We explore the $\textit{average-case deterministic query complexity}$ of boolean functions under the $\textit{uniform distribution}$, denoted by $\mathrm{D}_\mathrm{ave}(f)$, the minimum average depth of zero-error decision tree computing a boolean function $f$. This measure found several applications across diverse fields. We study $\mathrm{D}_\mathrm{ave}(f)$ of several common functions, including penalty shoot-out functions, symmetric functions, linear threshold functions and tribes functions. Let $\mathrm{wt}(f)$ denote the number of the inputs on which $f$ outputs $1$. We prove that $\mathrm{D}_\mathrm{ave}(f) \le \log \frac{\mathrm{wt}(f)}{\log n} + O\left(\log \log \frac{\mathrm{wt}(f)}{\log n}\right)$ when $\mathrm{wt}(f) \ge 4 \log n$ (otherwise, $\mathrm{D}_\mathrm{ave}(f) = O(1)$), and that for almost all fixed-weight functions, $\mathrm{D}_\mathrm{ave}(f) \geq \log \frac{\mathrm{wt}(f)}{\log n} - O\left( \log \log \frac{\mathrm{wt}(f)}{\log n}\right)$, which implies the tightness of the upper bound up to an additive logarithmic term. We also study $\mathrm{D}_\mathrm{ave}(f)$ of circuits. Using H\r{a}stad's switching lemma or Rossman's switching lemma [Comput. Complexity Conf. 137, 2019], one can derive upper bounds $\mathrm{D}_\mathrm{ave}(f) \leq n\left(1 - \frac{1}{O(k)}\right)$ for width-$k$ CNFs/DNFs and $\mathrm{D}_\mathrm{ave}(f) \leq n\left(1 - \frac{1}{O(\log s)}\right)$ for size-$s$ CNFs/DNFs, respectively. For any $w \ge 1.1 \log n$, we prove the existence of some width-$w$ size-$(2^w/w)$ DNF formula with $\mathrm{D}_\mathrm{ave} (f) = n \left(1 - \frac{\log n}{\Theta(w)}\right)$, providing evidence on the tightness of the switching lemmas.

翻译：我们研究了在均匀分布下布尔函数的平均情况确定性查询复杂度，记为$\mathrm{D}_\mathrm{ave}(f)$，即计算布尔函数$f$的零错误决策树的最小平均深度。该度量在多个不同领域有应用。我们研究了几种常见函数的$\mathrm{D}_\mathrm{ave}(f)$，包括点球大战函数、对称函数、线性阈值函数和部落函数。设$\mathrm{wt}(f)$表示函数$f$输出为$1$的输入数量。我们证明当$\mathrm{wt}(f) \ge 4 \log n$时，$\mathrm{D}_\mathrm{ave}(f) \le \log \frac{\mathrm{wt}(f)}{\log n} + O\left(\log \log \frac{\mathrm{wt}(f)}{\log n}\right)$（否则，$\mathrm{D}_\mathrm{ave}(f) = O(1)$），并且对于几乎所有固定权重的函数，$\mathrm{D}_\mathrm{ave}(f) \geq \log \frac{\mathrm{wt}(f)}{\log n} - O\left( \log \log \frac{\mathrm{wt}(f)}{\log n}\right)$，这表明该上界在加法对数项意义下是紧的。我们还研究了电路的$\mathrm{D}_\mathrm{ave}(f)$。利用Håstad切换引理或Rossman切换引理[Comput. Complexity Conf. 137, 2019]，对于宽度为$k$的CNF/DNF公式可推导出上界$\mathrm{D}_\mathrm{ave}(f) \leq n\left(1 - \frac{1}{O(k)}\right)$，对于规模为$s$的CNF/DNF公式可推导出上界$\mathrm{D}_\mathrm{ave}(f) \leq n\left(1 - \frac{1}{O(\log s)}\right)$。对于任意$w \ge 1.1 \log n$，我们证明了存在某些宽度为$w$、规模为$(2^w/w)$的DNF公式满足$\mathrm{D}_\mathrm{ave} (f) = n \left(1 - \frac{\log n}{\Theta(w)}\right)$，为切换引理的紧性提供了证据。