Abstract
We prove that constant depth circuits of size nlogO(1)n over the basis AND, OR, PARITY are no more powerful than circuits of this size with depth four. Similar techniques are used to obtain several other depth reduction theorems; in particular, we show every set in AC0 can be recognized by a family of depth-three threshold circuits of size nlogO(1)n. The size bound nlogO(1)n is optimal when considering depth reduction over AND, OR, and PARITY. Most of our results hold for both the uniform and the nonuniform case.
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