Equivalence classes of random boolean trees and application to the catalan satisfiability problem

Abstract

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. © 2014 Springer-Verlag Berlin Heidelberg.