Catalan satisfiability problem

TL;DR

This paper introduces a new model for Catalan trees with variable numbers of variables, analyzing the probability distributions and satisfiability problem as the size grows, extending existing combinatorial theories.

Contribution

It presents the first model of Catalan trees with variable variable counts and analyzes the satisfiability problem within this framework, extending Kozik's pattern theory.

Findings

Describes probability distributions depending on variable count functions k_n

Provides a comprehensive analysis of satisfiability in Catalan trees

Extends Kozik's pattern theory to variable-k models

Abstract

An and/or tree is usually a binary plane tree, with internal nodes labelled by logical connectives, and with leaves labelled by literals chosen in a fixed set of k variables and their negations. In the present paper, we introduce the first model of such Catalan trees, whose number of variables k_n is a function of n, the size of the expressions. We describe the whole range of the probability distributions depending on the function 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.