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One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., P 6⊆ NC1). Karchmer, Raz, and Wigderson [KRW95] 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 6⊆ 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 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-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 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. 

We show that algebraic proofs are hard to find: Given an unsatisfiable CNF formula F, it is NP-hard to find a refutation of F in the Nullstellensatz, Polynomial Calculus, or Sherali--Adams proof systems in time polynomial in the size of the shortest such refutation. Our work extends, and gives a simplified proof of, the recent breakthrough of Atserias and Muller (FOCS 2019) that established an analogous result for Resolution.