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$x,y \in \{0,1\}^n$，布尔函数$F(x,y)$称为XOR函数，若存在某个$n$输入比特函数$f$使得$F(x,y)=f(x\oplus y)$，其中$\oplus$表示逐位异或。XOR函数在通信复杂度领域具有重要意义，部分原因在于其允许傅里叶分析技术的应用。对于完全XOR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与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（2018）证明$F$的无限制通信复杂度与$f$的奇偶决策树复杂度呈多项式相关。本文首次研究了部分函数的类似关联。我们发现对于单向通信复杂度，这些度量是否相等取决于$f$的未定义输入数量。一方面，若$D_{cc}^{\rightarrow}(F)=t$且$f$至多在$O(\frac{2^{n-t}}{\sqrt{n-t}})$个输入上未定义，则$NADT^{\oplus}(f)=t$。另一方面，对于$D_{cc}^{\rightarrow}(F)$与$NADT^{\oplus}(f)$的广泛取值范围（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，我们给出了一个两类度量存在指数级差距的函数。这一分离结果同样适用于双向通信复杂度情形，尤其表明Hatami等（2018）的结论无法推广至部分函数。此前完全函数的结果高度依赖布尔傅里叶分析，而该技术无法直接迁移至部分函数。我们通过构建线性代数框架完成结果证明，分离结果则通过归约到覆盖码（covering codes）实现。