Toward Better Formula Lower Bounds: The Composition of a Function and a Universal Relation

One of the major open problems in complexity theory is proving superlogarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving superpolynomial circuit lower bounds. Karchmer, Raz, and Wigderson [Comput. Complexity, 5 (1995), pp. 191--204] suggested approaching this problem by proving the following conjecture: given two Boolean functions $f$ and $g$, the depth complexity of the composed function $g\diamond f$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^1$. As a starting point for studying the composition of functions, they introduced a relation called “the universal relation” and suggested studying the composition of universal relations. Thi...