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

Abstract

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\textbf{P} \not\subseteq \textbf{NC}^{1}$). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4):191–204, 1995) 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 $\textbf{P} \not\subseteq \textbf{NC}^{1}$.

As a way to realize this program, Edmonds et al. (Computational Complexity 10(3):210–246, 2001) suggested to study the ``multiplexor relation'' MUX. In this paper, 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 ◊ MUX for every function f, then this would imply $\textbf{P} \not\subseteq \textbf{NC}^{1}$.

• ○ 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.