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., P ⊈ NC¹). 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 ⋄f is roughly the sum of the depth complexities of f and g. They showed that the validity of this conjecture would imply that P ⊈ NC₁. 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. This suggestion proved fruitful, and an analogue of the Karchmer-Raz-Wigderson (KRW) conjecture for the universal relation was proved by Edmonds et al. [Comput. Complexity, 10 (2001), pp. 210-246]. An alternative proof was given later by Håstad and Wigderson [in Advances in Computational Complexity Theory, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 13, AMS, Providence, RI, 1993, pp. 119-134]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still an open question. In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation.