Specifically I'm thinking about NC$$^1$$/poly and NC$$^1$$/rpoly (randomized advice). Are there any statements like
"If $$\{C_n\}$$ is a family of NC$$^1$$/(r)poly circuits with depth $$C\log n$$, then there exists a family $$\{\tilde{C}_n\}$$ of NC$$^1$$/(r)poly circuits with depth $$\tilde{C}\log n$$ such that $$\tilde{C} < 1$$ and $$C_n(x) = \tilde{C}_n(x)$$"?
• As far as I can tell, with usual advice, i.e., non-uniform $\mathrm{NC}^1$ circuits, the statement is as false as possible: for any $d<n-\log n$ or so (such as $C\log n$), there exist functions computable by circuits of depth $d+1$, but not by circuits of depth $d$. Proof: if not, then all Boolean functions are computable by circuits of depth $d$, and therefore size $2^d$, as you can just take an arbitrary circuit, and keep reducing subcircuits of depth $d+1$ to $d$ until none remain. However, almost all functions require circuit size $\Omega(2^n/n)$. May 4, 2022 at 7:04
• Of course, with $\tilde C<1$ as written, it’s much easier to disprove: a circuit of depth $\tilde C\log n$ with $\tilde C<1$ has size at most $n^{\tilde C}<n$, hence it cannot even depend on all bits of the input. Thus, even simple functions such as $\mathrm{AND}(x_1,\dots,x_n)$ have no circuits of depth $\tilde C\log n$. May 4, 2022 at 9:03