École normale supérieure de Cachan

About: École normale supérieure de Cachan is a education organization based out in Cachan, Île-de-France, France. It is known for research contribution in the topics: Decidability & Finite element method. The organization has 2717 authors who have published 5585 publications receiving 175925 citations.

Reachability in MDPs: Refining Convergence of Value Iteration

Abstract: Markov Decision Processes (MDP) are a widely used model including both non-deterministic and probabilistic choices. Minimal and maximal probabilities to reach a target set of states, with respect to a policy resolving non-determinism, may be computed by several methods including value iteration. This algorithm, easy to implement and efficient in terms of space complexity, consists in iteratively finding the probabilities of paths of increasing length. However, it raises three issues: (1) defining a stopping criterion ensuring a bound on the approximation, (2) analyzing the rate of convergence, and (3) specifying an additional procedure to obtain the exact values once a sufficient number of iterations has been performed. The first two issues are still open and for the third one a “crude” upper bound on the number of iterations has been proposed. Based on a graph analysis and transformation of MDPs, we address these problems. First we introduce an interval iteration algorithm, for which the stopping criterion is straightforward. Then we exhibit convergence rate. Finally we significantly improve the bound on the number of iterations required to get the exact values.

Interactions between moderately close inclusions for the Laplace equation

Abstract: The presence of small inclusions modifies the solution of the Laplace equation posed in a reference domain Ω0. This question has been studied extensively for a single inclusion or well-separated inclusions. In two-dimensional situations, we investigate the case where the distance between the holes tends to zero but remains large with respect to their characteristic size. We first consider two perfectly insulated inclusions. In this configuration we give a complete multiscale asymptotic expansion of the solution to the Laplace equation. We also address the situation of a single inclusion close to a singular perturbation of the boundary ∂Ω0. We also present numerical experiments implementing a multiscale superposition method based on our first order expansion.

Lossy counter machines decidability cheat sheet

Abstract: Lossy counter machines (LCM's) are a variant of Minsky counter machines based on weak (or unreliable) counters in the sense that they can decrease nondeterministically and without notification. This model, introduced by R. Mayr [TCS 297:337-354 (2003)], is not yet very well known, even though it has already proven useful for establishing hardness results. In this paper we survey the basic theory of LCM's and their verification problems, with a focus on the decidability/undecidability divide.

Augmenting ATL with strategy contexts

Abstract: We study the extension of the alternating-time temporal logic (ATL) with strategy contexts: contrary to the original semantics, in this semantics the strategy quantifiers do not reset the previously selected strategies.We show that our extension ATL s c is very expressive, but that its decision problems are quite hard: model checking is k-EXPTIME-complete when the formula has k nested strategy quantifiers; satisfiability is undecidable, but we prove that it is decidable when restricting to turn-based games. Our algorithms are obtained through a very convenient translation to QCTL (the computation-tree logic CTL extended with atomic quantification), which we show also applies to Strategy Logic, as well as when strategy quantification ranges over memoryless strategies.

Quantum statistics of light transmitted through an intracavity Rydberg medium

Abstract: We theoretically investigate the quantum statistical properties of light transmitted through an atomic medium with strong optical nonlinearity induced by Rydberg-Rydberg van der Waals interactions. In our setup, atoms are located in a cavity and nonresonantly driven on a two-photon transition from their ground state to a Rydberg level via an intermediate state by the combination of the weak signal field and a strong control beam. To characterize the transmitted light, we compute the second-order correlation function $g^{(2)}(\tau)$. The simulations we obtained on the specific case of rubidium atoms suggest that the bunched or antibunched nature of the outgoing beam can be chosen at will by tuning the physical parameters appropriately.