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) 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 $\mathbf{P}\not\subseteq\mathbf{NC}^1$.

As a way to realize this program, Edmonds et. al. (Computational Complexity 10, 3) suggested to study the "multiplexor relation" $MUX$, which is a simplification of functions. In this note, we present two results regarding this relation:

- The multiplexor relation is "complete" for the approach of Karchmer et. al.

in the following sense: if we could prove (a variant of) their conjecture

for the composition $f \diamond MUX$ for every function $f$, then this would

imply $\mathbf{P}\not\subseteq\mathbf{NC}^1$.

- A simpler proof of a lower bound for the multiplexor relation due

to Edmonds et. al. Our proof has the additional benefit of fitting

better with the machinery used in previous works on the subject.

Some additional fixes following the submission to the journal. In particular, fixed an error in the proof of the Kovari-Sos-Turan theorem.

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) 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 $\mathbf{P}\not\subseteq\mathbf{NC}^1$.

As a way to realize this program, Edmonds et. al. (Computational Complexity 10, 3) suggested to study the "multiplexor relation" $MUX$, which is a simplification of functions. In this note, we present two results regarding this relation:

- The multiplexor relation is "complete" for the approach of Karchmer et. al.

in the following sense: if we could prove (a variant of) their conjecture

for the composition $f \diamond MUX$ for every function $f$, then this would

imply $\mathbf{P}\not\subseteq\mathbf{NC}^1$.

- A simpler proof of a lower bound for the multiplexor relation due

to Edmonds et. al. Our proof has the additional benefit of fitting

better with the machinery used in previous works on the subject.

Some minor fixes, and added a new "open problems" section.

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) 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 $\mathbf{P}\not\subseteq\mathbf{NC}^1$.

As a way to realize this program, Edmonds et. al. (Computational Complexity 10, 3) suggested to study the "multiplexor relation" $MUX$, which is a simplification of functions. In this note, we present two results regarding this relation:

- The multiplexor relation is "complete" for the approach of Karchmer et. al.

in the following sense: if we could prove (a variant of) their conjecture

for the composition $f \diamond MUX$ for every function $f$, then this would

imply $\mathbf{P}\not\subseteq\mathbf{NC}^1$.

- A simpler proof of a lower bound for the multiplexor relation due

to Edmonds et. al. Our proof has the additional benefit of fitting

better with the machinery used in previous works on the subject.