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Consider a balanced non triangular two-color Pólya-Eggenberger urn process, assumed to be large which means that the ratio σ of the replacement matrix eigenvalues satisfies 1/2 < σ < 1. The composition vector of both discrete time and continuous time models admits a drift which is carried by the principal direction of the replacement matrix. In the second… (More)

- Antoine Genitrini, Bernhard Gittenberger, Veronika Kraus, Cécile Mailler
- Electr. J. Comb.
- 2012

We study the asymptotic relation between the probability and the complexity of Boolean functions in the implicational fragment which are generated by large random Boolean expressions involving variables and implication, as the number of variables tends to infinity. In contrast to models studied in the literature so far, we consider two expressions to be… (More)

- Antoine Genitrini, Bernhard Gittenberger, Veronika Kraus, Cécile Mailler
- Theor. Comput. Sci.
- 2015

Since the 90’s, several authors have studied a probability distribution on the set of Boolean functions on n variables induced by some probability distributions on formulas built upon the connectors And and Or and the literals {x1, x̄1, . . . , xn, x̄n}. These formulas rely on plane binary labelled trees, known as Catalan trees. We extend all the results,… (More)

We define a probability distribution over the set of Boolean functions of k variables induced by the tree representation of Boolean expressions. The law we are interested in is inspired by the growth model of Binary Search Trees: we call it the growing tree law. We study it over different logical systems and compare the results we obtain to already known… (More)

Quantitative logic has been the subject of an increasing interest since a seminal paper by Chauvin et al. in 2004, which presented the first Analytic Combinatorics approach of the subject. Since then, the understanding of random Boolean trees has been deeply widened, even if the question of Shannon effect remains open for the majority of the models. We… (More)

- Antoine Genitrini, Cécile Mailler
- LATIN
- 2014

- Brigitte Chauvin, Danièle Gardy, Cécile Mailler
- Random Struct. Algorithms
- 2015

We define a new probability distribution for Boolean functions of k variables. Consider the random Binary Search Tree of size n, and label its internal nodes by connectives and its leaves by variables or their negations. This random process defines a random Boolean expression which represents a random Boolean function. Finally, let n tend to infinity: the… (More)

- Antoine Genitrini, Cécile Mailler
- Algorithmica
- 2016

This article is motivated by the following satisfiability question: pick uniformly at random an $$\mathsf {and/or}$$ and / or Boolean expression of length n, built on a set of $$k_n$$ 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… (More)

We consider a probability distribution on the set of Boolean functions in n variables which is induced by random Boolean expressions. Such an expression is a random rooted plane tree where the internal vertices are labelled with connectives And and OR and the leaves are labelled with variables or negated variables. We study limiting distribution when the… (More)

- Brigitte Chauvin, Danièle Gardy, Cécile Mailler
- ANALCO
- 2011

We define a probability distribution over the set of Boolean functions of k variables induced by the tree representation of Boolean expressions. The law we are interested in is inspired by the growth model of Binary Search Trees: we call it the growing trees law. We study it over different logical systems and compare the results we obtain to already known… (More)