Lower bounds for circuits with MOD_m gates

• Authors: Arkadev Chattopadhyay;Navin Goyal;Pavel Pudlak;Denis Therien
• Venue: FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
• Year: 2006

Let CC_{o(n)} \left[ m \right] be the class of circuits that have size o(n) and in which all gates are MOD_m gates. --We show that CC[m] circuits cannot compute MOD_q in sub-linear size when m, q \ge 1 are co-prime integers. No non-trivial lower bounds were known before on the size of CC[m] circuits of constant depth for computing MOD_q. On the other hand, our results show circuits of type MAJ \circCC_{o(n)} \left[ m \right] need exponential size to compute MOD_q. Using Bourgain's recent breakthrough result on estimates of exponential sums, we extend our bound to the case where small fan-in AND gates are allowed at the bottom of such circuits i.e. circuits of type MAJ \circCC_{o(n)} \left[ m \right] \circ AND_{ \in \log n,}where \in 0 is a sufficiently small constant. --CC[m] circuits of constant depth need superlinear number of wires to compute both the AND and MOD_q functions. To prove this, we show that any circuit computing such functions has a certain connectivity property that is similar to that of superconcentration. We show a superlinear lower bound on the number of edges of such graphs extending results on superconcentrators.