Jos de Bruijn, Free University of Bozen-Bolzano, Italy
Dissertation defended at the University of Innsbruck in April 2008.
With this dissertation we address the problem of combining the ontology and rule representation paradigms and the meta-modeling feature for the semantic Web. We assume that ontologies are typically represented using Description Logics (e.g., OWL DL) and rules are represented using nonmonotonic logic programs (LP). With meta-modeling we mean the possibility to use the same identifier as individual, class, and property. We assume that a given ontology and a given ruleset may be concerned
with the same domain, and thus we cannot restrict the use of particular predicates in either the ontology or the ruleset. For example, we cannot distinguish between “rule” and “ontology” predicates.
We observe that there is an important difference between the LP (and database) and DL worlds in the way integrity constraints are typically handled. Take as an example the constraint “every value of the property hasChild is a member of the class Person”. In the LP world, this constraint is violated if it is not known that a child is a person. In the DL world, the constraint is only violated it is known that the child is not a person; otherwise, it is derived that every child is a person. In technical terms, the approaches are very similar: an interpretation that violates the constraint is not the model. The difference stems from the fact that in the LP world, one considers only the minimal or preferred models, whereas in the DL world one considers all possible models.
We devised a Web language, called WSML, which allows expressing rules, as well as Description Logic — even arbitrary first-order logic — axioms, which allows meta-modeling (i.e., classes-as-instances), and which incorporates both kinds of above-mentioned constraints. We called the complete language WSML-Full and identified two subsets based on the DL and LP paradigms, called WSML-DL and WSML-Rule, respectively. The subsets correspond to, respectively, the description logic SHIQ(D) and logic programming with negation under the stable model semantics with
F-Logic extensions for modeling classes and properties. In turn, WSML-DL and WSML-Rule have a common subset, called WSML-Core.
It was clear how to define the semantics of WSML-DL and WSML-Rule such that existing DL and LP reasoners can be used for reasoning with these subsets. Two major challenges remained, namely, the difference between the DL-style and F-Logic-style of ontology modeling and the combination of theories in classical logic (i.e., DL ontologies) and nonmonotonic logic programs. The former needed to be solved in order to arrive to an adequate definition of the semantics of WSML-Core and for uncovering the precise relationship between WSML-Core and WSML-DL, respectively -Rule, thereby allowing interaction between the two paradigms through a common subset. The latter needed to be solved to obtain a semantics for WSML-Full, allowing interaction between the two paradigms through a common superset.
To address the first challenge, we studied a straightforward translation t() from standard FOL to F-Logic, where unary atoms A(x) are translated to membership formulas x:A and binary atoms R(x,y) are translated to property value statements x[R -> y]. We first observed that this translation does not preserve entailment in the general case. A translated theory may have strictly more entailments than the original one. However, we showed that if a theory is cardinal, i.e., we can essentially assume that every interpretation has at least as many elements as there are symbols in the language, the translation does preserve entailment. We then showed that the novel class of E-safe theories is cardinal. It turns out that the DL SHOIQ is not DL-safe, but SHIQ (and thus WSML-DL) is. This shows that one can take a horn subset of SHIQ (i.e., WSML-Core) and its straightforward translation into F-Logic (essentially making it a subset of WSML-Rule) preserves entailment. Therefore, WSML-DL and WSML-Rule are proper extensions of WSML-Core.
To address the second challenge, we studied embeddings into an expressive nonmonotonic logic, called first-order autoepistemic logic (FO-AEL). The combination of classical logic ontologies and nonmonotonic logic program can be realized through embedding into FO-AEL. As FO-AEL extends FOL, such theories can be trivially embedded. We then extended a number of embeddings of ground logic programs to the non-ground case, and studied these embeddings. We uncovered the correspondences and differences between the embeddings under combination with various classes of FOL theories. These differences and correspondences provide a useful guide to selecting a particular embedding in a particular combination scenario. WSML-Full may be realized as the set theoretic union of a classical logic theory and one of the studied embeddings of a logic program.
To put all the pieces together, we defined an extension of FO-AEL with F-Logic constructs and concrete domains to serve as a semantic framework for all of WSML. We showed that the Core, Rule, DL, and Full fragments of WSML can be seen as subsets of this language, by exploiting the earlier results about the relationship between FOL and F-Logic and the embeddings of logic programs. Finally, we showed the properties of language layering, i.e., that every Core entailment is a Rule entailment and a DL entailment, and every Rule or DL entailment is a Full entailment.
The full thesis is available at: