Average-case deterministic query complexity of boolean functions with fixed weight

Abstract

We study the $\textit{average-case deterministic query complexity}$ of boolean functions under a $\textit{uniform input distribution}$, denoted by $\mathrm{D}_\mathrm{ave}(f)$, the minimum average depth of zero-error decision trees that compute a boolean function $f$. This measure has found several applications across diverse fields, yet its understanding is limited. We study boolean functions with fixed weight, where weight is defined as the number of inputs on which the output is $1$. We prove $\mathrm{D}_\mathrm{ave}(f) \le \max \left\{ \log \frac{\mathrm{wt}(f)}{\log n} + O(\log \log \frac{\mathrm{wt}(f)}{\log n}), O(1) \right\}$ for every $n$-variable boolean function $f$, where $\mathrm{wt}(f)$ denotes the weight. 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 fixed weight $\mathrm{wt}(f) = m(n)$. H\r{a}stad's switching lemma or Rossman's switching lemma [Comput. Complexity Conf. 137, 2019] implies $\mathrm{D}_\mathrm{ave}(f) \leq n\left(1 - \frac{1}{O(w)}\right)$ or $\mathrm{D}_\mathrm{ave}(f) \le n\left(1 - \frac{1}{O(\log s)}\right)$ for CNF/DNF formulas of width $w$ or size $s$, respectively. We show there exists a DNF formula of width $w$ and size $\lceil 2^w / w \rceil$ such that $\mathrm{D}_\mathrm{ave}(f) = n \left(1 - \frac{\log n}{\Theta(w)}\right)$ for any $w \ge 2\log n$.