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arXiv · 1507.08448

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

Antoine Genitrini·Cécile Mailler
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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.

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BibTeXRIS

Antoine Genitrini, Cécile Mailler. 2015-07-30. Generalised and Quotient Models for Random And/Or Trees and Application to Satisfiability. https://arxiv.org/abs/1507.08448

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