  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 $f$.
