And/Or Trees Revisited

Source abstract

We consider Boolean functions over $n$ variables. Any such function can be represented (and computed) by a complete binary tree with and or or in the internal nodes and a literal in the external nodes, and many different trees can represent the same function, so that a fundamental question is related to the so-called complexity of a Boolean function: $L(f):=$ minimal size of a tree computing $f$ . The existence of a limiting probability distribution $P(\cdot)$ on the set of and/or trees was shown by Lefmann and Savický [8]. We give here an alternative proof, which leads to effective computation in simple cases. We also consider the relationship between the probability $P(f)$ and the complexity $L(f)$ of a Boolean function $f$ . A detailed analysis of the functions enumerating some sub-families of trees, and of their radius of convergence, allows us to improve on the upper bound of $P(f)$ , established by Lefmann and Savický.