Variation ranks of communication matrices and lower bounds for depth two circuits having symmetric gates with unbounded fan-in

An exponential lower bound for depth two circuits with arbitrary symmetric gates in the bottom level and with a MOD/sub m/-gate in the top level is proved. This solves a problem posed by R. Smolensky (1990). The method uses the variation rank of communication matrices. A variant of this method is used for deriving lower bounds for the size of depth-two circuits having a threshold gate at the top.<
>