Facing the growing amount of information in today's society, the task of specifying human knowledge in a way that can be unambiguously processed by computers becomes more and more important. Two acknowledged fields in this evolving scientific area of Knowledge Representation are Description Logics (DL) and Formal Concept Analysis (FCA). While DL concentrates on characterizing domains via logical statements and inferring knowledge from these characterizations, FCA builds conceptual hierarchies on the basis of present data. This work introduces Relational Exploration, a method for acquiring complete relational knowledge about a domain of interest by successively consulting a domain expert without ever asking redundant questions. This is achieved by combining DL and FCA: DL formalisms are used for defining FCA attributes while FCA exploration techniques are deployed to obtain or refine DL knowledge specifications.

Description Logics (DLs) are a class of knowledge representation formalisms that can represent terminological and assertional knowledge using a well-defined semantics. Often, knowledge engineers are experts in their own fields, but not in logics, and require assistance in the process of ontology design. This thesis presents three methods that can extract terminological knowledge from existing data and thereby assist in the design process. They are based on similar formalisms from Formal Concept Analysis (FCA), in particular the Next-Closure Algorithm and Attribute-Exploration. The first of the three methods computes terminological knowledge from the data, without any expert interaction. The two other methods use expert interaction where a human expert can confirm each terminological axiom or refute it by providing a counterexample. These two methods differ only in the way counterexamples are provided.

Description Logics (DLs) are increasingly successful knowledge representation formalisms, useful for any application requiring implicit derivation of knowledge from explicitly known facts. A prominent example domain benefiting from these formalisms since the 1990s is the biomedical field. This area contributes an intangible amount of facts and relations between low- and high-level concepts such as the constitution of cells or interactions between studied illnesses, their symptoms and remedies. DLs are well-suited for handling large formal knowledge repositories and computing inferable coherences throughout such data, relying on their well-founded first-order semantics. In particular, DLs of reduced expressivity have proven a tremendous worth for handling large ontologies due to their computational tractability. In spite of these assets and prevailing influence, classical DLs are not well-suited to adequately model some of the most intuitive forms of reasoning. The capability for abductive reasoning is imperative for any field subjected to incomplete knowledge and the motivation to complete it with typical expectations. When such default expectations receive contradicting evidence, an abductive formalism is able to retract previously drawn, conflicting conclusions. Common examples often include human reasoning or a default characterisation of properties in biology, such as the normal arrangement of organs in the human body. Treatment of such defeasible knowledge must be aware of exceptional cases - such as a human suffering from the congenital condition situs inversus - and therefore accommodate for the ability to retract defeasible conclusions in a non-monotonic fashion. Specifically tailored non-monotonic semantics have been continuously investigated for DLs in the past 30 years. A particularly promising approach, is rooted in the research by Kraus, Lehmann and Magidor for preferential (propositional) logics and Rational Closure (RC). The biggest advantages of RC are its well-behaviour in terms of formal inference postulates and the efficient computation of defeasible entailments, by relying on a tractable reduction to classical reasoning in the underlying formalism. A major contribution of this work is a reorganisation of the core of this reasoning method, into an abstract framework formalisation. This framework is then easily instantiated to provide the reduction method for RC in DLs as well as more advanced closure operators, such as Relevant or Lexicographic Closure. In spite of their practical aptitude, we discovered that all reduction approaches fail to provide any defeasible conclusions for elements that only occur in the relational neighbourhood of the inspected elements. More explicitly, a distinguishing advantage of DLs over propositional logic is the capability to model binary relations and describe aspects of a related concept in terms of existential and universal quantification. Previous approaches to RC (and more advanced closures) are not able to derive typical behaviour for the concepts that occur within such quantification. The main contribution of this work is to introduce stronger semantics for the lightweight DL EL_bot with the capability to infer the expected entailments, while maintaining a close relation to the reduction method. We achieve this by introducing a new kind of first-order interpretation that allocates defeasible information on its elements directly. This allows to compare the level of typicality of such interpretations in terms of defeasible information satisfied at elements in the relational neighbourhood. A typicality preference relation then provides the means to single out those sets of models with maximal typicality. Based on this notion, we introduce two types of nested rational semantics, a sceptical and a selective variant, each capable of deriving the missing entailments under RC for arbitrarily nested quantified concepts. As a proof of versatility for our new semantics, we also show that the stronger Relevant Closure, can be imbued with typical information in the successors of binary relations. An extensive investigation into the computational complexity of our new semantics shows that the sceptical nested variant comes at considerable additional effort, while the selective semantics reside in the complexity of classical reasoning in the underlying DL, which remains tractable in our case.

We make a first step towards adapting an existing approach for privacy-preserving publishing of linked data to Description Logic (DL) ontologies. We consider the case where both the knowledge about individuals and the privacy policies are expressed using concepts of the DL EL , which corresponds to the setting where the ontology is an EL instance store. We introduce the notions of compliance of a concept with a policy and of safety of a concept for a policy, and show how optimal compliant (safe) generalizations of a given EL concept can be computed. In addition, we investigate the complexity of the optimality problem.

