Combining Description Logics, Description Graphs, and Rules

Abstract

The Web Ontology Language (OWL) is a well-known language for ontology modeling in the Semantic Web [9]. The World Wide Web Consortium (W3C) is currently working on a revision of OWL—called OWL 2 [2]—whose main goal is to address some of the limitations of OWL. The formal underpinnings of OWL and OWL 2 are provided by description logics (DLs)[1]–knowledge representation formalisms with well-understood formal properties.

DLs are often used to describe structured objects—objects whose parts are interconnected in complex ways. Such objects abound in molecular biology and the clinical sciences, and clinical ontologies such as GALEN, the Foundational Model of Anatomy (FMA), and the National Cancer Institute (NCI) Thesaurus describe numerous structured objects. For example, FMA models the human hand as consisting of the fingers, the palm, various bones, blood vessels, and so on, all of which are highly interconnected.

Modeling structured objects poses numerous problems to DLs and the OWL family of languages. The design of DLs has been driven by the desire to provide practically useful knowledge modeling primitives while ensuring decidability of the core reasoning problems. To achieve the latter goal, the modeling constructs available in DLs are usually carefully crafted so that the resulting language exhibits a variant of the tree-model property [10]: each satisfiable DL ontology always has at least one model whose elements are connected in a tree-like manner. This property can be used to derive a decision procedure; however, it also prevents one from accurately describing (usually non-tree-like) structured objects since, whenever a model exists, at least one model does not reflect the intended structure. This technical problem has severe consequences in practice [6]. In search of the “correct” way of describing structured objects, modelers often create overly complex descriptions; however, since the required expressive power is actually missing, such descriptions do not entail the consequences that would follow if the descriptions accurately captured the intended structure.

In order to address this lack of expressivity, we extended DLs with description graphs, which can be understood as schema-level descriptions of structured objects. To allow for the representation of conditional statements about structured objects, we also incorporated first-order rules [3] into our extension. In this way we obtain a powerful and versatile knowledge representation formalism. It allows us, for example, to describe the structure of the hand using description graphs, statements such as “if a bone in the hand is fractured, then the hand is fractured as well” using rules, and nonstructural aspects of the domain such as “a medical doctor is a person with an MD degree” using DLs.

To study the computational properties of our formalism, we base the DL component on the $\mathcal{SHOIQ}^+$ description logic, as this DL provides the semantic underpinning of OWL 2. The resulting formalism is quite expressive, and it is unsurprising that it is undecidable. We investigate restrictions under which the formalism becomes decidable. In particular, we have observed that structured objects can often be described by a possibly large, yet bounded number of parts. For example, a human body consists of organs all of which can be decomposed into smaller parts; however, further decomposition will eventually lead to parts that one does not want or know how to describe any further. In this vein, FMA describes the skeleton of the hand, but it does not describe the internal structure of the distal phalanges of the fingers. The number of parts needed to describe the hand is therefore determined by the granularity of the hierarchical decomposition of the hand. This decomposition naturally defines an acyclic hierarchy of description graphs. For example, the fingers can be described by description graphs that are subordinate to that of the hand; however, the description graph for the hand is not naturally subordinate to the description graphs for the fingers. We used this observation to define an acyclicity restriction on description graphs. Acyclicity bounds the number of parts that one needs to reason with, which, provided that there are no DL axioms, can be used to obtain a decision procedure for the basic reasoning problems.

If description graphs are used in combination with DL axioms, the acyclicity condition alone does not ensure decidability due to possible interactions between DL axioms, graphs, and rules [5]. To obtain decidability, we limit this interaction by imposing an additional condition on the usage of roles: the roles (i.e., the binary predicates) that can be used in DL axioms must be separated from the roles that can be used in rules. We developed a hypertableau-based [7] reasoning algorithm that decides the satisfiability problem for our formalism, together with tight complexity bounds.

All proofs and additional decidability and complexity results for the case when DL axioms are expressed in $\mathcal{SHOIQ}^+$ can be found in [8].