Multiparty Communication Complexity and Threshold Circuit Size of AC^0

We prove an n^¿(-1)/4^k lower bound on the randomized k-party communication complexity of depth 4 AC⁰ functions in the number-on-forehead (NOF) model for up to ¿(log n) players. These are the first non-trivial lower bounds for general NOF multiparty communication complexity for any AC⁰ function for ¿ (log log n) players. For non-constant k the bounds are larger than all previous lower bounds for any AC⁰ function even for simultaneous communication complexity. Our lower bounds imply the first superpolynomial lower bounds for the simulation of AC⁰ by MAJ o SYMM o AND circuits, showing that the well-known quasipolynomial simulations of AC⁰ by such circuits are qualitatively optimal, even for formulas of small constant depth. We also exhibit a depth 5 formula in NP_k ^cc - BPP_k ^cc for k up to ¿(log n) and derive an ¿(2¿(log n/ ¿(k))) lower bound on the randomized k-party NOF communication complexity of set disjointness for up to ¿(log^1/3 n) players which is significantly larger than the O (log log n) players allowed in the best previous lower bounds for multiparty set disjointness. We prove other strong results for depth 3 and 4 AC⁰ functions.