On the decidability and complexity of integrating ontologies and rules

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Abstract

We define the formal framework of r-hybrid knowledge bases (KBs) integrating ontologies and rules. A r-hybrid KB has a structural component (ontology) and a rule component. Such a framework is very general, in the sense that: (i) the construction is parametric with respect to the logic used to specify the structural component; (ii) the rule component is very expressive, since it consists of a Datalog¬ program, i.e., a Datalog program with negation as failure and disjunction, (iii) the rule component is constrained in its interaction with the structural component according to a safeness condition: such a safe interaction between rules and structural KB captures (and is a generalization of) several previous proposals. As a consequence, we are able to show that such a framework of r-hybrid KBs comprises many systems proposed for combining rules and Description Logics. Then, we study reasoning in r-hybrid KBs. We provide a general algorithm for reasoning in r-hybrid KBs, and prove that, under very general conditions, decidability of reasoning is preserved when we add safe Datalog¬ rules to a KB: in other words, if reasoning in the logic L used to specify the structural component T is decidable, then reasoning in the extension of T with safe Datalog¬ rules is still decidable. We also show that an analogous property holds for the complexity of reasoning in r-hybrid KBs. Our decidability and complexity results generalize in a broad sense previous results obtained in recent research on this topic. In particular, we prove that reasoning in r-hybrid KBs whose structural component is specified in the Web Ontology Language OWL-DL is decidable.

Introduction

Research in knowledge-based systems has in the last years dealt with the problem of overcoming the limitations imposed by a single knowledge representation language. Hybrid systems [15] have thus been proposed, which are constituted of two or more subsystems, each of which deals with a distinct portion of the knowledge base and uses specific representation formalisms and reasoning procedures. The improvement in the deductive power of hybrid systems is in terms of both the inferences the system is able to make, and the efficiency of the reasoning process, since any subsystem can take advantage of the inferential power of the other subsystems, whereas the use of specialized reasoning procedures allows for improving the efficiency of the deduction process.

The idea of adding rules to structured knowledge representation systems follows this line of research, and dates back to early Description Logic systems like CLASSIC [33], LOOM [29], and CLASP [36]. Then, the idea of building hybrid systems combining rules and structured representation of information has been pursued in a formally more rigorous and coherent way. The first approaches in this direction [27], [6], [9] studied the problem of integrating Datalog rules with Description Logics (DLs). Informally, the basic idea underlying hybrid formalisms integrating rules and descriptions is to deal with knowledge bases (KBs) constituted by a rule component (a Datalog program) and a structural component (a DL knowledge base). The interaction between the two subsystems is obtained by allowing some variables in Datalog rules to range over the set of instances of a specified concept of the DL knowledge base.

Such hybrid formalisms proved well-suited for the construction of tools for accessing heterogeneous information systems. In particular, one of such formalisms (carin) has been used in the Information Manifold, a system developed at AT&T for integrated access to different structured information sources on the World Wide Web [24].

More recently, a renewed interest towards the integration of structured KBs and rules has emerged in the research on ontologies and the Semantic Web [21], [1]. DLs are also playing a central role in this field, since they are currently the most used formalisms for building ontologies, and have been proposed as standard languages for the specification of ontologies in the Semantic Web [32].

However, as shown by the first studies in this field [27], decidability (and complexity) of reasoning is a crucial issue in systems combining DL KBs and rules. In fact, the interaction does not preserve decidability, i.e., starting from a KB in which reasoning is decidable and a rule KB in which reasoning is decidable, reasoning in the KB obtained by integrating the two components may not be a decidable problem.

In this paper, we study reasoning in description logic knowledge bases augmented with rules expressed in Datalog (and its nonmonotonic extensions).

We start by defining r-hybrid knowledge bases (KBs). A r-hybrid KB has a structural component T, that is a theory in a subset of first-order logic (for instance, a DL knowledge base), and a rule component P. Such a framework is very general, in the sense that:

  • 1.

    the construction is parametric with respect to the structural language, i.e., the logic used to specify the structural component. The only condition imposed by the framework is that the logic is a subset of function-free first-order logic. In particular, any description logic can be chosen as the structural language;

  • 2.

    the rule component is very expressive, since it consists of a Datalog¬ program, i.e., a Datalog program in which negation as failure in the body of rules and disjunction in the head of rules are allowed.

