发表机构
University of Colorado Boulder; Santa Fe Institute; Duke University（科罗拉多大学博尔德分校; 圣塔菲研究所; 杜克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出自顶向下的切割博弈模型，结合多项式近似方法得到AC⁰电路的经典下界，研究其k-局部变体并证明仿射策略下的下界，还提出相关图论类比猜想。
AI 中文摘要
AC⁰电路的经典下界证明采用自底向上的方法，从输入层开始对门电路进行简化或近似。我们引入一种互补的自顶向下模型，称为“切割博弈”，由对手Spoiler（破坏者）和Duplicator（复制者）在布尔函数的0输入集与1输入集上进行博弈。每一轮中，Spoiler保留其中一侧至少1/m的比例，Duplicator则任意限制另一侧；Spoiler的目标是最小化轮数，Duplicator的目标是最大化轮数，直到某个坐标能分离剩余的两个集合。每个深度为d、扇入为m的电路都为Spoiler诱导出d轮获胜策略，而能在d轮中存活的Duplicator策略则形式化了自顶向下的下界论证。通过切割博弈并结合多项式近似方法，我们首先以自顶向下的方式得到了深度为d的AC⁰电路的经典下界。随后我们考虑了切割博弈的k-局部变体，该变体放松了Spoiler的获胜条件，要求在每个半径为k的汉明球内存在一个分离坐标，而非全局的单个坐标。我们提出一个猜想：在d≪k≪n的范围内，PARITY（奇偶函数）的d轮k-局部切割博弈需要m = n^ω(1)。当Spoiler被限制在所谓的仿射策略（一类能达到已知最优上界的策略）时，我们证明了这样的下界m ≥ n^Ω(k^(1/d)/d)。最后，我们在围长>2k的n-正则图上构建了k-局部切割博弈的版本，并提出了“PARITY ∉ AC⁰”的图论类比猜想。
英文摘要
Classical lower bounds for $\mathrm{AC^0}$ circuits proceed bottom-up by simplifying or approximating gates beginning at the input layer. We introduce a complementary top-down model called the Chopping Game, played by adversaries Spoiler and Duplicator on the sets of $0$- and $1$-inputs of a Boolean function. In each round, Spoiler keeps at least a $1/m$-fraction of one side, and Duplicator arbitrarily restricts the other; Spoiler seeks to minimize (and Duplicator to maximize) the number of rounds until some coordinate separates the two remaining sets. Every depth-$d$, fan-in-$m$ circuit induces a $d$-round winning strategy for Spoiler, while Duplicator strategies that survive $d$ rounds formalize top-down lower-bound arguments. Through the Chopping Game and using the polynomial-approximation method, we first obtain the classical lower bound for depth-$d$ $\mathrm{AC^0}$ circuits in a top-down fashion. We then consider a $k$-local variant of the Chopping Game, which relaxes Spoiler's win condition by requiring a separating coordinate within each Hamming ball of radius $k$, rather than a single coordinate globally. We put forward a conjecture that the $d$-round $k$-local Chopping Game for $\mathrm{PARITY}$ requires $m = n^{ω(1)}$ in the regime $d \ll k \ll n$. We prove such a lower bound $m \ge n^{Ω(k^{1/d}/d)}$ when Spoiler is restricted to so-called affine strategies, a class of strategies that achieves the best known upper bounds. Finally, we formulate a version of the $k$-local Chopping Game on $n$-regular graphs of girth $>2k$, and we conjecture a graph-theoretic analogue of ``$\mathrm{PARITY} \notin \mathrm{AC^0}$''.
Comments22 pages
