Exploring the Average Values of Boolean Functions via Asymptotics and Experimentation

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 2 n − 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 th