Equivalence Classes of Random Boolean Trees and Application to the Catalan Satisfiability Problem
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.
KeywordsRandom Boolean expressions Boolean formulas Boolean function Probability distribution Satisfiability Analytic combinatorics
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