One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^{1}$). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995) suggested to approach this problem by proving that depth complexity of a composition of functions $f\diamond g$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^{1}$. The intuition that underlies the KRW conjecture is that the composition $f\diamond g$ should behave like a "direct-sum problem", in a certain sense, and therefore the depth complexity of $f\diamond g$ should be the sum of the individual depth complexities. Nevertheless, there are two obstacles toward turning this intuition into a proof: first, we do not know how to prove that $f\diamond g$ must behave like a direct-sum problem; second, we do not know how to prove that the complexity of the latter direct-sum problem is indeed the sum of the individual complexities. In this work, we focus on the second obstacle. To this end, we study a notion called "strong composition", which is the same as $f\diamond g$ except that it is forced to behave like a direct-sum problem. We prove a variant of the KRW conjecture for strong composition, thus overcoming the above second obstacle. This result demonstrates that the first obstacle above is the crucial barrier toward resolving the KRW conjecture. Along the way, we develop some general techniques that might be of independent interest.

翻译：复杂度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 $\mathbf{P}\not\subseteq\mathbf{NC}^{1}$）。Karchmer、Raz 和 Wigderson (Computational Complexity 5(3/4), 1995) 提出通过证明函数组合 $f\diamond g$ 的深度复杂度大致等于 $f$ 和 $g$ 的深度复杂度之和来攻克该问题。他们指出，该猜想若成立，则可推出 $\mathbf{P}\not\subseteq\mathbf{NC}^{1}$。KRW 猜想背后的直觉是，组合 $f\diamond g$ 在某种意义上应表现为“直接和问题”，因此 $f\diamond g$ 的深度复杂度应为各自深度复杂度之和。然而，将这一直觉转化为证明面临两大障碍：首先，我们无法证明 $f\diamond g$ 必然表现出直接和问题的行为；其次，我们无法证明该直接和问题的复杂度确实是各自复杂度之和。本文聚焦于第二项障碍。为此，我们研究了一种称为“强组合”的概念，它与 $f\diamond g$ 相同，但被强制表现为直接和问题。我们证明了针对强组合的 KRW 猜想变体，从而克服了上述第二项障碍。这一结果表明，第一项障碍是解决 KRW 猜想的关键瓶颈。在此过程中，我们发展了一些可能具有独立价值的一般性技巧。