Toward the KRW composition conjecture: Cubic formula lower bounds via communication complexity

One of the major challenges of the research in circuit complexity is proving super-polynomial lower bounds for de-Morgan formulas. Karchmer, Raz, and Wigderson [20] suggested to approach this problem by proving that formula complexity behaves "as expected" with respect to the composition of functions f ⋄ g. They showed that this conjecture, if proved, would imply superpolynomial formula lower bounds. The first step toward proving the KRW conjecture was made by Edmonds et al. [10], who proved an analogue of the conjecture for the composition of "universal relations". In this work, we extend the argument of [10] further to f ⋄ g where f is an arbitrary function and g is the parity function. While this special case of the KRW conjecture was already proved implicitly in Håstad's work on random restrictions [14], our proof seems more likely to be generalizable to other cases of the conjecture. In particular, our proof uses an entirely different approach, based on communication complexity technique of Karchmer and Wigderson [21]. In addition, our proof gives a new structural result, which roughly says that the naive way for computing f ⋄ g is the only optimal way. Along the way, we obtain a new proof of the state-of-the-art formula lower bound of n3-o(1) due to [14].