通过提升的KRW组合定理
KRW Composition Theorems via Lifting
AI总结：
本文通过引入半单调组合概念并利用查询到通信的提升定理，显著扩展了KRW猜想中可处理的内部函数范围，包括s-t连通性、团和生成函数等，从而推进了证明电路深度超对数下界的研究。
摘要：
复杂性理论中一个主要的未解决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（Computational Complexity 5(3/4), 1995）建议通过证明深度复杂性在函数组合 $f\diamond g$ 方面表现得“如预期”来接近这一问题。他们表明，该猜想的成立将意味着 $\mathbf{P}\not\subseteq\mathbf{NC}^1$。\n\n若干工作通过证明特殊情况在解决这一猜想方面取得了进展。特别是，这些工作证明了每个外部函数 $f$ 的KRW猜想，但仅对少数内部函数 $g$ 成立。因此，证明更广泛内部函数的KRW猜想是一个重要挑战。\n\n在这项工作中，我们显著扩展了可处理的内部函数范围。首先，我们考虑KRW猜想的单调版本。我们证明它对每个单调内部函数 $g$ 成立，只要其深度复杂性可以通过查询到通信的提升定理下界化。这使我们能够处理若干新的且被广泛研究的函数，例如 $s\textbf{-}t$-连通性、团和生成函数。\n\n为了将这一进展带回非单调情形，我们引入了一种新的半单调组合概念，它将外部函数 $f$ 的非单调复杂性与内部函数 $g$ 的单调复杂性结合起来。在这一设定下，我们针对类似的内部函数 $g$ 选择证明了KRW猜想，但仅针对外部函数 $f$ 的特定选择。
英文摘要：
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 behaves "as expected" with respect to the composition of functions $f\diamond g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^1$. Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function $f$, but only for few inner functions $g$. Thus, it is an important challenge to prove the KRW conjecture for a wider range of inner functions. In this work, we extend significantly the range of inner functions that can be handled. First, we consider the $\textit{monotone}$ version of the KRW conjecture. We prove it for every monotone inner function $g$ whose depth complexity can be lower bounded via a query-to-communication lifting theorem. This allows us to handle several new and well-studied functions such as the $s\textbf{-}t$-connectivity, clique, and generation functions. In order to carry this progress back to the $\textit{non-monotone}$ setting, we introduce a new notion of $\textit{semi-monotone}$ composition, which combines the non-monotone complexity of the outer function $f$ with the monotone complexity of the inner function $g$. In this setting, we prove the KRW conjecture for a similar selection of inner functions $g$, but only for a specific choice of the outer function $f$.
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