Model-theoretic characterisations of description logics

The growing need for computer aided processing of knowledge has led to an increasing interest in description logics (DLs), which are applied to encode knowledge in order to make it explicit and accessible to logical reasoning. DLs and in particular the family around the DL ALC have therefore been thoroughly investigated w.r.t. their complexity theory and proof theory. The question arises which expressiveness these logics actually have. The expressiveness of a logic can be inferred by a model theoretic characterisation. On concept level, these DLs are akin to modal logics whose model theoretic properties have been investigated. Yet the model theoretic investigation of the DLs with their TBoxes, which are an original part of DLs usually not considered in context of modal logics, have remained unstudied. This thesis studies the model theoretic properties of ALC, ALCI, ALCQ, as well as ALCO, ALCQO, ALCQIO and EL. It presents model theoretic properties, which characterise these logics as fragments of the first order logic (FO). The characterisations are not only carried out on concept level and on concept level extended by the universal role, but focus in particular on TBoxes. The properties used to characterise the logics are `natural' notions w.r.t. the logic under investigation: On the concept-level, each of the logics is characterised by an adapted form of bisimulation and simulation, respectively. TBoxes of ALC, ALCI and ALCQ are characterised as fragments of FO which are invariant under global bisimulation and disjoint unions. The logics ALCO, ALCQO and ALCQIO, which incorporate individuals, are characterised w.r.t. to the class K of all interpretations which interpret individuals as singleton sets. The characterisations for TBoxes of ALCO and ALCQO both require, additionally to being invariant under the appropriate notion of global bisimulation and an adapted version of disjoint unions, that an FO-sentence is, under certain circumstances, preserved under forward generated subinterpretations. FO-sentences equivalent to ALCQIO-TBoxes, are - due to ALCQIO's inverse roles - characterised similarly to ALCO and ALCQO but have as third additional requirement that they are preserved under generated subinterpretations. EL as sub-boolean DL is characterised on concept level as the FO-fragment which is preserved under simulation and preserved under direct products. Equally valid is the characterisation by being preserved under simulation and having minimal models. For EL-TBoxes, a global version of simulation was not sufficient but FO-sentences of EL-TBoxes are invariant under global equi-simulation, disjoint unions and direct products. For each of these description logics, the characteristic concepts are explicated and the characterisation is accompanied by an investigation under which notion of saturation the logic in hand enjoys the Hennessy-and-Milner-Property. As application of the results we determine the minimal globally bisimilar companion w.r.t. ALCQO-bisimulation and introduce the L1-to-L2-rewritability problem for TBoxes, where L1 and L2 are (description) logics. The latter is the problem to decide whether or not an L1-TBox can be equivalently expressed as L2-TBox. We give algorithms which decide ALCI-to-ALC-rewritability and ALC-to-EL-rewritability

Computing Crisp Bisimulations for Fuzzy Structures

Fuzzy structures such as fuzzy automata, fuzzy transition systems, weighted social networks and fuzzy interpretations in fuzzy description logics have been widely studied. For such structures, bisimulation is a natural notion for characterizing indiscernibility between states or individuals. There are two kinds of bisimulations for fuzzy structures: crisp bisimulations and fuzzy bisimulations. While the latter fits to the fuzzy paradigm, the former has also attracted attention due to the application of crisp equivalence relations, for example, in minimizing structures. Bisimulations can be formulated for fuzzy labeled graphs and then adapted to other fuzzy structures. In this article, we present an efficient algorithm for computing the partition corresponding to the largest crisp bisimulation of a given finite fuzzy labeled graph. Its complexity is of order $O((m\log{l} + n)\log{n})$, where $n$, $m$ and $l$ are the number of vertices, the number of nonzero edges and the number of different fuzzy degrees of edges of the input graph, respectively. We also study a similar problem for the setting with counting successors, which corresponds to the case with qualified number restrictions in description logics and graded modalities in modal logics. In particular, we provide an efficient algorithm with the complexity $O((m\log{m} + n)\log{n})$ for the considered problem in that setting

Most specific consequences in the description logic EL

The notion of a most specific consequence with respect to some terminological box is introduced, conditions for its existence in the description logic EL and its variants are provided, and means for its computation are developed. Algebraic properties of most specific consequences are explored. Furthermore, several applications that make use of this new notion are proposed and, in particular, it is shown how given terminological knowledge can be incorporated in existing approaches for the axiomatization of observations. For instance, a procedure for an incremental learning of concept inclusions from sequences of interpretations is developed

Constructing and Extending Description Logic Ontologies using Methods of Formal Concept Analysis

Description Logic (abbrv. DL) belongs to the field of knowledge representation and reasoning. DL researchers have developed a large family of logic-based languages, so-called description logics (abbrv. DLs). These logics allow their users to explicitly represent knowledge as ontologies, which are finite sets of (human- and machine-readable) axioms, and provide them with automated inference services to derive implicit knowledge. The landscape of decidability and computational complexity of common reasoning tasks for various description logics has been explored in large parts: there is always a trade-off between expressibility and reasoning costs. It is therefore not surprising that DLs are nowadays applied in a large variety of domains: agriculture, astronomy, biology, defense, education, energy management, geography, geoscience, medicine, oceanography, and oil and gas. Furthermore, the most notable success of DLs is that these constitute the logical underpinning of the Web Ontology Language (abbrv. OWL) in the Semantic Web. Formal Concept Analysis (abbrv. FCA) is a subfield of lattice theory that allows to analyze data-sets that can be represented as formal contexts. Put simply, such a formal context binds a set of objects to a set of attributes by specifying which objects have which attributes. There are two major techniques that can be applied in various ways for purposes of conceptual clustering, data mining, machine learning, knowledge management, knowledge visualization, etc. On the one hand, it is possible to describe the hierarchical structure of such a data-set in form of a formal concept lattice. On the other hand, the theory of implications (dependencies between attributes) valid in a given formal context can be axiomatized in a sound and complete manner by the so-called canonical base, which furthermore contains a minimal number of implications w.r.t. the properties of soundness and completeness. In spite of the different notions used in FCA and in DLs, there has been a very fruitful interaction between these two research areas. My thesis continues this line of research and, more specifically, I will describe how methods from FCA can be used to support the automatic construction and extension of DL ontologies from data