Unification in description logics (DLs) has been introduced as a novel inference service that can be used to detect redundancies in ontologies, by finding different concepts that may potentially stand for the same intuitive notion. Together with the special case of matching, they were first investigated in detail for the DL FL0, where these problems can be reduced to solving certain language equations. In this thesis, we extend this service in two directions. In order to increase the recall of this method for finding redundancies, we introduce and investigate the notion of approximate unification, which basically finds pairs of concepts that “almost” unify, in order to account for potential small modelling errors. The meaning of “almost” is formalized using distance measures between concepts. We show that approximate unification in FL0 can be reduced to approximately solving language equations, and devise algorithms for solving the latter problem for particular distance measures. Furthermore, we make a first step towards integrating background knowledge, formulated in so-called TBoxes, by investigating the special case of matching in the presence of TBoxes of different forms. We acquire a tight complexity bound for the general case, while we prove that the problem becomes easier in a restricted setting. To achieve these bounds, we take advantage of an equivalence characterization of FL0 concepts that is based on formal languages. In addition, we incorporate TBoxes in computing concept distances. Even though our results on the approximate setting cannot deal with TBoxes yet, we prepare the framework that future research can build on. Before we journey to the technical details of the above investigations, we showcase our program in the simpler setting of the equational theory ACUI, where we are able to also combine the two extensions. In the course of studying the above problems, we make heavy use of automata theory, where we also derive novel results that could be of independent interest.

Concrete domains have been introduced in the area of Description Logic to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. Unfortunately, in the presence of general concept inclusions (GCIs), which are supported by all modern DL systems, adding concrete domains may easily lead to undecidability. To regain decidability of the DL ALC in the presence of GCIs, quite strong restrictions, in sum called ω-admissibility, were imposed on the concrete domain. On the one hand, we generalize the notion of ω-admissibility from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate ω-admissibility to well-known notions from model theory. In particular, we show that finitely bounded homogeneous structures yield ω-admissible concrete domains. This allows us to show ω-admissibility of concrete domains using existing results from model theory. When integrating concrete domains into lightweight DLs of the E L family, achieving decidability is not enough. One wants reasoning in the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. We investigate padmissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical p-admissible concrete domain based on the rational numbers. Althoughω-admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into ALC yields decidable DLs.

The description logic ALCQI extends the 'standard' description logic ALC by qualifying number restrictions and converse roles. We show that concept satisfiability for this DL is still decidable in polynomial space. The presented algorithm combines techniques from [Tob99] to deal with qualifying number restrictions and from [HST99] to deal with converse roles.

Aus der Einleitung: „As widely argued [Horrocks&Gough,1997; Sattler,1996], transitive roles play an important role in the adequate representation of aggregated objects: they allow these objects to be described by referring to their parts without specifying a level of decomposition. In [Horrocks&Gough,1997], the Description Logic (DL) ALCHR+ is presented, which extends ALC with transitive roles and a role hierarchy. It is argued in [Sattler,1998] that ALCHR+ is well-suited to the representation of aggregated objects in applications that require various part-whole relations to be distinguished, some of which are transitive. However, ALCHR+ allows neither the description of parts by means of the whole to which they belong, or vice versa. To overcome this limitation, we present the DL ALCHIR+ which allows the use of, for example, has part as well as is part of. To achieve this, ALCHR+ was extended with inverse roles. It could be argued that, instead of defining yet another DL, one could make use of the results presented in [De Giacomo&Lenzerini,1996] and use ALC extended with role expressions which include transitive closure and inverse operators. The reason for not proceeding like this is the fact that transitive roles can be implemented more eficiently than the transitive closure of roles (see [Horrocks&Gough,1997]), although they lead to the same complexity class (ExpTime-hard) when added, together with role hierarchies, to ALC. Furthermore, it is still an open question whether the transitive closure of roles together with inverse roles necessitates the use of the cut rule [DeGiacomo&Massacci,1998], and this rule leads to an algorithm with very bad behaviour. We will present an algorithm for ALCHIR+ without such a rule.” / An abridged version will appear in the Proceedings of the International Workshop on Description Logics, Trento, Italy, 1998.

Matching of concepts against patterns is a new inference task in Description Logics, which was originally motivated by applications of the CLASSIC system. Consequently, the work on this problem was until now mostly concerned with sublanguages of the Classic language, which does not allow for existential restrictions. Motivated by an application in chemical process engineering, which requires a description language with existential restrictions, this paper investigates the matching problem in Description Logics with existential restrictions. It turns out that existential restrictions make matching more complex in two respects. First, whereas matching in sublanguages of CLASSIC is polynomial, deciding the existence of matchers is an NP-complete problem in the presence of existential restrictions. Second, whereas in sublanguages of Classic solvable matching problems have a unique least matcher, this is not the case for languages with existential restrictions. Thus, it is not a priori clear which of the (possibly infinitely many) matchers should be returned by a matching algorithm. After determining the complexity of the decision problem, the present paper first investigates the question of what are 'interesting' sets of matchers, and then describes algorithms for computing these sets for the languages EL (which allows for conjunction and existential restrictions) and ALE (which additionally allows for value restrictions, primitive negation, and the bottom concept). / An abridged version of this technical report has been submitted to KR 2000.

We investigate the computational complexity of several decision, enumeration and counting problems related to pseudo-intents. We show that given a formal context and a set of its pseudo-intents, checking whether this context has an additional pseudo-intent is in conp and it is at least as hard as checking whether a given simple hypergraph is saturated. We also show that recognizing the set of pseudo-intents is also in conp and it is at least as hard as checking whether a given hypergraph is the transversal hypergraph of another given hypergraph. Moreover, we show that if any of these two problems turns out to be conp-hard, then unless p = np, pseudo-intents cannot be enumerated in output polynomial time. We also investigate the complexity of finding subsets of a given Duquenne-Guigues Base from which a given implication follows. We show that checking the existence of such a subset within a specified cardinality bound is np-complete, and counting all such minimal subsets is #p-complete.