And/Or Trees Revisited

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 (·) 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.

The fraction of large random trees representing a given Boolean function in implicational logic

“…For a fixed, (very) small number of Boolean variables, explicit computation of the limiting ratios is feasible by writing, then solving, an algebraic system; see [14] for an overview of the mathematical technology involved and [4] for the application to And/Or trees. However, the fact that size of the system grows exponentially in k severely restricts hand-made evaluation.…”

“…It is known that a tree is almost surely finite in this model [1]. This probability distribution has been introduced in [4] on And/Or trees and can be obviously adapted to the case of implication trees (here the labelling of the internal node is not at random because there is a single label). So for a tree A, we get: π k (A) = P(structure of A) · P(labelling of A) = 1 2 2|A|−1 k |A| .…”

Associative and commutative tree representations for Boolean functions

“…The limit of this distribution when m tends to infinity (cf. Definition 2.3), denoted by P n has already been studied, in particular by Lefmann and Savický [19], Chauvin et al [4] and Kozik [18], who has shown the following theorem.…”

“…First, Lefmann and Savický [19] established some bounds for the probability of a function, bounds that are linked to the complexity of the functions. These bounds have been improved by Chauvin et al [4] where other models based on Galton-Watson branching processes have been studied as well. Then Kozik [18] has developed a powerful tool based on pattern languages that allows to classify and count large trees according to some structural constraints.…”

The growing tree distribution on Boolean functions.

“…It gives a distribution over F k denoted by π k . It has been shown [CFGG04,GG10] that every Boolean function is weighted, but Boolean functions with lower complexity are more weighted by both µ k and π k .…”