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- W4287727082 abstract "One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $mathbf{P}notsubseteqmathbf{NC}^1$). 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 $fdiamond g$. They showed that the validity of this conjecture would imply that $mathbf{P}notsubseteqmathbf{NC}^1$. 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 $stextbf{-}t$-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 $f$." @default.
- W4287727082 created "2022-07-26" @default.
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- W4287727082 date "2020-07-06" @default.
- W4287727082 modified "2023-10-16" @default.
- W4287727082 title "KRW Composition Theorems via Lifting" @default.
- W4287727082 doi "https://doi.org/10.48550/arxiv.2007.02740" @default.
- W4287727082 hasPublicationYear "2020" @default.
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