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.

LTL with the freeze quantifier and register automata

Abstract: A data word is a sequence of pairs of a letter from a finite alphabet and an element from an infinite set, where the latter can only be compared for equality. To reason about data words, linear temporal logic is extended by the freeze quantifier, which stores the element at the current word position into a register, for equality comparisons deeper in the formula. By translations from the logic to alternating automata with registers and then to faulty counter automata whose counters may erroneously increase at any time, and from faulty and error-free counter automata to the logic, we obtain a complete complexity table for logical fragments defined by varying the set of temporal operators and the number of registers. In particular, the logic with future-time operators and 1 register is decidable but not primitive recursive over finite data words. Adding past-time operators or 1 more register, or switching to infinite data words, causes undecidability.

Multiple Counters Automata, Safety Analysis and Presburger Arithmetic

Abstract: We consider automata with counters whose values are updated according to signals sent by the environment. A transition can be fired only if the values of the counters satisfy some guards (the guards of the transition). We consider guards of the form yi#yj + ci,j where yi is either xi or xi, the values of the counter i respectively after and before the transition, and # is any relational symbol in {=,≤,≥,>,<}. We show that the set of possible counter values which can be reached after any number of iterations of a loop is definable in the additive theory of ℕ (or ℤ or ℝ depending on the type of the counters). This result can be used for the safety analysis of multiple counters automata.

Edge Detection by Helmholtz Principle

Abstract: We apply to edge detection a recently introduced method for computing geometric structures in a digital image, without any a priori information. According to a basic principle of perception due to Helmholtz, an observed geometric structure is perceptually “meaningful” if its number of occurences would be very small in a random situation: in this context, geometric structures are characterized as large deviations from randomness. This leads us to define and compute edges and boundaries (closed edges) in an image by a parameter-free method. Maximal detectable boundaries and edges are defined, computed, and the results compared with the ones obtained by classical algorithms.

On the singular set of stationary harmonic maps.

Abstract: LetM andN be compact riemannian manifolds, andu a stationary harmonic map fromM toN. We prove thatH n−2 (Σ)=0, wheren=dimM and Σ is the singular set ofu. This is a generalization of a result of C. Evans [7], where this is proved in the special caseN is a sphere. We also prove that, ifu is a weakly harmonic map inW 1,n (M, N), thenu is smooth. This extends results of F. Helein for the casen=2, or the caseN is a sphere ([9], [10]).