For any Boolean functions $f$ and $g$, the question whether $R(f\circ g) = \tilde{\Theta}(R(f)R(g))$, is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether $\widetilde{deg}(f\circ g) = \tilde{\Theta}(\widetilde{deg}(f)\cdot\widetilde{deg}(g))$. These questions are two of the most important and well-studied problems, and yet we are far from answering them satisfactorily. It is known that the measures compose if one assumes various properties of the outer function $f$ (or inner function $g$). This paper extends the class of outer functions for which $\text{R}$ and $\widetilde{\text{deg}}$ compose. A recent landmark result (Ben-David and Blais, 2020) showed that $R(f \circ g) = \Omega(noisyR(f)\cdot R(g))$. This implies that composition holds whenever $noisyR(f) = \Tilde{\Theta}(R(f))$. We show two results: (1)When $R(f) = \Theta(n)$, then $noisyR(f) = \Theta(R(f))$. (2) If $\text{R}$ composes with respect to an outer function, then $\text{noisyR}$ also composes with respect to the same outer function. On the other hand, no result of the type $\widetilde{deg}(f \circ g) = \Omega(M(f) \cdot \widetilde{deg}(g))$ (for some non-trivial complexity measure $M(\cdot)$) was known to the best of our knowledge. We prove that $\widetilde{deg}(f\circ g) = \widetilde{\Omega}(\sqrt{bs(f)} \cdot \widetilde{deg}(g)),$ where $bs(f)$ is the block sensitivity of $f$. This implies that $\widetilde{\text{deg}}$ composes when $\widetilde{\text{deg}}(f)$ is asymptotically equal to $\sqrt{\text{bs}(f)}$. It is already known that both $\text{R}$ and $\widetilde{\text{deg}}$ compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function.

翻译：对任意布尔函数 $f$ 和 $g$，$R(f\circ g) = \tilde{\Theta}(R(f)R(g))$ 是否成立被称为随机查询复杂度的组合问题。类似地，近似度数的组合问题询问 $\widetilde{deg}(f\circ g) = \tilde{\Theta}(\widetilde{deg}(f)\cdot\widetilde{deg}(g))$ 是否成立。这两个问题是最重要且被广泛研究的问题之一，但离令人满意的解答仍相距甚远。已知当外层函数 $f$（或内层函数 $g$）满足某些性质时，这些测度具有组合性质。本文扩展了使得 $\text{R}$ 和 $\widetilde{\text{deg}}$ 具有组合性质的外层函数类。近期一项里程碑式结果（Ben-David and Blais, 2020）表明 $R(f \circ g) = \Omega(noisyR(f)\cdot R(g))$。这意味着当 $noisyR(f) = \Tilde{\Theta}(R(f))$ 时组合性质成立。我们给出两个结果：(1) 当 $R(f) = \Theta(n)$ 时，$noisyR(f) = \Theta(R(f))$。(2) 若 $\text{R}$ 相对于某个外层函数具有组合性质，则 $\text{noisyR}$ 也相对于该外层函数具有组合性质。另一方面，据我们所知，此前尚无 $\widetilde{deg}(f \circ g) = \Omega(M(f) \cdot \widetilde{deg}(g))$（其中 $M(\cdot)$ 为非平凡复杂度测度）类型的结果。我们证明 $\widetilde{deg}(f\circ g) = \widetilde{\Omega}(\sqrt{bs(f)} \cdot \widetilde{deg}(g))$，其中 $bs(f)$ 是 $f$ 的块灵敏度。这表明当 $\widetilde{\text{deg}}(f)$ 渐近等于 $\sqrt{\text{bs}(f)}$ 时，$\widetilde{\text{deg}}$ 具有组合性质。已知当外层函数对称时，$\text{R}$ 和 $\widetilde{\text{deg}}$ 均具有组合性质。我们还将这些结果推广到外层函数更弱的对称性情形。