Simultaneous Multiparty Communication Complexity of Composed Functions

The Number On the Forehead (NOF) model is a multiparty communication game between k players that collaboratively want to evaluate a given function F : X1 × · · · ×Xk → Y on some input (x1, . . . , xk) by broadcasting bits according to a predetermined protocol. The input is distributed in such a way that each player i sees all of it except xi (as if xi is written on the forehead of player i). In the Simultaneous Message Passing (SMP) model, the players have the extra condition that they cannot speak to each other, but instead send information to a referee. The referee does not know the players’ inputs, and cannot give any information back. At the end, the referee must be able to recover F (x1, . . . , xk) from what she obtained from the players. A central open question in the simultaneous NOF model, called the logn barrier, is to find a function which is hard to compute when the number of players is polylog(n) or more (where the xi’s have size poly(n)). This has an important application in circuit complexity, as it could help to separate ACC from other complexity classes [HG91, BGKL04]. One of the candidates for breaking the logn barrier belongs to the family of composed functions. The input to these functions in the k-party NOF model is represented by a k× (t ·n) boolean matrix M , whose row i is the number xi on the forehead of player i and t is a block-width parameter. A symmetric composed function acting on M is specified by two symmetric nand kt-variate functions f and g (respectively), that output f ◦ g(M) = f(g(B1), . . . , g(Bn)) where Bj is the j-th block of width t of M . As the majority function Maj is conjectured to be outside of ACC, Babai et. al. [BKL95, BGKL04] suggested to study the composed function Maj ◦Majt, with t large enough, for breaking the logn barrier (where Majt outputs 1 if at least kt/2 bits of the input block are set to 1). So far, it was only known that block-width t = 1 is not enough for Maj ◦Majt to break the logn barrier in the simultaneous NOF model [BGKL04] (Chattopadhyay and Saks [CS14] found an efficient protocol for t ≤ polyloglog(n), but it requires randomness to be simultaneous). In this paper, we extend this result to any constant block-width t > 1 by giving a deterministic simultaneous protocol of cost 2( t) log t+1 (n) for any symmetric composed function f ◦g (which includes Maj ◦ Majt) when there are more than 2 Ω(2) logn players.