And/Or Trees Revisited

  • Authors:

  • B. Chauvin;P. Flajolet;D. Gardy;B. Gittenberger

  • Affiliations:

  • LAMA, CNRS UMR 8100, Université de Versailles Saint-Quentin, 78035 Versailles Cedex, France (e-mail: chauvin@math.uvsq.fr);INRIA Rocquencourt, Projet Algorithmes, Domaine de Voluceau, 78153 Le Chesnay, France (e-mail: Philippe.Flajolet@inria.fr);PRISM, CNRS UMR 8144, Université de Versailles Saint-Quentin, 78035 Versailles Cedex, France (e-mail: Daniele.Gardy@prism.uvsq.fr);Department of Geometry, Technische Universität Wien, Wiedner Hauptstraße 8-10/113, A-1040 Wien, Austria (e-mail: Bernhard.Gittenberger@tuwien.ac.at)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2004

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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.