Generalised and Quotient Models for Random And/Or Trees and Application to Satisfiability
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 kn 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 (kn)n≥ 1. The fundamental breakthrough of this paper is to generalise the previous results to any (reasonable) sequence of integers (kn)n≥ 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 kn = n/ln n.
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