Discrepancy and the Power of Bottom Fan-in in Depth-three Circuits

We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty ´number on the forehead´ model. Our method is based on the notion of voting polynomial degree of functions and extends the degree-discrepancy lemma in the recent work of Sherstov (2007). Using this we prove that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fan-in at the base layer i.e. circuits of type MAJ o SYMM o ANY_O(1) cannot simulate the circuit class AC⁰ in sub-exponential size. Further, even if the fan-in of the bottom ANY gates are increased to o(log log n), such circuits cannot simulate AC⁰ in quasi-polynomial size. This is in contrast to the classical result of Yao and Beigel-Tarui that shows that such circuits, having only MAJORITY gales, can simulate the class ACC⁰ in quasi-polynomial size when the bottom fan-in is increased to poly-logarithmic size. In the second part, we simplify the arguments in the breakthrough work of Bourgain (2005) for obtaining exponentially small upper bounds on the correlation between the boolean function MOD_q and functions represented bv polynomials of small degree over Z_m, when m,q ges 2 are co-prime integers. Our calculation also shows similarity with techniques used to estimate discrepancy of functions in the multiparty communication setting. This results in a slight improvement of the estimates of Bourgain et al. (2005). It is known that such estimates imply that circuits of type MAJ o MOD_m o AND_{isin log n} cannot compute the MOD_q function in sub-exponential size. It remains a major open question to determine if such circuits can simulate ACC⁰ in polynomial size when the bottom fan-in is increased to poly-logarithmic size.