  • 3.

    the rule component is constrained in its interaction with the structural component according to a safeness condition: such a safe interaction between rules and structural KB captures (and is a generalization of) several previous proposals [9], [34], [31].

As a consequence, we are able to show that the framework of r-hybrid KBs comprises many systems proposed for integrating rules and Description Logics.

Then, we study reasoning in r-hybrid KBs. First, we provide a general algorithm for reasoning in r-hybrid KBs, and prove that, very often, decidability of reasoning is preserved when we add safe Datalog¬ rules to a KB: in other words, under very general conditions, if reasoning in the logic L used to specify the structural component T is decidable, then reasoning in the extension of T with safe Datalog¬ rules is still decidable. We also show that an analogous property holds for the complexity of reasoning in r-hybrid KBs.

Our decidability and complexity results generalize in a broad sense previous results shown in [9], [34], [31]. In particular, [31] established decidability of reasoning in the description logic SHOIN enhanced with safe, positive Datalog rules. Our results imply that we can extend such a framework to nonmonotonic Datalog¬ rules and to more expressive, decidable DLs, and preserve decidability of reasoning. Notably, one such DL, SHOIN(D), is equivalent to the Web Ontology language OWL-DL, which is currently playing a crucial role in the Semantic Web initiative [32], since it is a W3C recommendation language for ontology representation in the Semantic Web: therefore, our results immediately imply that extending OWL-DL ontology specifications with safe Datalog¬rules preserves decidability of reasoning.

Finally, our algorithm highlights that reasoning in r-hybrid KBs can be done by strongly separating reasoning about the structural component and reasoning about the rule component. This is a very important property, which allows for reusing deductive techniques (and implemented systems) developed for the structural language and for Datalog¬[12].

The paper is structured as follows. In Section 2 we define r-hybrid KBs. In Section 3 we study reasoning in r-hybrid KBs: we first define an algorithm for satisfiability of r-hybrid KBs, then address decidability and complexity of reasoning with r-hybrid KBs. We discuss related work in Section 4. Finally, we draw some conclusions in Section 5.

Section snippets

Framework

In this section we define syntax and semantics of r-hybrid KBs. We introduce both a monotonic, first-order semantics and a nonmonotonic semantics based on stable models.

Reasoning in r-hybrid KBs

In this section we study reasoning in r-hybrid KBs. In particular, we study satisfiability of r-hybrid KBs, which is the basic reasoning task: as in many other logics, in r-hybrid KBs many important reasoning tasks (e.g., skeptical entailment) can be easily reduced to (un)satisfiability.

In the following, we first define an algorithm for deciding satisfiability of r-hybrid KBs; then, based on such an algorithm, we analyze decidability and complexity of reasoning in r-hybrid KBs, and examine in

Related work

In this section we relate our approach to recent work in integrating ontologies and rules. We divide such studies in two main streams: (i) studies that deal with forms of “safe” (or loose) interaction between the structural and the rule components, and hence close, at least conceptually, to our proposal; (ii) studies concerning forms of “non-safe” (or strict) interaction.

Conclusions

The results presented in this paper can be summarized as follows:

  • we have defined a very general framework for integrating ontologies and rules;

  • we have defined a general, modular reasoning method for r-hybrid KBs;

  • based on such a method, we have shown that the safe combination of decidable first-order KBs and Datalog¬ rules preserves decidability of reasoning, under very general conditions. An analogous general property holds for the complexity of reasoning in r-hybrid KBs;

  • as a byproduct of our

Acknowledgements

The author wishes to thank Diego Calvanese, Maurizio Lenzerini and Daniele Nardi for helpful discussions on this topic. The author is also thankful to the anonymous reviewers for their precious comments. This research has been partially supported by the projects INFOMIX (IST-2001-33570), SEWASIE (IST-2001-34825) and INTEROP Network of Excellence (IST-508011) funded by the EU, by the project “Società dell’Informazione” subproject SP1 “Reti Internet: Efficienza, Integrazione e Sicurezza” funded

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