Exploring the Average Values of Boolean Functions via Asymptotics and Experimentation

Abstract

In recent years, there has been a great interest in studying Boolean functions by studying their analogous Boolean trees (with internal nodes labeled by Boolean gates; leaves viewed as inputs to the Boolean function). Many of these investigations consider Boolean functions of n variables and m leaves. Our study is related but has a quite different flavor. We investigate the mean output Xn of a Boolean function defined by a complete Boolean tree of depth n. Each internal node of such a tree is labeled with a Boolean gate, via 2n – 1 IID fair coin flips. The value of the input at each leaf can be simply fixed at 1/2, so the randomness of Xn derives only from the selection of the gates at the internal nodes. For each n, there are 2(2n – 1) possible Boolean binary trees to consider, so we cannot expect to obtain a complete description of the probability distribution of Xn for large n. Therefore, we perform a twofold investigation of the Xn, using both asymptotics and experiments. We prove that, with probability 1, Xn → 0 or Xn → 1. Then we directly compute the asymptotics of the first four moments of Xn. Writing Zn = Xn(1 – Xn), we also prove that E(Zn) and E(Zn) are both Θ(1/n). Finally, we utilize C++ and a significant amount of computation and experimentation to obtain a more descriptive understanding of Xn for small values of n (say, n ≤ 100).