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We consider boolean functions over n variables. Any such function can be represented (and computed) by a complete binary tree with and or or in the internal nodes and a literal in the external nodes, and many different trees can represent the same function, so that a fundamental question is related to the so-called complexity of a boolean function: L(f) :=(More)
A classical random walk (S t , t ∈ N) is defined by S t := t n=0 X n , where (X n) are i.i.d. When the increments (X n) n∈N are a one-order Markov chain, a short memory is introduced in the dynamics of (S t). This so-called " persistent " random walk is nolonger Markovian and, under suitable conditions, the rescaled process converges towards the integrated(More)
The purpose of this article is to show that the distribution of the longest fragment in the random multisection problem after k steps and the height of m-ary search trees (and some extensions) are not only closely related in a formal way but both can be asymptotically described with the same distribution function that has to be shifted in a proper way(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)
Infinite random sequences of letters can be viewed as stochastic chains or as strings produced by a source, in the sense of information theory. The relationship between Variable Length Markov Chains (VLMC) and probabilistic dynamical sources is studied. We establish a probabilistic frame for context trees and VLMC and we prove that any VLMC is a dynamical(More)
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)
Common assumptions on the source producing the words inserted in a suffix trie with n leaves lead to a ln n height and saturation level. We provide an example of a suffix trie whose height increases faster than a power of n and another one whose saturation level is negligible with respect to ln n. Both are built from VLMC (Variable Length Markov Chain)(More)