Algebraic optimization of recursive queries

Over the past few years, much attention has been paid to deductive databases. They offer a logic-based interface, and allow formulation of complex recursive queries. However, they do not offer appropriate update facilities, and do not support existing applications. To overcome these problems an SQL-like interface is required besides a logic-based interface.\ud \ud In the PRISMA project we have developed a tightly-coupled distributed database, on a multiprocessor machine, with two user interfaces: SQL and PRISMAlog. Query optimization is localized in one component: the relational query optimizer. Therefore, we have defined an eXtended Relational Algebra that allows recursive query formulation and can also be used for expressing executable schedules, and we have developed algebraic optimization strategies for recursive queries. In this paper we describe an optimization strategy that rewrites regular (in the context of formal grammars) mutually recursive queries into standard Relational Algebra and transitive closure operations. We also describe how to push selections into the resulting transitive closure operations.\ud \ud The reason we focus on algebraic optimization is that, in our opinion, the new generation of advanced database systems will be built starting from existing state-of-the-art relational technology, instead of building a completely new class of systems

Independence Logic and Abstract Independence Relations

We continue the work on the relations between independence logic and the model-theoretic analysis of independence, generalizing the results of [15] and [16] to the framework of abstract independence relations for an arbitrary AEC. We give a model-theoretic interpretation of the independence atom and characterize under which conditions we can prove a completeness result with respect to the deductive system that axiomatizes independence in team semantics and statistics

A Categorical View on Algebraic Lattices in Formal Concept Analysis

Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or representability. The latter are commonly studied in denotational semantics and domain theory and captured most prominently by the notion of algebraicity, e.g. of lattices. In this paper, we explore the notion of algebraicity in formal concept analysis from a category-theoretical perspective. To this end, we build on the the notion of approximable concept with a suitable category and show that the latter is equivalent to the category of algebraic lattices. At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating well-known structures such as Scott information systems with further formalisms from logic, topology, domains and lattice theory.Comment: 36 page

Automated Generation of Geometric Theorems from Images of Diagrams

We propose an approach to generate geometric theorems from electronic images of diagrams automatically. The approach makes use of techniques of Hough transform to recognize geometric objects and their labels and of numeric verification to mine basic geometric relations. Candidate propositions are generated from the retrieved information by using six strategies and geometric theorems are obtained from the candidates via algebraic computation. Experiments with a preliminary implementation illustrate the effectiveness and efficiency of the proposed approach for generating nontrivial theorems from images of diagrams. This work demonstrates the feasibility of automated discovery of profound geometric knowledge from simple image data and has potential applications in geometric knowledge management and education.Comment: 31 pages. Submitted to Annals of Mathematics and Artificial Intelligence (special issue on Geometric Reasoning

Developing a labelled object-relational constraint database architecture for the projection operator

Current relational databases have been developed in order to improve the handling of stored data, however, there are some types of information that have to be analysed for which no suitable tools are available. These new types of data can be represented and treated as constraints, allowing a set of data to be represented through equations, inequations and Boolean combinations of both. To this end, constraint databases were defined and some prototypes were developed. Since there are aspects that can be improved, we propose a new architecture called labelled object-relational constraint database (LORCDB). This provides more expressiveness, since the database is adapted in order to support more types of data, instead of the data having to be adapted to the database. In this paper, the projection operator of SQL is extended so that it works with linear and polynomial constraints and variables of constraints. In order to optimize query evaluation efficiency, some strategies and algorithms have been used to obtain an efficient query plan. Most work on constraint databases uses spatiotemporal data as case studies. However, this paper proposes model-based diagnosis since it is a highly potential research area, and model-based diagnosis permits more complicated queries than spatiotemporal examples. Our architecture permits the queries over constraints to be defined over different sets of variables by using symbolic substitution and elimination of variables.Ministerio de Ciencia y Tecnología DPI2006-15476-C02-0

Towards an Efficient Evaluation of General Queries

Database applications often require to evaluate queries containing quantifiers or disjunctions, e.g., for handling general integrity constraints. Existing efficient methods for processing quantifiers depart from the relational model as they rely on non-algebraic procedures. Looking at quantified query evaluation from a new angle, we propose an approach to process quantifiers that makes use of relational algebra operators only. Our approach performs in two phases. The first phase normalizes the queries producing a canonical form. This form permits to improve the translation into relational algebra performed during the second phase. The improved translation relies on a new operator - the complement-join - that generalizes the set difference, on algebraic expressions of universal quantifiers that avoid the expensive division operator in many cases, and on a special processing of disjunctions by means of constrained outer-joins. Our method achieves an efficiency at least comparable with that of previous proposals, better in most cases. Furthermore, it is considerably simpler to implement as it completely relies on relational data structures and operators

Proving Finite Satisfiability of Deductive Databases

It is shown how certain refutation methods can be extended into semi-decision procedures that are complete for both unsatisfiability and finite satisfiability. The proposed extension is justified by a new characterization of finite satisfiability. This research was motivated by a database design problem: Deduction rules and integrity constraints in definite databases have to be finitely satisfiabl