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：复杂性理论中的一个主要开放问题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（《计算复杂性》5(3/4), 1995）建议通过证明函数组合 $f\diamond g$ 的深度复杂度大致等于 $f$ 与 $g$ 深度复杂度之和来解决此问题。他们指出，该猜想的有效性将蕴含 $\mathbf{P}\not\subseteq\mathbf{NC}^{1}$。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 $f\diamond g$ 的深度复杂度应是各独立深度复杂度之和。然而，将这一直觉转化为证明存在两大障碍：首先，我们无法证明 $f\diamond g$ 必然表现为直接和问题；其次，我们无法证明后者的直接和问题的复杂度确实是各独立复杂度之和。本文聚焦于第二个障碍。为此，我们研究了一种称为“强组合”的概念，它与 $f\diamond g$ 完全相同，但强制其表现为直接和问题。我们证明了强组合的 KRW 猜想变体，从而克服了上述第二个障碍。这一结果表明，第一个障碍是解决 KRW 猜想的关键瓶颈。在过程中，我们发展了一些可能具有独立价值的通用技术。