Boolean functions with a large number of subfunctions and small complexity and depth

Abstract

If f(x₁,...,x_n) is a Boolean function on the variables x₁,...,x_n then f(*₁,...,*_n) where *_i ∈ {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).