Average-case deterministic query complexity of boolean functions with fixed weight (2403.03530v2)

Abstract: We explore the $\textit{average-case deterministic query complexity}$ of boolean functions under a $\textit{uniform distribution}$, denoted by $\mathrm{D}\mathrm{ave}(f)$, the minimum average depth of zero-error decision tree computing a boolean function $f$. This measure has found several applications across diverse fields, yet its understanding is limited. We study $\mathrm{D}\mathrm{ave}(f)$ of several functions, including the penalty shoot-out function, symmetric functions, linear threshold functions and the tribes functions. We prove $\mathrm{D}\mathrm{ave}(f) \le \max { \log \frac{\mathrm{wt}(f)}{\log n} + O(\log \log \frac{\mathrm{wt}(f)}{\log n}), O(1) }$ for every $n$-variable boolean function $f$, where $\mathrm{wt}(f)$ denotes the weight (the number of inputs on which $f$ outputs $1$). For any $4\log n \le m(n) \le 2^{n-1}$, we prove the upper bound is tight up to an additive logarithmic term for almost all $n$-variable boolean functions with weight $\mathrm{wt}(f) = m(n)$. Using H\r{a}stad's switching lemma or Rossman's switching lemma [Comput. Complexity Conf. 137, 2019], one can derive $\mathrm{D}\mathrm{ave}(f) \leq n(1 - \frac{1}{O(w)})$ or $\mathrm{D}\mathrm{ave}(f) \le n(1 - \frac{1}{O(\log s)})$ for CNF/DNF formulas of width $w$ or size $s$, respectively. We show that, for any $w \ge \log n + \log \log n + 3$, there exists a DNF formula of width $w$ and size $\lceil 2^w / w \rceil$ such that $\mathrm{D}\mathrm{ave}(f) = n (1 - \frac{\log n}{\Theta(w)})$. In other words, we show the criticality upper bounds $O(w)$ and $O(\log s)$ are tight up to a multiplicative $\log n$ factor, providing evidence on the tightness of the switching lemmas.