Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e.~functions obtained by composing a base function with itself a number of times. Let $h^d$ denote the standard $d$-fold composition of the base function $h$. The main result of this work is to show that the approximate degree composes if either of the following conditions holds: \begin{itemize} \item The outer function $f:\{0,1\}^n\to \{0,1\}$ is a recursive function of the form $h^d$, with $h$ being any base function and $d= \Omega(\log\log n)$. \item The inner function is a recursive function of the form $h^d$, with $h$ being any constant arity base function (other than AND and OR) and $d= \Omega(\log\log n)$, where $n$ is the arity of the outer function. \end{itemize} In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be \emph{efficiently eliminated} if the inner or outer function is a recursive function.

翻译：布尔函数的近似度复合性判定仍然是布尔函数复杂度领域一个重要的未解问题。近几十年来，研究者们主要致力于证明特殊类型内函数与外函数的近似度复合性。其中一类重要且被广泛研究的函数是递归函数，即通过将基函数与自身多次复合而得到的函数。令 $h^d$ 表示基函数 $h$ 的标准 $d$ 次复合。本文的主要结果表明，在以下任一条件成立时近似度具有复合性：\begin{itemize} \item 外函数 $f:\{0,1\}^n\to \{0,1\}$ 是形如 $h^d$ 的递归函数，其中 $h$ 为任意基函数且 $d= \Omega(\log\log n)$。 \item 内函数是形如 $h^d$ 的递归函数，其中 $h$ 为任意常元数基函数（除 AND 与 OR 外）且 $d= \Omega(\log\log n)$，此处 $n$ 表示外函数的元数。 \end{itemize} 在证明技术层面，我们首先观察到可通过在内函数与外函数之间引入多数函数来获得复合性下界。随后我们证明，当内函数或外函数为递归函数时，多数函数可以被\emph{高效消除}。