Olivier Bournez

Approximate Reachability Analysis of Piecewise-Linear Dynamical Systems

In this paper we describe an experimental system called d/dt for approximating reachable states for hybrid systems whose continuous dynamics is defined by linear differential equations. We use an approximation algorithm whose accumulation of errors during the continuous evolution is much smaller than in previously-used methods. The d/dt system can, so far, treat non-trivial continuous systems, hybrid systems, convex differential inclusions and controller synthesis problems.

Orthogonal Polyhedra: Representation and Computation

In this paper we investigate orthogonal polyhedra, i.e. polyhedra which are finite unions of full-dimensional hyper-rectangles. We define representation schemes for these polyhedra based on their vertices, and show that these compact representation schemes are canonical for all (convex and non-convex) polyhedra in any dimension. We then develop efficient algorithms for membership, face-detection and Boolean operations for these representations.

Deciding stability and mortality of piecewise affine dynamical systems

In this paper we study problems such as: given a discrete time dynamical system of the form x(t + 1)= f(x(t)) where f: R-n --> R-n is a piecewise affine function, decide whether all trajectories converge to 0. We show in our main theorem that this Attractivity Problem is undecidable as soon as n greater than or equal to2. The same is true of two related problems: Stability (is the dynamical system globally asymptotically stable?) and Mortality (do all trajectories go through 0?). We then show that AM-activity and Stability become decidable in dimension 1 for continuous functions

The Stability of Saturated Linear Dynamical Systems Is Undecidable

We prove that several global properties (global convergence, global asymptotic stability, mortality, and nilpotence) of particular classes of discrete time dynamical systems are undecidable. Such results had been known only for point-to-point properties. We prove these properties undecidable for saturated linear dynamical systems, and for continuous piecewise affine dynamical systems in dimension 3. We also describe some consequences of our results on the possible dynamics of such systems.

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature.

Using Local Planar Geometric Invariants to Match and Model Images of Line Segments

Image matching consists of finding features in different images that represent the same feature of the observed scene. It is a basic process in vision whenever several images are used. This paper describes a matching algorithm for lines segments in two images. The key idea of the algorithm is to assume that the apparent motion between the two images can be approximated by a planar geometric transformation (a similarity or an affine transformation) and to compute such an approximation. Under such an assumption, local planar invariants related the kind of transformation used as approximation, should have the same value in both images. Such invariants are computed for simple segment configurations in both images and matched according to their values. A global constraint is added to ensure a global coherency between all the possible matches: all the local matches must define approximately the same geometric transformation between the two images. These first matches are verified and completed using a better and more global approximation of the apparent motion by a planar homography and an estimate of the epipolar geometry. If more than two images are considered, they are initially matched pairwise; then global matches are deduced in a second step. Finally, from a set of images representing different aspects of an object, it is possible to compare them and to compute a model of each aspect using the matching algorithm. This work uses in a new way many elements already known in vision; some of the local planar invariants used here were presented as quasi-invariants by Binford and studied by Ben-Arie in his work on thepeaking effect. The algorithm itself uses other ideas coming from the geometric hashing and the Hough algorithms. Its main limitations come from the invariants used. They are really stable when they are computed for a planar object or for many man-made objects which contain many coplanar facets and elements. On the other hand, the algorithm will probably fail when used with images of very general polyhedrons. Its main advantages are that it still works even if the images are noisy and the polyhedral approximation of the contours is not exact, if the apparent motion between the images is not infinitesimal, if they are several different motions in the scene, and if the camera is uncalibrated and its motion unknown. The basic matching algorithm is presented in Section 2, the verification and completion stages in Section 3, the matching of several images is studied in Section 4 and the algorithm to model the different aspects of an object is presented in Section 5. Results obtained with the different algorithms are shown in the corresponding sections.

Proving positive almost-sure termination

In order to extend the modeling capabilities of rewriting systems, it is rather natural to consider that the firing of rules can be subject to some probabilistic laws. Considering rewrite rules subject to probabilities leads to numerous questions about the underlying notions and results. We focus here on the problem of termination of a set of probabilistic rewrite rules. A probabilistic rewrite system is said almost surely terminating if the probability that a derivation leads to a normal form is one. Such a system is said positively almost surely terminating if furthermore the mean length of a derivation is finite. We provide several results and techniques in order to prove positive almost sure termination of a given set of probabilistic rewrite rules. All these techniques subsume classical ones for non-probabilistic systems.

Rewriting logic and probabilities

Rewriting Logic has shown to provide a general and elegant framework for unifying a wide variety of models, including concurrency models and deduction systems. In order to extend the modeling capabilities of rule based languages, it is natural to consider that the firing of rules can be subject to some probabilistic laws. Considering rewrite rules subject to probabilities leads to numerous questions about the underlying notions and results. In this paper, we discuss whether there exists a notion of probabilistic rewrite system with an associated notion of probabilistic rewriting logic.

The Mortality Problem for Matrices of Low Dimensions

In this paper we discuss the existence of an algorithm to decide if a given set of 2 \times 2 matrices is mortal. A set F={A1,\ldots,Am} of square matrices is said to be mortal if there exist an integer k \geq 1 and some integers i1,i2,\ldots,ik ∈ {1, \ldots, m} with Ai1 Ai2 ⋅s Aik=0 . We survey this problem and propose some new extensions. We prove the problem to be BSS-undecidable for real matrices and Turing-decidable for two rational matrices. We relate the problem for rational matrices to the entry-equivalence problem, to the zero-in-the-corner problem, and to the reachability problem for piecewise-affine functions. Finally, we state some NP-completeness results.

Probabilistic Rewrite Strategies. Applications to ELAN

Recently rule based languages focussed on the use of rewriting as a modeling tool which results in making specifications executable. To extend the modeling capabilities of rule based languages, we explore the possibility of making the rule applications subject to probabilistic choices.We propose an extension of the ELAN strategy language to deal with randomized systems.We argue through several examples that we propose indeed a natural setting to model systems with randomized choices.This leads us to interesting new problems, and we address the generalization of the usual concepts in abstract reduction systems to randomized systems.