and $0<\epsilon<1$, any $f$ with $deg_p(f)\le(1-\epsilon)\log n$ must satisfy $Q^*_{1/3}(F) = \Omega(\frac{n^{\epsilon/2}}{\log n})$. Here, $deg_p(f)$ denotes the degree of the multilinear polynomial of $f$ over $\F_p$. * For any $f$ such that there exists primes $p$ and $q$ with $deg_q(f) \ge \Omega(deg_p(f)^\delta)$ for $\delta > 2$, the deterministic communication complexity - $D(F)$ and $Q^*_{1/3}(F)$ are polynomially related. In particular, this holds when $deg_p(f) = O(1)$. Thus, for this class of functions, this answers an open question (see Buhrman and deWolf (2001)) about the relation between the two measures. We construct linear transformation $A$, such that $g$ satisfies, $alt(f) \le 2s(g)+1$. Using this, we exhibit a family of Boolean functions that rule out a potential approach to settle the XOR Log-Rank conjecture via a proof of Sensitivity conjecture [Hao Huang (2019)].
翻译:$\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\F\\\\\\\\\F\\\\\\F\\\\\F\\\\F\\\F\\\\\\F\\\\F\\\\\F\\\\\\F\\\\F\\\\\\\\\\F\\\\F\\\\美元, 变色美元是$\\\\美元\\\\美元\\美元\\\\美元\\美元\\美元\\\\F\\\F\\\\\\\\\\\\\\\\\\\美元\美元\\\\美元\\\\\美元\\\\\\美元\\\\\\\\\\\\\\\\\\\\\\\\美元\美元\美元\美元\美元\美元\美元\美元\美元\美元\美元\美元\美元\美元\\\\美元\\\\\\\\\美元\美元\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\美元\\美元\美元\\\\\\\\\\\\\美元\美元\美元\美元\美元\美元\美元\美元\美元\\\\\美元\美元\c\\\\\\\\\一个一个一个一个一个一个一个一个一个在变变的变美元\美元(美元(美元\一个一个一个一个一个一个一个一个一个一个一个一个一个一个一个一个一个一个一个一个一个一个一个一个一个一个在美元(美元\一个一个一个一个一个一个一个或美元和美元\美元的变美元的变美元(美元(美元(美元(美元, 美元\美元\一个或美元(美元\c\