On the Communication Complexity of Read-Once AC^0 Formulae

We study the 2-party randomized communication complexity of read-once AC⁰ formulae. For balanced AND-OR trees T with n inputs and depth d, we show that the communication complexity of the function f^T(x, y) = T(x omicron y) is Omega(n/4^d) where (x omicron y)_i is defined so that the resulting tree also has alternating levels of AND and OR gates. For each bit of x, y, the operation omicron is either AND or OR depending on the gate in T to which it is an input. Using this, we show that for general AND-OR trees T with n inputs and depth d, the communication complexity of f^T(x, y) is n/2^{Omega(d log d)}. These results generalize classical results on the communication complexity of set-disjointness (where T is an OR -gate) and recent results on the communication complexity of the TRIBES functions (where T is a depth-2 read-once formula). Our techniques build on and extend the information complexity methodology for proving lower bounds on randomized communication complexity. Our analysis for trees of depth d proceeds in two steps: (1) reduction to measuring the information complexity of binary depth-d trees, and (2) proving lower bounds on the information complexity of binary trees. In order to execute this program, we carefully construct input distributions under which both these steps can be carried out simultaneously. We believe the tools we develop will prove useful in further studies of information complexity in particular, and communication complexity in general.