One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $$\textbf{P} \not\subseteq \textbf{NC}^{1}$$ P ⊈ NC 1 ). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4):191–204, 1995) suggested to approach this problem by proving that depth complexity behaves ``as expected'' with respect to the composition of functions f ◊ g . They showed that the validity of this conjecture would imply that $$\textbf{P} \not\subseteq \textbf{NC}^{1}$$ P ⊈ NC 1 . As a way to realize this program, Edmonds et al. (Computational Complexity 10(3):210–246, 2001) suggested to study the ``multiplexor relation'' MUX. In this paper, we present two results regarding this relation: ○ The multiplexor relation is ``complete'' for the approach of Karchmer et al. in the following sense: if we could prove (a variant of) their conjecture for the composition f ◊ MUX for every function f , then this would imply $$\textbf{P} \not\subseteq \textbf{NC}^{1}$$ P ⊈ NC 1 . ○ A simpler proof of a lower bound for the multiplexor relation due to Edmonds et al. Our proof has the additional benefit of fitting better with the machinery used in previous works on the subject.

中文翻译：

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走向更好的深度下界：多路复用器关系的两个结果

复杂性理论中的主要开放问题之一是证明电路深度的超对数下界（即 $$\textbf{P} \not\subseteq \textbf{NC}^{1}$$ P ⊈ NC 1 ）。Karchmer、Raz 和 Wigderson（Computational Complexity 5(3/4):191–204, 1995）建议通过证明深度复杂性在函数 f ◊ g 的组合方面表现“如预期”来解决这个问题。他们表明，这个猜想的有效性意味着 $$\textbf{P} \not\subseteq \textbf{NC}^{1}$$ P ⊈ NC 1 。作为实现该程序的一种方式，Edmonds 等人。(Computational Complexity 10(3):210–246, 2001) 建议研究“多路复用器关系”MUX。在本文中，我们提出了关于这种关系的两个结果： ○ 对于 Karchmer 等人的方法，多路复用器关系是“完整的”。在以下意义上：如果我们可以证明他们对每个函数 f 的组合 f ◊ MUX 的猜想（的变体），那么这将意味着 $$\textbf{P} \not\subseteq \textbf{NC}^{1 }$$ P ⊈ NC 1 . ○ Edmonds 等人提供的多路复用器关系下界的更简单证明。我们的证明还有一个额外的好处，那就是更适合之前关于该主题的工作中使用的机器。