Functions with Bounded Symmetric Communication Complexity, Programs over Commutative Monoids, and ACC

Abstract We answer the question: What are the Boolean functions that can be computed with a constant number of bit-exchange in a two-processor environment no matter how the input bits are distributed among the processors? The characterization uses “programs over a monoid M ,” a construction introduced by D. Barrington. We prove that if the symmetric communication complexity of a Boolean function f is at most c (i.e., the communication complexity is at most c for all possible partitions of the input into two parts) then there is a commutative monoid M of size at most exp(exp(exp(exp(exp c )))) such that a program over the monoid M can be built that computes f . We also give size and depth upper bounds for synchronous circuits that compute functions with bounded symmetric communication complexity, as well as width upper bounds for read-only once branching programs that compute these functions.