On the Decidability of Semilinearity for Semialgebraic Sets and Its Implications for Spatial Databases

AbstractSeveral authors have suggested using first-order logic over the real numbers to describe spatial database applications. Geometric objects are then described by polynomial inequalities with integer coefficients involving the coordinates of the objects. Such geometric objects are called semialgebraic sets. Similarly, queries are expressed by polynomial inequalities. The query language thus obtained is usually referred to as FO+poly. From a practical point of view, it has been argued that a linear restriction of this so-called polynomial model is more desirable. In the so-called linear model, geometric objects are described by linear inequalities and are called semilinear sets. The language of the queries expressible by linear inequalities is usually referred to as FO+linear. As part of a general study of the feasibility of the linear model, we show in this paper that semilinearity is decidable for semialgebraic sets. In doing so, we point out important subtleties related to the type of the coefficients in the linear inequalities used to describe semilinear sets. An important concept in the development of the paper is regular stratification. We point out the geometric significance, as well as its significance in the context of FO+linear and FO+poly computations. The decidability of semilinearity of semialgebraic sets has an important consequence. It has been shown that it is undecidable whether a query expressible in FO+poly is linear, i.e., maps spatial databases of the linear model into spatial databases of the linear model. It follows now that, despite this negative result, there exists a syntactically definable language precisely expressing the linear queries expressible in FO+poly

Affine-Invariant Triangulation of Spatio-Temporal Data with an Application to Image Retrieval

In the geometric data model for spatio-temporal data, introduced by Chomicki and Revesz , spatio-temporal data are modelled as a finite collection of triangles that are transformed by time-dependent affinities of the plane. To facilitate querying and animation of spatio-temporal data, we present a normal form for data in the geometric data model. We propose an algorithm for constructing this normal form via a spatio-temporal triangulation of geometric data objects. This triangulation algorithm generates new geometric data objects that partition the given objects both in space and in time. A particular property of the proposed partition is that it is invariant under time-dependent affine transformations, and hence independent of the particular choice of coordinate system used to describe the spatio-temporal data in. We can show that our algorithm works correctly and has a polynomial time complexity (of reasonably low degree in the number of input triangles and the maximal degree of the polynomial functions that describe the transformation functions). We also discuss several possible applications of this spatio-temporal triangulation. The application of our affine-invariant spatial triangulation method to image indexing and retrieval is discussed and an experimental evaluation is given in the context of bird images

A query processing system for very large spatial databases using a new map algebra

Dans cette thèse nous introduisons une approche de traitement de requêtes pour des bases de donnée spatiales. Nous expliquons aussi les concepts principaux que nous avons défini et développé: une algèbre spatiale et une approche à base de graphe utilisée dans l'optimisateur. L'algèbre spatiale est défini pour exprimer les requêtes et les règles de transformation pendant les différentes étapes de l'optimisation de requêtes. Nous avons essayé de définir l'algèbre la plus complète que possible pour couvrir une grande variété d'application. L'opérateur algébrique reçoit et produit seulement des carte. Les fonctions reçoivent des cartes et produisent des scalaires ou des objets. L'optimisateur reçoit la requête en expression algébrique et produit un QEP (Query Evaluation Plan) efficace dans deux étapes: génération de QEG (Query Evaluation Graph) et génération de QEP. Dans première étape un graphe (QEG) équivalent de l'expression algébrique est produit. Les règles de transformation sont utilisées pour transformer le graphe a un équivalent plus efficace. Dans deuxième étape un QEP est produit de QEG passé de l'étape précédente. Le QEP est un ensemble des opérations primitives consécutives qui produit les résultats finals (la réponse finale de la requête soumise au base de donnée). Nous avons implémenté l'optimisateur, un générateur de requête spatiale aléatoire, et une base de donnée simulée. La base de donnée spatiale simulée est un ensemble de fonctions pour simuler des opérations spatiales primitives. Les requêtes aléatoires sont soumis à l'optimisateur. Les QEPs générées sont soumis au simulateur de base de données spatiale. Les résultats expérimentaux sont utilisés pour discuter les performances et les caractéristiques de l'optimisateur.Abstract: In this thesis we introduce a query processing approach for spatial databases and explain the main concepts we defined and developed: a spatial algebra and a graph based approach used in the optimizer. The spatial algebra was defined to express queries and transformation rules during different steps of the query optimization. To cover a vast variety of potential applications, we tried to define the algebra as complete as possible. The algebra looks at the spatial data as maps of spatial objects. The algebraic operators act on the maps and result in new maps. Aggregate functions can act on maps and objects and produce objects or basic values (characters, numbers, etc.). The optimizer receives the query in algebraic expression and produces one efficient QEP (Query Evaluation Plan) through two main consecutive blocks: QEG (Query Evaluation Graph) generation and QEP generation. In QEG generation we construct a graph equivalent of the algebraic expression and then apply graph transformation rules to produce one efficient QEG. In QEP generation we receive the efficient QEG and do predicate ordering and approximation and then generate the efficient QEP. The QEP is a set of consecutive phases that must be executed in the specified order. Each phase consist of one or more primitive operations. All primitive operations that are in the same phase can be executed in parallel. We implemented the optimizer, a randomly spatial query generator and a simulated spatial database. The query generator produces random queries for the purpose of testing the optimizer. The simulated spatial database is a set of functions to simulate primitive spatial operations. They return the cost of the corresponding primitive operation according to input parameters. We put randomly generated queries to the optimizer, got the generated QEPs and put them to the spatial database simulator. We used the experimental results to discuss on the optimizer characteristics and performance. The optimizer was designed for databases with a very large number of spatial objects nevertheless most of the concepts we used can be applied to all spatial information systems."--Résumé abrégé par UMI

Dagstuhl News January - December 2002

"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic

Fibred contextual quantum physics

Inspired by the recast of the quantum mechanics in a toposical framework, we develop a contextual quantum mechanics via the geometric mathematics to propose a quantum contextuality adaptable in every topos. The contextuality adopted corresponds to the belief that the quantum world must only be seen from the classical viewpoints à la Bohr consequently putting forth the notion of a context, while retaining a realist understanding. Mathematically, the cardinal object is a spectral Stone bundle Σ → B (between stably-compact locales) permitting a treatment of the kinematics, fibre by fibre and fully point-free. In leading naturally to a new notion of points, the geometricity permits to understand those of the base space B as the contexts C — the commutative C*–algebras of a incommutative C*–algebras — and those of the spectral locale Σ as the couples (C, ψ), with ψ a state of the system from the perspective of such a C. The contexts are furnished with a natural order, the aggregation order which is installed as the specialization on B and Σ thanks to (one part of) the Priestley's duality adapted geometrically as well as to the effectuality of the lax descent of the Stone bundles along the perfect maps