BOOLEAN FUNCTIONS WITH A LARGE NUMBER OF SUBFUNCTIONS AND SMALL COMPLEXITY AND DEPTH

If f(x1,...,x(n)) is a Boolean function on the variables x1,...,x(n) then f(*1,...,*n) where *i is-an-element-of {0, 1, x(i)}, i = 1,...,n, is called subfunction of f. The number of subfunctions of f is at most 3n. Intuition suggests that a Boolean function with a large number of subfunctions has a large (combinatorial) complexity and a large depth. We show that this intuition is wrong. There exist Boolean functions with about 3n subfunctions (i. e. about the maximal number of subfunctions) and with a very small complexity and depth (about 2n and log2n, respectively).