Constructing and Extending Description Logic Ontologies using Methods of Formal Concept Analysis
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DISSERTATION AND HABILITATION ABSTRACTS
Constructing and Extending Description Logic Ontologies using Methods of Formal Concept Analysis A Dissertation Summary Francesco Kriegel1 Received: 24 January 2020 / Accepted: 6 June 2020 © The Author(s) 2020
Abstract My thesis describes how methods from Formal Concept Analysis can be used for constructing and extending description logic ontologies. In particular, it is shown how concept inclusions can be axiomatized from data in the description logics EL , M , 𝖧𝗈𝗋𝗇-M , and 𝖯𝗋𝗈𝖻-EL . All proposed methods are not only sound but also complete, i.e., the result not only consists of valid concept inclusions but also entails each valid concept inclusion. Moreover, a lattice-theoretic view on the description logic EL is provided. For instance, it is shown how upper and lower neighbors of EL concept descriptions can be computed and further it is proven that the set of EL concept descriptions forms a graded lattice with a non-elementary rank function. Keywords Description logic · Formal concept analysis · Axiomatization · Concept inclusion
1 Introduction Description Logic (abbrv. DL) [1] belongs to the field of knowledge representation and reasoning. DL researchers have developed a large family of logic-based languages, socalled 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 [1]: 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) [5] in the Semantic Web. * Francesco Kriegel francesco.kriegel@tu‑dresden.de 1
Formal Concept Analysis (abbrv. FCA) [3] is 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 [3]. 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 [4], which furthermore contains a minimal number of implicat
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