Abstract
Proving super-logarithmic lower bounds on the depth of circuits is one of the main frontiers of circuit complexity. In 1991, Karchmer, Raz and Wigderson observed that we could resolve this question by proving the following conjecture: Given two boolean functions f,g, the depth complexity of their composition is about the sum of their individual depth complexities. While we cannot prove the conjecture yet, there has been some exciting progress toward such a proof, some of it in the last few years.
In this talk, I will survey the known results and propose future directions for research on the KRW conjecture. In particular, I will present some open problems which I believe are both tractable and important toward making further progress on the conjecture.