A general theory is developed for constructing the asymptotically shallowest networks and the asymptotically smallest networks (with respect to formula size) for the carry save addition of n numbers using any given basic carry save adder as a building block. Using these optimal carry save addition networks the shallowest known multiplication circuits and the shortest formulae for the majority function (and many other symmetric Boolean functions) are obtained. In this paper, simple basic carry save adders are described, using which multiplication circuits of depth 3:71 log n (the result of which is given as the sum of two numbers) and majority formulae of size O(n 3:21 ) are constructed. Using more complicated basic carry save adders, not described here, these results could be further improved. Our best bounds are currently 3:57 log n for depth and O(n 3:13 ) for formula size. 1. Introduction The question `How fast can we multiply?' is one of the fundamental questions...

Consider the following communication problem. Alice gets a word x 2 f0; 1g n and Bob gets a word y 2 f0; 1g n . Alice and Bob are told that x 6= y. Their goal is to find an index 1 i n such that x i 6= y i (the index i should be known to both of them). This problem is one of the most basic communication problems. It arises naturally from the correspondence between circuit depth and communication complexity discovered by Karchmer and Wigderson. We present three protocols using which Alice and Bob can solve the problem by exchanging at most n + 2 bits. One of this protocols is due to Rudich and Tardos. These protocols improve the previous upper bound of n + log n, obtained by Karchmer. We also show that any protocol for solving the problem must exchange, in the worst case, at least n+ 1 bits. This improves a simple lower bound of n \Gamma 1 obtained by Karchmer. Our protocols, therefore, are at most one bit away from optimality. The three n + 2 bit protocols use two completely d...

Abstract. In this paper we report preliminary results of experiments with finding efficient circuits (over binary bases) using SAT-solvers. We present upper bounds for functions with constant number of inputs as well as general upper bounds that were found automatically. We focus mainly on MOD-functions. Besides theoretical interest, these functions are also interesting from a practical point of view as they are the core of the residue number system. In particular, we present a circuit of size 3n + c over the full binary basis computing MOD n 3. 1