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

Formal Foundations of 3D Geometry to Model Robot Manipulators

International audienceWe are interested in the formal specification of safety properties of robot manipulators down to the mathematical physics. To this end, we have been developing a formalization of the mathematics of rigid body transformations in the COQ proof-assistant. It can be used to address the forward kinematics problem, i.e., the computation of the position and orientation of the end-effector of a robot manipulator in terms of the link and joint parameters. Our formalization starts by extending the Mathematical Components library with a new theory for angles and by developing three-dimensional geometry. We use these theories to formalize the foundations of robotics. First, we formalize a comprehensive theory of three-dimensional rotations, including exponentials of skew-symmetric matrices and quaternions. Then, we provide a formalization of the various representations of rigid body transformations: isometries, homogeneous representation, the Denavit-Hartenberg convention, and screw motions. These ingredients make it possible to formalize robot manipulators: we illustrate this aspect by an application to the SCARA robot manipulator

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

Étude dynamique des processus de gélification dans des matériaux hybrides

Les processus de gélification d'un matériau hybride sont étudiés par plusieurs techniques. Ces différentes mesures sont nécessaires pour décrire la structure et le comportement dynamique de ces matériaux complexes. Cette approche permet de préciser à la fois les mouvements moléculaires et macromoléculaires. La spectroscopie de photo-corrélation permet d'observer la formation du squelette du gel. La diffusion Rayleigh Forcée permet d'appréhender la microstructure de ces matériaux. L'utilisation de la RMN permet de mesurerles influences relatives des phases 'liquide' et 'solide'. De très faibles changements sont observés de part et d'autre du point de gel, alors que l'évolution amène à des changements considérables au bout d'un long délai

Dynamique du directeur des nématiques en rotation dans un champ magnétique. Comportement orientationnel au voisinage de l'angle magique

L'étude théorique de l'intensité des bandes de rotation obtenues en RMN lors de la rotation macroscopique des cristaux liquides nématiques est décrite. Les expressions théoriques ont été évaluées par une méthode numérique à partir des données expérimentales : couplage dipolaire, anisotropie de déplacement chimique et vitesse de rotation. Ces expressions permettent par comparaison avec l'expérience de revenir à la dynamique de réorientation du directeur du cristal liquide.Dans le cas d'un cristal liquide de Δχ< 0 (Merk ZLI 1167), entre 0° et 52° le directeur est perpendiculaire à l'axe de rotation, autour de l'angle magique celui-ci se distribue dans un cône, alors qu'à l'angle magique il est distribué isotropiquement. Entre 56° et 90°, le directeur est orienté parallèlement à l'axe de rotation

Automated Certified Proofs with CiME3

We present the rewriting toolkit CiME3. Amongst other original features, this version enjoys two kinds of engines: to handle and discover proofs of various properties of rewriting systems, and to generate COQ scripts from proofs traces input in certification proof format (CPF) (an XML format widely accepted by the certified rewriting community) in order to certify them with a skeptical proof assistant like COQ. CiME3 may thus be used to add automation to proofs of termination or confluence in a formal development in the COQ proof assistant

Proton-decoupled carbon-13 NMR spectroscopy in a lyotropic chiral nematic solvent as an analytical tool for the measurement of the enantiomeric excess

International audienceOrganic solutions of poly-ç-(benzyl-L-glutamate) (PBLG) generate a sufficient differential ordering effect (DOE) to discriminate enantiomers using proton decoupled carbon-13 NMR in natural abundance. Discrimination between enantiomers is observed through the carbon-13 chemical shift anisotropy (CSA) differences. This method is successfully applied to a large number of chiral molecules including a case of axial chirality and offers the advantage that no labeling or chemical modification of molecules is needed. In most cases, the chemical shift differences are large enough to measure the enantiomeric excess with accuracy. We show that this new tool is an attractive and powerful alternative to the existing enantiomeric analytical techniques

Enantiomeric and enantiotopic analysis of cone-shaped compounds with C(3) and C(3)(v) symmetry using NMR spectroscopy in chiral anisotropic solvents

We describe the enantiomeric and enantiotopic analysis of the NMR spectra of compounds derived from the functionalized cone-shaped core, cyclotriveratrylenes (CTV), dissolved in weakly oriented lyotropic chiral liquid crystals (CLCs) based on organic solutions of poly-γ-benzyl-l-glutamate. The CTV core lacks prostereogenic as well as stereogenic tetrahedral centers. However, depending on the pattern of substitution, chiral and achiral compounds with different symmetries can be obtained. Thus, symmetrically nonasubstituted CTVs (C3 symmetry) are optically active and exhibit enantiomeric isomers, while symmetrically hexasubstituted (C3v symmetry) derivatives are prochiral and possess enantiotopic elements. In the first part we use 2H and 13C NMR to study two nonasubstituted (−OH or −OCH3) CTVs, where the ring methylenes are fully deuterated, and show for the first time that the observation of enantiomeric discrimination of chiral molecules with a 3-fold symmetry axis is possible in a CLC. It is argued that this discrimination reflects different orientational ordering of the M and P isomers, rather than specific chiral short-range solvent−solute interactions that may affect differently the magnetic parameters of the enantiomers or even their geometry. In the second part we present similar measurements on hexasubstituted CTV with flexible side groups (−OC(O)CH3 and the, partially deuterated bidentate, −OCH2CH2O−), having on the average C3v symmetry. No spectral discrimination of enantiotopic sites was detected for the −OC(O)CH3 derivative. This is consistent with a recent theoretical work (J. Chem. Phys. 1999, 111, 6890) that indicates that in C3v molecules no chiral discrimination between enantiotopic elements, based on ordering, is possible. In contrast, a clear splitting was observed in the 2H spectra of the enantiotopic deuterons of the side groups in the tri(dioxyethylene)−CTV. It is argued that this discrimination reflects different ordering characteristics of the various, rapidly (on the NMR time scale) interconverting conformers of this compound. Assuming two twisted structures for each of the dioxyethylene side groups, four different conformers are expected, comprising two sets of enantiomeric pairs with, respectively, C3 and C1 symmetries. Differential ordering and/or fractional population imbalance of these enantiomeric pairs leads to the observed spectral discrimination of sites in the side chains that on average form enantiotopic pairs