Anjolina Grisi de Oliveira

Modelling and querying geographical data warehouses

A number of proposals for integrating geographical (Geographical Information Systems-GIS) and multidimensional (data warehouse-DW and online analytical processing-OLAP) processing are found in the database literature. However, most of the current approaches do not take into account the use of a GDW (geographical data warehouse) metamodel or query language to make available the simultaneous specification of multidimensional and spatial operators. To address this, this paper discusses the UML class diagram of a GDW metamodel and proposes its formal specifications. We then present a formal metamodel for a geographical data cube and propose the Geographical Multidimensional Query Language (GeoMDQL) as well. GeoMDQL is based on well-known standards such as the MultiDimensional eXpressions (MDX) language and OGC simple features specification for SQL and has been specifically defined for spatial OLAP environments based on a GDW. We also present the GeoMDQL syntax and a discussion regarding the taxonomy of GeoMDQL query types. Additionally, aspects related to the GeoMDQL architecture implementation are described, along with a case study involving the Brazilian public healthcare system in order to illustrate the proposed query language.

A set of aggregation functions for spatial measures

A number of studies have been developed in recent years aimed at integrating pertinent concepts and technologies for analytical multidimensional (OLAP) and geographic (GIS) processing environments. This type of integrated environment has been identified as SOLAP (Spatial OLAP). However, due to the fact that these two technologies were conceived with different purposes in mind, the interaction of the two environments is not an easy task and even with so much research being developed, there remain unresolved issues that merit exploration. One such issue refers to aggregation functions for measures. These functions are currently used in the definition of multidimensional and geographic data cubes. The aim of this paper is to present a set of aggregation functions for geographic measures. We also show these functions in practice, by taking into account their use with a SOLAP architecture prototype. This SOLAP prototype is based on a model for Geographic Data Warehouse (GDW), a data cube model and a geographic multidimensional query language.

The impact of spatial data redundancy on SOLAP query performance

Geographic Data Warehouses (GDW) are one of the main technologies used in decision-making processes and spatial analysis, and the literature proposes several conceptual and logical data models for GDW. However, little effort has been focused on studying how spatial data redundancy affects SOLAP (Spatial On-Line Analytical Processing) query performance over GDW. In this paper, we investigate this issue. Firstly, we compare redundant and non-redundant GDW schemas and conclude that redundancy is related to high performance losses. We also analyze the issue of indexing, aiming at improving SOLAP query performance on a redundant GDW. Comparisons of the SB-index approach, the star-join aided by R-tree and the star-join aided by GiST indicate that the SB-index significantly improves the elapsed time in query processing from 25% up to 99% with regard to SOLAP queries defined over the spatial predicates of intersection, enclosure and containment and applied to roll-up and drill-down operations. We also investigate the impact of the increase in data volume on the performance. The increase did not impair the performance of the SB-index, which highly improved the elapsed time in query processing. Performance tests also show that the SB-index is far more compact than the star-join, requiring only a small fraction of at most 0.20% of the volume. Moreover, we propose a specific enhancement of the SB-index to deal with spatial data redundancy. This enhancement improved performance from 80 to 91% for redundant GDW schemas.

A Metamodel for the Specification of Geographical Data

The decision-making processes can be supported by many tools such as Data Warehouse (DW), On-Line Analytical Processing (OLAP) and Geographical Information System (GIS). Much research found in literature is aimed at integrating these technologies, although most of these approaches do not provide formal definitions for a Geographical Data Warehouse (GDW) nor there is a consensus regarding the design of spatial dimensional schemas for GDW. To address this, GeoDWFrame was proposed as a set of guidelines to design spatial dimensional schemas. However, there is not a metamodel that implements its specifications, nor there is a metamodel that integrates GDW concepts with some standards. Then, in this paper we propose GeoDWM, which is a formally specified metamodel, that extends GeoDWFrame with spatial measures. We have instantiated our formal metamodel based on CWM and OGC standards to assist the conceptual modelling needs found in the development of GDW applications. Finally, some issues concerning the development of a CASE tool that is based on the GeoDWM metamodel and that provides a means of validating the application conceptual data model are given together with a discussion on GDW conceptual design aspects and some descriptions for future work.

Intuitionistic N-Graphs

The geometric system of deduction called N-Graphs was introduced by De Oliveira in 2001. The proofs in this system are represented by means of digraphs and, while its derivations are mostly based on Gentzen’s sequent calculus, the system gets its inspiration from geometrically based systems, such as the Kneales’ tables of development, Statman’s proofs-as-graphs, Buss’ logical flow graphs and Girard’s proof-nets. Given that all these geometric systems appeal to the classical symmetry between premises and conclusions, providing an intuitionistic version of any of these is an interesting exercise in extending the range of applicability of the geometric system in question. In this paper we produce an intuitionistic version of N-Graphs, based on Maehara’s LJ’ system, as described by Takeuti. Recall that LJ’ has multiple conclusions in all but the essential intuitionistic rules e.g. implication right and negation right. We show soundness and completeness of our intuitionistic N-Graphs with respect to LJ’. We also discuss how we expect to extend this work to a version of N-Graphs corresponding to the intuitionistic logic system FIL (Full Intuitionistic Logic) of de Paiva and Pereira and sketch future developments.

Proof-graphs: a Thorough Cycle Treatment, Normalization and Subformula Property

A normalization procedure is presented for a classical natural deduction (ND) proof system. This proof system, called N-Graphs, has a multiple conclusion proof structure, where cycles are allowed. With this, we have developed a thorough treatment of cycles, including cycles normalization via an algorithm. We also demonstrate the usefulness of the graphical framework of N-Graphs, where derivations are seen as digraphs. We use geometric perspective techniques to establish the normalization mechanism, thus giving a direct normalization proof. Moreover, the subformula and separation properties are determined.

Transformations via Geometric Perspective Techniques Augmented with Cycles Normalization

A normalization procedure is presented for a classical natural deduction (ND) proof system. This proof system, called N-Graphs, has a multiple conclusion proof structure where cycles are allowed. With this, we have developed a thorough treatment of cycles, including cycles normalization via an algorithm. We also demonstrate the usefulness of the graphical framework of N-Graphs, where derivations are seen as digraphs. We use geometric perspective techniques to establish the normalization mechanism, thus giving a direct normalization proof.

Natural Deduction for Equality: The Missing Entity

The conception of the very first decision procedures for first-order sentences brought about the need for giving citizenship to function symbols (e.g. Skolem functions). We argue that a closer look at proof procedures for first-order sentences with equality brings about the need for introducing (function) symbols for rewrites. This is appropriately done via the framework of labelled natural deduction which allows to formulate a proof theory for the “logical connective” of propositional equality. The basic idea is that when analysing an equality sentence into (i) proof conditions (introduction) and (ii) immediate consequences (elimination), it becomes clear that we need to bring in identifiers (i.e. function symbols) for sequences of rewrites, and this is what we claim is the missing entity in P. Martin-Lof’s equality types (both intensional and extensional). What we end up with is a formulation of what appears to be a middle ground solution to the ‘intensional’ versus ‘extensional’ dichotomy which permeates most of the work on characterizing propositional equality in natural deduction style. (Part of this material was presented at the Logical Methods in the Humanities Seminar, Stanford University, and the authors would like to thank Solomon Feferman and Grigori Mints for their comments and suggestions.)

On the Identity Type as the Type of Computational Paths

We introduce a new way of formalizing the intensional identity type based on the fact that a entity known as computational paths can be interpreted as terms of the identity type. Our approach enjoys the fact that our elimination rule is easy to understand and use. We make this point clear constructing terms of some relevant types using our proposed elimination rule. We also show that the identity type, as defined by our approach, induces a groupoid structure. This result is on par with the fact that the traditional identity type induces a groupoid, as exposed by Hofmann \& Streicher (1994).

Linear Time Proof Verification on N-Graphs: A Graph Theoretic Approach

This paper presents a linear time algorithm for proof verification on N-Graphs. This system, introduced by de Oliveira, incorporates the geometrical techniques from the theory of proof-nets to present a multiple-conclusion calculus for classical propositional logic. The soundness criterion is based on the one given by Danos and Regnier for Linear Logic. We use a DFS-like search to check the validity of the cycles in a proof graph, and some properties from trees to check the connectivity of every switching a concept similar to D-R graph. Since the soundness criterion in proof graphs is analogous to Danos-Regniers procedure, the algorithm can also be extended to check proofs in the multiplicative linear logic without units MLL- with linear time complexity.