And/Or Trees Revisited

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(.) on the set of and/or trees was shown by Lefmann and Savicky [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 Savicky.

• Type: Journal
• Year: 2004
• Where: CPC
• Authors: Brigitte Chauvin, Philippe Flajolet, Danièle Gardy, Bernhard Gittenberger
• Keywords: Boolean Function, Complete Binary Tree, CPC 2004, External Nodes