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TR19-120 | 11th September 2019 17:37

Toward Better Depth Lower Bounds: Two Results on the Multiplexor Relation



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.

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