A theorem on probabilistic constant depth Computations

1. In t roduc t ion I{t~txml, ly there has been much interest ill tht, comput:Ltional power of circuits o1' bounded dt;pth. In particular Furst, Saxe and Sipser [FSS], and indcpentlently Ajtai [Aj] in a different form, have shown that no polynomial size circuits of bounded depth can compute the parity function of n boolean variables (and other related functions such as the majority function, the exact number of ones in the input, etc.). On the other hand, Stockmeyer [St] showed that probabilistic bounded depth circuits can approximate the exact number of ones in the input with very low probability of error. Tlmse results lead to the following interesting question: (*) Are probabilistic constant depth circuits more powerful than deterministic ones? While it is well known [BG] that (non uniform) deterministic polynomial size circuits are as powerful as probabilistic ones, the question is still open for bounded depth circuits. This is so because the reduction given by Bennet and Gill [BG]