Lower Bounds for Circuits with Few Modular and Symmetric Gates

We consider constant depth circuits augmented with few modular, or more generally, arbitrary symmetric gates. We prove that circuits augmented with o(log n) symmetric gates must have size n n) to compute a certain (complicated) function in ACC. This function is also hard on the average for circuits of size n logn augmented with o(log n) symmetric gates, and as a consequence we can get a pseudorandom generator for circuits of size m containing o( √ logm) symmetric gates. For a composite integer m having r distinct prime factors, we prove that circuits augmented with s MODm gates must have size n Ω( 1 s log 1 r−1 n) to compute MAJORITY or MODl, if l has a prime factor not dividing m. For proving the latter result we introduce a new notion of representation of boolean function by polynomials, for which we obtain degree lower bounds that are of independent interest.