KRW Composition Theorems via Lifting

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Abstract
One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., P\nsubseteq NC 1 ). Karchmer, Raz, and Wigderson [13] 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 P\nsubseteq 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, but only for few inner functions. 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 monotone version of the KRW conjecture. We prove it for every monotone inner function 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-t-connectivity, clique, and generation functions. In order to carry this progress back to the non-monotone setting, we introduce a new notion of semi-monotone composition, which combines the non-monotone complexity of the outer function with the monotone complexity of the inner function. In this setting, we prove the KRW conjecture for a similar selection of inner functions, but only for a specific choice of the outer function f.
Funding Information
  • European Research Council (279611)
  • Knut and Alice Wallenberg (KAW 2016.0066,KAW 2018.0371,KAW 2016.006)
  • Israel Science Foundation (1445/16)
  • Swedish Research Council (2016-00782)
  • Independent Research Fund Denmark (9040-00389B)
  • NSERC
  • National Science Foundation (CCF-1900460)

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