On the Power of Circuits with Gates of Low L1 Norms

We examine the power of Boolean functions with low L 1 norms in several settings. In large part of the recent literature, the degree of a polynomial which represents a Boolean function in some way was chosen to be the measure of the complexity of the Boolean function (see, e. have high degree, but small L 1 norms. So, in conjunction with communication complexity, instead of the degree, the L 1 norm can be an important measure of hardness. We conjecture that the randomized communication complexity of any Boolean function is bounded by the polylogarithm of its L 1 norm. We can prove only a weaker statement: we present a two-party, randomized, common-coin communication protocol for computing functions with O(L 2 1) bits of communication, with error-probability of exp(?cc), (even with large degree or exponential number of terms). Then we present several applications of this theorem for circuit lower bounds (both for bounded-and unbounded depth), and a decision-tree lower bound.