We prove that constant depth circuits of size 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 AC can be recognized by a family of depth three . The size bound n is optimal when considering depth reduction over AND, OR, and PARITY. Most of our results hold both for the uniform and the nonuniform case.

It is well-known which symmetric Boolean functions can be computed by constant depth, polynomial size, unbounded fan-in circuits, i.e. which are contained in the complexity class AC0. This result is sharpened. Symmetric Boolean functions in AC0 can be computed by unbounded fan-in circuits with the following properties. If the optimal depth of AC0-circuits is d, the depth is at most d +2, the number of wires is almost linear, namely n log O(1) n, and the number of gates is subpolynomial (but superpolylogarithmic), namely 2 O(log n) for some <1. Warning: Essentially this paper has been published in Information and Computation and is hence subject to copyright restrictions. It is for personal use only. Supported in part by DFG grants No. We 1066/2-1 and Me 872/1-2 1