A Framework for Certified Self-Stabilization

We propose a general framework to build certified proofs of distributed self-stabilizing algorithms with the proof assistant Coq. We first define in Coq the locally shared memory model with composite atomicity, the most commonly used model in the self-stabilizing area. We then validate our framework by certifying a non trivial part of an existing silent self-stabilizing algorithm which builds a $k$-hop dominating set of the network. We also certified a quantitative property related to the output of this algorithm. Precisely, we show that the computed $k$-hop dominating set contains at most $\lfloor \frac{n-1}{k+1} \rfloor + 1$ nodes, where $n$ is the number of nodes in the network. To obtain these results, we also developed a library which contains general tools related to potential functions and cardinality of sets

NF-κB: A lesson in family values

A set of mobile robots (represented as points) is distributed in the Cartesian plane. The collection contains an unknown subset of byzantine robots which are indistinguishable from the reliable ones. The reliable robots need to gather, i.e., arrive to a configuration in which at the same time, all of them occupy the same point on the plane. The robots are equipped with GPS devices and at the beginning of the gathering process they communicate the Cartesian coordinates of their respective positions to the central authority. On the basis of this information, without the knowledge of which robots are faulty, the central authority designs a trajectory for every robot. The central authority aims to provide the trajectories which result in the shortest possible gathering time of the healthy robots. The efficiency of a gathering strategy is measured by its competitive ratio, i.e., the maximal ratio between the time required for gathering achieved by the given trajectories and the optimal time required for gathering in the offline case, i.e., when the faulty robots are known to the central authority in advance. The role of the byzantine robots, controlled by the adversary, is to act so that the gathering is delayed and the resulting competitive ratio is maximized. The objective of our paper is to propose efficient algorithms when the central authority is aware of an upper bound on the number of byzantine robots. We give optimal algorithms for collections of robots known to contain at most one faulty robot. When the proportion of byzantine robots is known to be less than one half or one third, we provide algorithms with small constant competitive ratios. We also propose algorithms with bounded competitive ratio in the case where the proportion of faulty robots is arbitrary

Lettre à la rédaction

L'étude RMN de l'isotope 129 du xénon dans un mélange de cristaux liquides nématiques de faible Δχ montre que cet atome est sensible à la transition de phase Δχ ˂ 0 → Δχ ˃ 0. Cependant l'environnement de l'atome de xénon à la transition n'est pas affecté

Distribution du directeur dans une mésophase lyotrope en rotation rapide dans un champ magnétique

Le comportement des mésophases lyotropes nématiques de Δχ < 0 en rotation rapide perpendiculairement au champ magnétique est décrit. D'une manière analogue aux mélanges de Δχ proche de zéro, le directeur, parallèle à l'axe de rotation pour des vitesses faibles, bascule dans une direction perpendiculaire à l'axe de rotation pour des vitesses élevées. Cependant la dynamique du processus est bien plus lente que dans les mélanges de nématiques.Ce basculement intervient lorsque la force d'inertie rotationnelle prend le pas sur la force magnétique.Le calcul des spectres expérimentaux permet de revenir à la largeur de la fonction de distribution du directeur. Cette largeur augmente avec la vitesse de rotation pour la fonction de distribution centrée sur l'axe de rotation, alors que pour la fonction de distribution centrée sur l’axe perpendiculaire sa largeur diminue avec la vitesse