Boolean function $F(x,y)$ for $x,y \in \{0,1\}^n$ is an XOR function if $F(x,y)=f(x\oplus y)$ for some function $f$ on $n$ input bits, where $\oplus$ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known that deterministic communication complexity of $F$ is closely related to parity decision tree complexity of $f$. Montanaro and Osbourne (2009) observed that one-sided communication complexity $D_{cc}^{\rightarrow}(F)$ of $F$ is exactly equal to nonadaptive parity decision tree complexity $NADT^{\oplus}(f)$ of $f$. Hatami et al. (2018) showed that unrestricted communication complexity of $F$ is polynomially related to parity decision tree complexity of $f$. We initiate the studies of a similar connection for partial functions. We show that in case of one-sided communication complexity whether these measures are equal, depends on the number of undefined inputs of $f$. On the one hand, if $D_{cc}^{\rightarrow}(F)=t$ and $f$ is undefined on at most $O(\frac{2^{n-t}}{\sqrt{n-t}})$, then $NADT^{\oplus}(f)=t$. On the other hand, for a wide range of values of $D_{cc}^{\rightarrow}(F)$ and $NADT^{\oplus}(f)$ (from constant to $n-2$) we provide partial functions for which $D_{cc}^{\rightarrow}(F) < NADT^{\oplus}(f)$. In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-sided communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on Boolean Fourier analysis and the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.

翻译：对于 $x,y \in \{0,1\}^n$，若存在一个 $n$ 输入比特的函数 $f$ 使得 $F(x,y)=f(x\oplus y)$，其中 $\oplus$ 表示按位异或运算，则布尔函数 $F(x,y)$ 称为异或函数。异或函数在通信复杂度研究中具有重要意义，部分原因在于其允许应用傅里叶分析技术。对于完全定义的异或函数，已知 $F$ 的确定性通信复杂度与 $f$ 的奇偶决策树复杂度密切相关。Montanaro 和 Osbourne（2009）指出 $F$ 的单向通信复杂度 $D_{cc}^{\rightarrow}(F)$ 恰好等于 $f$ 的非自适应奇偶决策树复杂度 $NADT^{\oplus}(f)$。Hatami 等人（2018）证明了 $F$ 的无限制通信复杂度与 $f$ 的奇偶决策树复杂度存在多项式关系。我们针对部分函数展开了类似关联性的研究。我们发现，在单向通信复杂度情形下，这些度量是否相等取决于 $f$ 未定义输入的数量。一方面，若 $D_{cc}^{\rightarrow}(F)=t$ 且 $f$ 的未定义输入不超过 $O(\frac{2^{n-t}}{\sqrt{n-t}})$，则 $NADT^{\oplus}(f)=t$。另一方面，对于 $D_{cc}^{\rightarrow}(F)$ 与 $NADT^{\oplus}(f)$ 的广泛取值区间（从常数到 $n-2$），我们构造了满足 $D_{cc}^{\rightarrow}(F) < NADT^{\oplus}(f)$ 的部分函数。特别地，我们给出了两个度量间存在指数级差距的函数实例。我们的分离结果同样适用于双向通信复杂度情形，这尤其表明 Hatami 等人（2018）的结果无法推广至部分函数。先前针对完全函数的研究严重依赖于布尔傅里叶分析技术，而该技术无法直接适用于部分函数。为此，我们构建了线性代数框架来证明所得结果，分离性证明则通过归约到覆盖码问题实现。