And/Or Trees Revisited

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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.
Brigitte Chauvin, Philippe Flajolet, Danièl
Added 17 Dec 2010
Updated 17 Dec 2010
Type Journal
Year 2004
Where CPC
Authors Brigitte Chauvin, Philippe Flajolet, Danièle Gardy, Bernhard Gittenberger
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