Extending the Calculus of Constructions with Tarski's fix-point theorem

We propose to use Tarski's least fixpoint theorem as a basis to define recursive functions in the calculus of inductive constructions. This widens the class of functions that can be modeled in type-theory based theorem proving tool to potentially non-terminating functions. This is only possible if we extend the logical framework by adding the axioms that correspond to classical logic. We claim that the extended framework makes it possible to reason about terminating and non-terminating computations and we show that common facilities of the calculus of inductive construction, like program extraction can be extended to also handle the new functions

Inductive and Coinductive Components of Corecursive Functions in Coq

In Constructive Type Theory, recursive and corecursive definitions are subject to syntactic restrictions which guarantee termination for recursive functions and productivity for corecursive functions. However, many terminating and productive functions do not pass the syntactic tests. Bove proposed in her thesis an elegant reformulation of the method of accessibility predicates that widens the range of terminative recursive functions formalisable in Constructive Type Theory. In this paper, we pursue the same goal for productive corecursive functions. Notably, our method of formalisation of coinductive definitions of productive functions in Coq requires not only the use of ad-hoc predicates, but also a systematic algorithm that separates the inductive and coinductive parts of functions.Comment: Dans Coalgebraic Methods in Computer Science (2008

Robust Mobile Object Tracking Based on Multiple Feature Similarity and Trajectory Filtering

This paper presents a new algorithm to track mobile objects in different scene conditions. The main idea of the proposed tracker includes estimation, multi-features similarity measures and trajectory filtering. A feature set (distance, area, shape ratio, color histogram) is defined for each tracked object to search for the best matching object. Its best matching object and its state estimated by the Kalman filter are combined to update position and size of the tracked object. However, the mobile object trajectories are usually fragmented because of occlusions and misdetections. Therefore, we also propose a trajectory filtering, named global tracker, aims at removing the noisy trajectories and fusing the fragmented trajectories belonging to a same mobile object. The method has been tested with five videos of different scene conditions. Three of them are provided by the ETISEO benchmarking project (http://www-sop.inria.fr/orion/ETISEO) in which the proposed tracker performance has been compared with other seven tracking algorithms. The advantages of our approach over the existing state of the art ones are: (i) no prior knowledge information is required (e.g. no calibration and no contextual models are needed), (ii) the tracker is more reliable by combining multiple feature similarities, (iii) the tracker can perform in different scene conditions: single/several mobile objects, weak/strong illumination, indoor/outdoor scenes, (iv) a trajectory filtering is defined and applied to improve the tracker performance, (v) the tracker performance outperforms many algorithms of the state of the art

The union of unit balls has quadratic complexity, even if they all contain the origin

We provide a lower bound construction showing that the union of unit balls in three-dimensional space has quadratic complexity, even if they all contain the origin. This settles a conjecture of Sharir.Comment: 5 pages, 5 figure

Improved Incremental Randomized Delaunay Triangulation

We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, and small memory occupation. The location structure is organized into several levels. The lowest level just consists of the triangulation, then each level contains the triangulation of a small sample of the levels below. Point location is done by marching in a triangulation to determine the nearest neighbor of the query at that level, then the march restarts from that neighbor at the level below. Using a small sample (3%) allows a small memory occupation; the march and the use of the nearest neighbor to change levels quickly locate the query.Comment: 19 pages, 7 figures Proc. 14th Annu. ACM Sympos. Comput. Geom., 106--115, 199

Continued Fraction Expansion of Real Roots of Polynomial Systems

We present a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis. It is inspired by existing subdivision methods in the Bernstein basis; it can be seen as generalization of the univariate continued fraction algorithm or alternatively as a fully analog of Bernstein subdivision in the monomial basis. The representation of the subdivided domains is done through homographies, which allows us to use only integer arithmetic and to treat efficiently unbounded regions. We use univariate bounding functions, projection and preconditionning techniques to reduce the domain of search. The resulting boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. An extension of Vincent's theorem to multivariate polynomials is proved and used for the termination of the algorithm. New complexity bounds are provided for a simplified version of the algorithm. Examples computed with a preliminary C++ implementation illustrate the approach.Comment: 10 page

Triangulating the Real Projective Plane

We consider the problem of computing a triangulation of the real projective plane P2, given a finite point set S={p1, p2,..., pn} as input. We prove that a triangulation of P2 always exists if at least six points in S are in general position, i.e., no three of them are collinear. We also design an algorithm for triangulating P2 if this necessary condition holds. As far as we know, this is the first computational result on the real projective plane

Concrete Domains

This paper introduces the theory of a particular kind of computation domains called concrete domains. The purpose of this theory is to find a satisfactory framework for the notions of coroutine computation and sequentiality of evaluation