Equivalence Classes of Random Boolean Trees and Application to the Catalan Satisfiability Problem

  • Antoine Genitrini
  • Cécile Mailler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)


An and/or tree is a binary plane tree, with internal nodes labelled by connectives, and with leaves labelled by literals chosen in a fixed set of k variables and their negations. We introduce the first model of such Catalan trees, whose number of variables k n is a function of n, its number of leaves. We describe the whole range of the probability distributions depending on the functions k n , as soon as it tends jointly with n to infinity. As a by-product we obtain a study of the satisfiability problem in the context of Catalan trees.

Our study is mainly based on analytic combinatorics and extends the Kozik’s pattern theory, first developed for the fixed-k Catalan tree model.


Random Boolean expressions Boolean formulas Boolean function Probability distribution Satisfiability Analytic combinatorics 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Antoine Genitrini
    • 1
    • 2
  • Cécile Mailler
    • 3
  1. 1.UMR 7606, LIP6Sorbonne Universités, UPMC Univ Paris 06ParisFrance
  2. 2.UMR 7606, LIP6CNRSParisFrance
  3. 3.Laboratoire de Mathématiques de Versailles; CNRS UMR 8100Université de Versailles Saint-Quentin-en-YvelinesVersaillesFrance

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