Checking Zenon Modulo Proofs in Dedukti

Dedukti has been proposed as a universal proof checker. It is a logical framework based on the lambda Pi calculus modulo that is used as a backend to verify proofs coming from theorem provers, especially those implementing some form of rewriting. We present a shallow embedding into Dedukti of proofs produced by Zenon Modulo, an extension of the tableau-based first-order theorem prover Zenon to deduction modulo and typing. Zenon Modulo is applied to the verification of programs in both academic and industrial projects. The purpose of our embedding is to increase the confidence in automatically generated proofs by separating untrusted proof search from trusted proof verification.Comment: In Proceedings PxTP 2015, arXiv:1507.0837

Accelerated Gossip in Networks of Given Dimension using Jacobi Polynomial Iterations

Consider a network of agents connected by communication links, where each agent holds a real value. The gossip problem consists in estimating the average of the values diffused in the network in a distributed manner. We develop a method solving the gossip problem that depends only on the spectral dimension of the network, that is, in the communication network set-up, the dimension of the space in which the agents live. This contrasts with previous work that required the spectral gap of the network as a parameter, or suffered from slow mixing. Our method shows an important improvement over existing algorithms in the non-asymptotic regime, i.e., when the values are far from being fully mixed in the network. Our approach stems from a polynomial-based point of view on gossip algorithms, as well as an approximation of the spectral measure of the graphs with a Jacobi measure. We show the power of the approach with simulations on various graphs, and with performance guarantees on graphs of known spectral dimension, such as grids and random percolation bonds. An extension of this work to distributed Laplacian solvers is discussed. As a side result, we also use the polynomial-based point of view to show the convergence of the message passing algorithm for gossip of Moallemi \& Van Roy on regular graphs. The explicit computation of the rate of the convergence shows that message passing has a slow rate of convergence on graphs with small spectral gap

Resampling: an improvement of Importance Sampling in varying population size models

Sequential importance sampling algorithms have been defined to estimate likelihoods in models of ancestral population processes. However, these algorithms are based on features of the models with constant population size, and become inefficient when the population size varies in time, making likelihood-based inferences difficult in many demographic situations. In this work, we modify a previous sequential importance sampling algorithm to improve the efficiency of the likelihood estimation. Our procedure is still based on features of the model with constant size, but uses a resampling technique with a new resampling probability distribution depending on the pairwise composite likelihood. We tested our algorithm, called sequential importance sampling with resampling (SISR) on simulated data sets under different demographic cases. In most cases, we divided the computational cost by two for the same accuracy of inference, in some cases even by one hundred. This study provides the first assessment of the impact of such resampling techniques on parameter inference using sequential importance sampling, and extends the range of situations where likelihood inferences can be easily performed

High-contrast imaging in polychromatic light with the self-coherent camera

Context. In the context of direct imaging of exoplanets, coronagraphs are commonly proposed to reach the required very high contrast levels. However, wavefront aberrations induce speckles in their focal plane and limit their performance. Aims. An active correction of these wavefront aberrations using a deformable mirror upstream of the coronagraph is mandatory. These aberrations need to be calibrated and focal-plane wavefront-sensing techniques in the science channel are being developed. One of these, the self-coherent camera, of which we present the latest laboratory results. Methods. We present here an enhancement of the method: we directly minimized the complex amplitude of the speckle field in the focal plane. Laboratory tests using a four-quadrant phase-mask coronagraph and a 32x32 actuator deformable mirror were conducted in monochromatic light and in polychromatic light for different bandwidths. Results. We obtain contrast levels in the focal plane in monochromatic light better than 3.10^-8 (RMS) in the 5 - 12 {\lambda}/D region for a correction of both phase and amplitude aberrations. In narrow bands (10 nm) the contrast level is 4.10^-8 (RMS) in the same region. Conclusions. The contrast level is currently limited by the amplitude aberrations on the bench. We identified several improvements that can be implemented to enhance the performance of our optical bench in monochromatic as well as in polychromatic light.Comment: 4 pages, 3 figures, accepted in Astronomy & Astrophysics (02/2014

Tight Bounds for Consensus Systems Convergence

We analyze the asymptotic convergence of all infinite products of matrices taken in a given finite set, by looking only at finite or periodic products. It is known that when the matrices of the set have a common nonincreasing polyhedral norm, all infinite products converge to zero if and only if all infinite periodic products with period smaller than a certain value converge to zero, and bounds exist on that value. We provide a stronger bound holding for both polyhedral norms and polyhedral seminorms. In the latter case, the matrix products do not necessarily converge to 0, but all trajectories of the associated system converge to a common invariant space. We prove our bound to be tight, in the sense that for any polyhedral seminorm, there is a set of matrices such that not all infinite products converge, but every periodic product with period smaller than our bound does converge. Our technique is based on an analysis of the combinatorial structure of the face lattice of the unit ball of the nonincreasing seminorm. The bound we obtain is equal to half the size of the largest antichain in this lattice. Explicitly evaluating this quantity may be challenging in some cases. We therefore link our problem with the Sperner property: the property that, for some graded posets, -- in this case the face lattice of the unit ball -- the size of the largest antichain is equal to the size of the largest rank level. We show that some sets of matrices with invariant polyhedral seminorms lead to posets that do not have that Sperner property. However, this property holds for the polyhedron obtained when treating sets of stochastic matrices, and our bound can then be easily evaluated in that case. In particular, we show that for the dimension of the space $n \geq 8$, our bound is smaller than the previously known bound by a multiplicative factor of $\frac{3}{2 \sqrt{\pi n}}$