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Outline

Title
Abstract
Figures
Introduction
Databases and Queries
Linear Datalog ±
Related Work
Database Schemata, Ontologies, and Chase
Datalog Extensions
Conclusion
References

Query Answering over Ontologies

Profile image of Georg GottlobGeorg Gottlob

Abstract

Ontologies and rules play a central role in the development of the Semantic Web. Recent research in this context focuses especially on highly scalable formalisms for the Web of Data, which may highly benefit from exploiting database technologies. In this paper, as a first step towards closing the gap between the Semantic Web and databases, we introduce a family of expressive extensions of Datalog, called Datalog±, as a new paradigm for query answering over ontologies. The Datalog± family admits existentially quantified variables in rule heads, and has suitable restrictions to ensure highly efficient ontology querying. We show in particular that different versions of Datalog± generalize the tractable description logic ℇℒ and the DL-Lite family of tractable description logics, which are the most common tractable ontology languages in the context of the Semantic Web and databases. We also show how stratified negation can be added to Datalog± while keeping ontology querying tractable. Furthermore, the Datalog± family is of interest in its own right, and can, moreover, be used in various contexts such as data integration and data exchange. It paves the way for applying results from databases to the context of the Semantic Web.

Figures (5)
Note that sets of guarded TGDs (with single-atom heads) are theories in the guarded fragment of first- order logic [4]. Note also that guardedness is a truly fundamental class ensuring decidability. As shown in [23], adding a single unguarded Datalog rule to a guarded Datalog* program may destroy decidability. [23], adding a single unguarded Datalog rule to a guarded Datalog~ program may destroy decidability. In the sequel, let 7 be a relational schema, D be a database for R, and & be a set of guarded TGDs on #. We first give some preliminary definitions as follows. The chase graph for D and & is the directed graph consisting of chase(D, ©) as the set of nodes and having an arrow from a to b iff b is obtained from a and possibly other atoms by a one-step application of a TGD o € &. Here, we mark a as guard iff a is the guard of o. The guarded chase forest for D and & contains (i) for every atom a € D, one node labeled with a, and (ii) for every node labeled with a € chase(D,%) and for every atom b € chase(D, X) that is obtained from a and possibly other atoms by a one-step application of a TGD o € » with a as guard, one node labeled with b along with an arrow from the node labeled with a. The subtree of a node v labeled with an atom a (also simply called subtree of a) in this forest, denoted subtree(a), is the restriction of the forest to all descendants of v. The type of an atom a, denoted type(a), is the set of all atoms b in chase(D,™) that have only constants and nulls from a as arguments. Note that the subtree of (a node labeled with) an atom a in the guarded chase forest depends only on the set of all atoms in the type of a (and no others).
Note that sets of guarded TGDs (with single-atom heads) are theories in the guarded fragment of first- order logic [4]. Note also that guardedness is a truly fundamental class ensuring decidability. As shown in [23], adding a single unguarded Datalog rule to a guarded Datalog* program may destroy decidability. [23], adding a single unguarded Datalog rule to a guarded Datalog~ program may destroy decidability. In the sequel, let 7 be a relational schema, D be a database for R, and & be a set of guarded TGDs on #. We first give some preliminary definitions as follows. The chase graph for D and & is the directed graph consisting of chase(D, ©) as the set of nodes and having an arrow from a to b iff b is obtained from a and possibly other atoms by a one-step application of a TGD o € &. Here, we mark a as guard iff a is the guard of o. The guarded chase forest for D and & contains (i) for every atom a € D, one node labeled with a, and (ii) for every node labeled with a € chase(D,%) and for every atom b € chase(D, X) that is obtained from a and possibly other atoms by a one-step application of a TGD o € » with a as guard, one node labeled with b along with an arrow from the node labeled with a. The subtree of a node v labeled with an atom a (also simply called subtree of a) in this forest, denoted subtree(a), is the restriction of the forest to all descendants of v. The type of an atom a, denoted type(a), is the set of all atoms b in chase(D,™) that have only constants and nulls from a as arguments. Note that the subtree of (a node labeled with) an atom a in the guarded chase forest depends only on the set of all atoms in the type of a (and no others).
of the forest. Here, a proof of an atom a from D and © is a subgraph 7 of the chase graph such that 7 contains the atom a as a vertex, and whenever 7 contains b as a vertex, where b ¢ D, then z also contains a set of “parent” atoms b1,...,b, for b such that there exists a TGD in © that applied to bi,..., b, yields b. The proof of Lemma 3 in Appendix A is done by induction on the derivation level of a. There, it is also stated how y is bounded in terms of the size of the relational schema R, namely double-exponentially in the general case, and single-exponentially in case of a fixed arity. Of course, for a fixed relational schema FR, is a constant.
of the forest. Here, a proof of an atom a from D and © is a subgraph 7 of the chase graph such that 7 contains the atom a as a vertex, and whenever 7 contains b as a vertex, where b ¢ D, then z also contains a set of “parent” atoms b1,...,b, for b such that there exists a TGD in © that applied to bi,..., b, yields b. The proof of Lemma 3 in Appendix A is done by induction on the derivation level of a. There, it is also stated how y is bounded in terms of the size of the relational schema R, namely double-exponentially in the general case, and single-exponentially in case of a fixed arity. Of course, for a fixed relational schema FR, is a constant.
Figure 4: The DL-Lite family of description logics.
Figure 4: The DL-Lite family of description logics.
Figure 6: Construction in the proof of Lemma 4.
Figure 6: Construction in the proof of Lemma 4.

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