Generalised and Quotient Models for Random And/Or Trees and Application to Satisfiability

TL;DR

This paper generalizes models of random Boolean expressions to variable-sized sets, analyzes their satisfiability probability asymptotically, and uncovers a threshold phenomenon related to expression complexity and variable count.

Contribution

It extends previous Boolean Catalan tree models to arbitrary variable sequences, enabling new satisfiability probability results and revealing a threshold at k_n = n/ln n.

Findings

Asymptotic satisfiability probability for general variable sequences

Introduction of a quotient model with a threshold phenomenon

Identification of a saturation point at k_n = n/ln n

Abstract

This article is motivated by the following satisfiability question: pick uniformly at random an and/or Boolean expression of length n, built on a set of k_n Boolean variables. What is the probability that this expression is satisfiable? asymptotically when n tends to infinity? The model of random Boolean expressions developed in the present paper is the model of Boolean Catalan trees, already extensively studied in the literature for a constant sequence (k_n)_{n\geq 1}. The fundamental breakthrough of this paper is to generalise the previous results to any (reasonable) sequence of integers (k_n)_{n\geq 1}, which enables us, in particular, to solve the above satisfiability question. We also analyse the effect of introducing a natural equivalence relation on the set of Boolean expressions. This new "quotient" model happens to exhibit a very interesting threshold (or saturation) phenomenon at k_n = n/ln n.