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

One of the major open problems in complexity theory is proving super-logarithmiclower bounds on the depth of circuits (i.e., P⊈ NC¹).Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4):191–204, 1995)suggested to approach this problem by proving that depth complexitybehaves ``as expected'' with respect to the composition of functions f ◊ g.They showed that the validity of this conjecture would imply that P⊈ NC<sup>1</sup>. As a way to realize this program, Edmonds et al. (Computational Complexity10(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 Karchmeret al. in the following sense: if we could prove (a variant of) theirconjecture for the composition f ◊ MUX for every function f, thenthis would imply P⊈ NC¹.○A simpler proof of a lower bound for the multiplexor relation dueto Edmonds et al. Our proof has the additional benefit of fittingbetter with the machinery used in previous works on the subject.