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：对于任意布尔函数$f$和$g$，不等式$R(f\circ g) = \tilde{\Theta}(R(f)R(g))$是否成立的问题被称为随机查询复杂度的复合性问题。类似地，近似度量的复合性问题则询问$\widetilde{deg}(f\circ g) = \tilde{\Theta}(\widetilde{deg}(f)\cdot\widetilde{deg}(g))$是否成立。这两个问题是该领域最重要且被广泛研究的难题之一，然而我们距离圆满解答仍相去甚远。已知若对外部函数$f$（或内部函数$g$）施加特定性质，则上述度量具有复合性。本文扩展了使得$\text{R}$和$\widetilde{\text{deg}}$具有复合性的外部函数类。近期一项里程碑式结果（Ben-David与Blais, 2020）表明$R(f \circ g) = \Omega(noisyR(f)\cdot R(g))$，这意味着当$noisyR(f) = \Tilde{\Theta}(R(f))$时复合性成立。我们给出两个结果：(1) 当$R(f) = \Theta(n)$时，有$noisyR(f) = \Theta(R(f))$；(2) 若$\text{R}$关于某外部函数具有复合性，则$\text{noisyR}$关于同一外部函数也具有复合性。另一方面，据我们所知，此前尚未有$\widetilde{deg}(f \circ g) = \Omega(M(f) \cdot \widetilde{deg}(g))$（其中$M(\cdot)$为非平凡复杂度度量）这类结果的报道。我们证明了$\widetilde{deg}(f\circ g) = \widetilde{\Omega}(\sqrt{bs(f)} \cdot \widetilde{deg}(g))$，其中$bs(f)$为$f$的块灵敏度。这表明当$\widetilde{\text{deg}}(f)$渐近等价于$\sqrt{\text{bs}(f)}$时，$\widetilde{\text{deg}}$具有复合性。已知当外部函数为对称函数时，$\text{R}$和$\widetilde{\text{deg}}$均具有复合性。本文进一步将这些结果推广至对外部函数施加更弱对称性的情形。