Multiparty Communication Complexity and Threshold Circuit Size of Ac

We prove an nΩ(1)/4k lower bound on the randomized k-party communicationcomplexity of depth 4 AC0 functions in the number-on-forehead (NOF) model for up to Θ(logn)players. These are the first non-trivial lower bounds for general NOF multiparty communicationcomplexity for any AC0 function for ω(log logn) players. For non-constant k the bounds are largerthan 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 byMAJ◦SYM◦AND circuits, showing that the well-known quasipolynomial simulations of AC0 by suchcircuits due to Allender (1989) and Yao (1990) are qualitatively optimal, even for formulas of smallconstant depth. We also exhibit a depth 5 formula in NPcck −BPP cck for k up to Θ(logn) and derive Ω(2√logn/√k)lower bound on the randomized k-party NOF communication complexity of set disjointness for upto Θ(log n) players which is significantly larger than the O(log logn) players allowed in the bestprevious lower bounds for multiparty set disjointness. We prove other strong results for depth 3 and4 AC0 functions.