Discrepancy and the Power of Bottom Fan-in in Depth-three Circuits

We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty `Number on the Forehead' model. Our method is based on the notion of voting polynomial degree of functions and extends the Degree-Discrepancy Lemma in the recent work of Sherstov [24]. Using this we prove that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fan-in at the base layer i.e. circuits of type MAJ SYMM ANYO(1) cannot simulate the circuit class AC0 in sub-exponential size. Further, even if the fan-in of the bottom ANY gates are increased to o(log log n), such circuits cannot simulate AC0 in quasi-polynomial size. This is in contrast to the classical result of Yao and Beigel-Tarui that shows that such circuits, having only MAJORITY gates, can simulate the class ACC0 in quasi-polynomial size when the bottom fan-in is increased ...

• Type: Conference
• Year: 2007
• Where: FOCS
• Authors: Arkadev Chattopadhyay
• Keywords: Boolean Function, FOCS 2007, Quasi-polynomial Size, Sub-exponential Size, Theoretical Computer Science