Computing majority with low-fan-in majority queries

In this paper we examine the problem of computing majority functionMAJn on n bits by depthtwo formula, where each gate is a majority function on at most k inputs. We present such formula that gives the first nontrivial upper bound for this problem, with k = 2 3n + 4. This answers an open question in [7]. We also look at this problem in adaptive setting — when we are allowed to query for value of MAJk on any subset, and wish to minimize the number of such queries. We give a simple lower bound for this setting with dn/ke queries, and we present two algorithms for this model: the first one makes ≈ 2 nk log k queries in the case when we are limited to the standard majority functions, and the second one makes nk log k queries when we are allowed to change the threshold of majority function. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes