Towards a logical reconstruction of a theory for locally closed databases

Abstract

The Closed World Assumption (CWA) on databases expresses the assumption that an atom not in the database is false. This assumption is applicable only in cases where the database has complete knowledge about the domain of discourse. In this paper, we investigate locally closed databases, that is: databases that are sound but partially incomplete about their domain. Such databases consist of a standard database instance, augmented with a collection of Local Closed World Assumptions (LCWAs). A LCWA is a 'local' form of the CWA, expressing that a database relation is complete in a certain area, called a window of expertise. In this work, we study locally closed databases both from a knowledge representation and from a computational perspective. At the representation level, the approach taken in this paper distinguishes between the data that is conveyed by a database and the meta-knowledge about the area in which the data is complete. We study the semantics of the LCWA's and relate it to several knowledge representation formalisms. At the reasoning level, we study the complexity of, and algorithms for two basic reasoning tasks: computing certain and possible answers to queries and determining whether a database has complete knowledge on a query. As the complexity of these tasks is unacceptably high, we develop efficient approximate methods for query answering. We also prove that for useful classes of queries and locally closed databases, these methods are optimal, and thus they solve the original query in a tractable way. As a result, we obtain classes of queries and locally closed databases for which query answering is tractable.

Supporting quantified queries in distributed databases

“…Semantically, our approach is based on the intuition, already presented in [13], that in a distributed setting each database should be considered as a correct but incomplete representation of the world. In particular, the domain of discourse of a quantified query should be evaluated over the overall system.…”

“…For instance, when A is count, A 0 is sum and when A is sum, A 0 ¼ A. Grouping can be dealt with using the common strategy for maintenance of materialised views: let < þ A;f ;B be the operator on relations R defined by: if two tuples t and t 0 are such that t½A ¼ t 0 ½A, then t modifies its B value according to aggregate f applies to t and t 0 , and t 0 is erased; if a tuple t has a unique A value, it is left unchanged. Then, GB A;f ðBÞ ðRÞ ¼ < 13 Note that this extends to right outerjoin and full outerjoin. For right outerjoin, we would distribute on the left relation.…”

“…We argue that such challenges arise from the fact that these queries require an explicit specification of the particular domain of discourse (intuitively, the set of objects that are presupposed by the query), and that said universe is difficult to specify or construct in distributed environments. This is due to the fact that each database traditionally follows the closed world assumption, which makes it difficult to account for objects existing in other databases [13]. Thus, we are faced with two issues: a semantic one (how do we extract a meaningful answer from distributed systems when the components behave as self-contained units?)…”

Lightweight Reasoning with Incomplete and Inconsistent Information: A Case Study

“…We assume that there is a partial order defined on modules (to be more specific, on module names). 5 External literal is an expression of the form ±m.r(arg list), where ±, r, arg list are described above, and m is a module name. 6 We say, that literal ±m.r(arg list) references module m.…”

“…by setting: 5 This relation divides modules into layers. 6 The intuitive meaning of m.r(arg list) is a query r(arg list) computed in context of module m. 7 The semantics of connectives is defined e.g.…”