Patrick Koopmann

Uniform Interpolation of \(\mathcal{ALC}\)-Ontologies Using Fixpoints

We present a method to compute uniform interpolants with fixpoints for ontologies specified in the description logic \(\mathcal{ALC}\). The aim of uniform interpolation is to reformulate an ontology such that it only uses a specified set of symbols, while preserving consequences that involve these symbols. It is known that in \(\mathcal{ALC}\) uniform interpolants cannot always be finitely represented. Our method computes uniform interpolants for the target language \(\mathcal{ALC}\mu\), which is \(\mathcal{ALC}\) enriched with fixpoint operators, and always computes a finite representation. If the result does not involve fixpoint operators, it is the uniform interpolant in \(\mathcal{ALC}\). The method focuses on eliminating concept symbols and combines resolution-based reasoning with an approach known from the area of second-order quantifier elimination to introduce fixpoint operators when needed. If fixpoint operators are not desired, it is possible to approximate the interpolant.

Count and Forget: Uniform Interpolation of \mathcal{SHQ}-Ontologies

We propose a method for forgetting concept symbols and non-transitive roles symbols of \(\mathcal{SHQ}\)-ontologies, or for computing uniform interpolants in \(\mathcal{SHQ}\). Uniform interpolants restrict the symbols occuring in an ontology to a specified set, while preserving all logical entailments that can be expressed using this set in the description logic under consideration. Uniform interpolation has applications in ontology reuse, information hiding and ontology analysis, but so far no method for computing uniform interpolants for expressive description logics with number restrictions has been developed. Our results are not only interesting because they allow to compute uniform interpolants of ontologies using a more expressive language. Using number restrictions also allows to preserve more information in uniform interpolants of ontologies in less complex logics, such as \(\mathcal{ALC}\) or \(\mathcal{EL}\). The presented method computes uniform interpolants on the basis of a new resolution calculus for \(\mathcal{SHQ}\). The output of our method is expressed using \(\mathcal{SHQ}\mu\), which is \(\mathcal{SHQ}\) extended with fixpoint operators, to always enable a finite representation of the uniform interpolant. If the uniform interpolant uses fixpoint operators, it can be represented in \(\mathcal{SHQ}\) without fixpoints operators using additional concept symbols or by approximation.

Forgetting Concept and Role Symbols in \(\mathcal{ALCH}\)-Ontologies

We develop a resolution-based method for forgetting concept and role symbols in \(\mathcal{ALCH}\) ontologies, or for computing uniform interpolants in \(\mathcal{ALCH}\). Uniform interpolants use only a restricted set of symbols, while preserving logical consequences of the original ontology involving these symbols. While recent work towards practical methods for uniform interpolation in expressive description logics limits attention to forgetting concept symbols, we believe most applications would benefit from the possibility to forget both role and concept symbols. We focus on the description logic \(\mathcal{ALCH}\), which allows for the formalisation of role hierarchies. Our approach is based on a recently developed resolution-based calculus for forgetting concept symbols in \(\mathcal{ALC}\) ontologies, which we extend by redundancy elimination techniques to make it practical for larger ontologies. Experiments on \(\mathcal{ALCH}\) fragments of real life ontologies suggest that our method is applicable in a lot of real-life applications.