Inconsistency, Logic Databases, and Ontologies

Inconsistency, Logic Databases, and Ontologies

José A. Alonso-Jiménez (Artificial Universidad de Sevilla, Spain), Joaquín Borrego-Díaz (Artificial Universidad de Sevilla, Spain) and Antonia M. Chávez-González (Artificial Universidad de Sevilla, Spain)
DOI: 10.4018/978-1-60566-242-8.ch049
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Abstract

Nowadays, data management on the World Wide Web needs to consider very large knowledge databases (KDB). The larger is a KDB, the smaller the possibility of being consistent. Consistency in checking algorithms and systems fails to analyse very large KDBs, and so many have to work every day with inconsistent information. Database revision—transformation of the KDB into another, consistent database—is a solution to this inconsistency, but the task is computationally untractable. Paraconsistent logics are also a useful option to work with inconsistent databases. These logics work on inconsistent KDBs but prohibit non desired inferences. From a philosophical (logical) point of view, the paraconsistent reasoning is a need that the self human discourse practices. From a computational, logical point of view, we need to design logical formalisms that allow us to extract useful information from an inconsistent database, taking into account diverse aspects of the semantics that are “attached” to deductive databases reasoning (see Table 1). The arrival of the semantic web (SW) will force the database users to work with a KDB that is expressed by logic formulas with higher syntactic complexity than are classic logic databases.
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Introduction

Nowadays, data management on the World Wide Web needs to consider very large knowledge databases (KDB). The larger is a KDB, the smaller the possibility of being consistent. Consistency in checking algorithms and systems fails to analyse very large KDBs, and so many have to work every day with inconsistent information.

Database revision—transformation of the KDB into another, consistent database—is a solution to this inconsistency, but the task is computationally untractable. Paraconsistent logics are also a useful option to work with inconsistent databases. These logics work on inconsistent KDBs but prohibit non desired inferences. From a philosophical (logical) point of view, the paraconsistent reasoning is a need that the self human discourse practices. From a computational, logical point of view, we need to design logical formalisms that allow us to extract useful information from an inconsistent database, taking into account diverse aspects of the semantics that are “attached” to deductive databases reasoning (see Table 1). The arrival of the semantic web (SW) will force the database users to work with a KDB that is expressed by logic formulas with higher syntactic complexity than are classic logic databases.

Table 1.
Semantics aspects to consider in logic databases
  • Classical semantics for First Order Logic
      Extended semantics for databases
      Reiter’s formalization of databases (Reiter, 1984). Closed World Assumption
      Relations among a KDB, queries and integrity constraints
      Expressive power of recursive definitions
      Consistency checking versus intentional part of the KDB
      Multivalued semantics
      Contextualized semantics for ontologies or data
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Background

Logic databases are based on the formalisms of first order logic (FOL); thus, they inherit a classical semantics that is based on models. Also, they can be interpreted within a proof–theoretic approach to logical consequence from the logic programming paradigm (Lloyd, 1987). The extended database semantics paradigm is developed to lay before the foundations of query-answering tasks and related questions (see Minker, 1999), but its aim is not to deal with inconsistencies. The data cleaning task may involve—in the framework of repairing logic databases— logical reasoning and automated theorem proving (Boskovitz, Goré, & Hegland, 2003).

On the other hand, new paradigms, such as SW, need new formalisms to reason about data. Description logics (DL) provide logic systems based on objects, concepts, and relationships, with which we can construct new concepts and relations for reasoning (Baader, Calvanese, McGuinnes, Nardi, & Patel- Schneider, 2003). Formally, DL are a subset of FOL, and the classical problems on consistency remains, but several sublogics of DL provide nice algorithms for reasoning services. The ontology web language (OWL; its DL-sublanguage) is a description logic language designed for automated reasoning, not only designed for the classical ask–tell paradigm. With languages such as OWL, ontologies exceed their traditional aspects (e.g., taxonomies and dictionaries) to be essential in frameworks as data integration.

Key Terms in this Chapter

Contextualizing Logics: Method to formally represent knowledge associated with a particular circumstance on which it has the intended meaning.

Paraconsistent Logics: Logic systems that limit, for example, the power of classical inference relations to retrieve non trivial information of inconsistent sets of formulas.

Description Logics: Logical formalism to represent structured concepts and the relationships among them. Formally, it is a subset of First Order Logic dealing with concepts (monadic predicates) and roles (binary predicates) which are useful to relate concepts. KDB in DL are composed of a Tbox (the intensional component) and an Abox (Box of asserts, the extensional component part).

OWL: Ontology Web Language. Language (based on Description Logics) designed to represent ontologies capable of being processed by machines. The World Wide Web Consortium released OWL as recommendation http://www.w3.org/2001/sw/webOnt.

Arguments: An argument in a KDB is a pair (B, F), where ? is a subset of the KDB such that ? entails F. The basic relation among arguments is rebutting.

Closed World Assumption: A principle that claims that every atom not entailed by the KDB is assumed to be false. This principle is sound on KDBs with simple syntax, as logic programs.

Logical Inconsistency: A logical theory is inconsistent if there is no logical model for it. In the logic database paradigm, the notion of inconsistency is usually restricted to express the violation of integrity constraints. This restriction makes sense when the data are atomic formulas.

Reiter’s Formalization of Database Theory: A set of axioms that, when they are added to a relational database, formalizes the reasoning with them. They are the Unique Names Principle, the Domain Closure Axiom, the Completion Axioms and Equality Axioms. This formalization permits to identify the answers with logical consequences.

Skolem Noise: A kind of anomalous answers obtained by resolution oriented theorem provers working on non clausal theories. The classical method of skolemization leads to new function symbols with no intended meaning. If these symbols appear in the output, the answer may not be consistently interpreted.

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