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 for 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) phenomena at kn=n/lnn.
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https://hal.sorbonne-universite.fr/hal-01329255
Contributor : Antoine Genitrini <>
Submitted on : Wednesday, June 8, 2016 - 9:56:06 PM
Last modification on : Friday, May 24, 2019 - 5:22:17 PM

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Antoine Genitrini, Cecile Mailler. Generalised and Quotient Models for Random And/Or Trees and Application to Satisfiability. Algorithmica, Springer Verlag, 2016, 1, pp.1-33. ⟨10.1007/s00453-016-0113-3⟩. ⟨hal-01329255⟩

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