Francesco Kriegel

We propose an interactive repair method for the description logic \(\mathcal{EL}\) that is based on the optimal-repair framework. The obtained repair might not be optimal in the theoretical sense, i.e. more than a minimal amount of consequences might have been removed—but from a practical perspective it is superior to a theoretically optimal repair as the interaction strategy enables the users to identify further faulty consequences connected to the initially reported errors.

@inproceedings{ Kr-KRRSAC2025,
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 40th ACM/SIGAPP Symposium On Applied Computing {(SAC 2025)}, March 31 -- April 4, 2025, Sicily, Italy},
  doi = {https://doi.org/10.1145/3672608.3707750},
  pages = {1019--1026},
  publisher = {Association for Computing Machinery},
  title = {Beyond Optimal: Interactive Identification of Better-than-optimal Repairs},
  year = {2025},
}

The \(\mathcal{EL}\) family of description logics facilitates efficient polynomial-time reasoning and has been standardized as the profile OWL 2 EL of the Web Ontology Language. \(\mathcal{EL}\) can represent and reason not only with symbolic knowledge but also with concrete knowledge expressed by numbers, strings, and other concrete datatypes. Such concrete domains must be convex to avoid introducing disjunctions “through the backdoor.” However, existing concrete domains provide only limited utility. In order to overcome this issue, we introduce a novel form of concrete domains based on semi-lattices. They are convex by design and can thus be integrated into Horn-DLs such as \(\mathcal{EL}\). Moreover, they allow for FBoxes to express dependencies between concrete features. We describe four instantiations concerned with real intervals, 2D-polygons, regular languages, and graphs.

@inproceedings{ Kr-FroCoS-2025,
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 15th International Symposium on Frontiers of Combining Systems {(FroCoS 2025)}},
  note = {To appear.},
  title = {Reasoning in OWL 2 EL with Hierarchical Concrete Domains},
  year = {2025},
}

Errors in knowledge bases (KBs) written in a Description Logic (DL) are usually detected when reasoning derives an inconsistency or a consequence that does not hold in the application domain modelled by the KB. Whereas classical repair approaches produce maximal subsets of the KB not implying the inconsistency or unwanted consequence, optimal repairs maximize the consequence sets. In this paper, we extend previous results on how to compute optimal repairs from the DL \(\mathcal{EL}\) to its extension \(\mathcal{EL}^{\bot}\), which in contrast to \(\mathcal{EL}\) can express inconsistency. The problem of how to deal with inconsistency in the context of optimal repairs was addressed previously, but in a setting where the (fixed) terminological part of the KB must satisfy a restriction on cyclic dependencies. Here, we consider a setting where this restriction is not required. We also show how the notion of optimal repairs obtained this way can be used in inconsistency- and error-tolerant reasoning.

@inproceedings{ BaKrNu-FoIKS2024,
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {Proceedings of the 13th International Symposium on Foundations of Information and Knowledge Systems {(FoIKS 2024)}, April 8--11, 2024, Sheffield, United Kingdom},
  doi = {https://doi.org/10.1007/978-3-031-56940-1_1},
  pages = {3--22},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {Inconsistency- and Error-Tolerant Reasoning w.r.t.\ Optimal Repairs of $\mathcal{EL}^{\bot}$ Ontologies},
  volume = {14589},
  year = {2024},
}

We present an FCA-based axiomatization method that produces a complete \(\mathcal{EL}\) TBox (the terminological part of an OWL 2 EL ontology) from a graph dataset in at most exponential time. We describe technical details that allow for efficient implementation as well as variations that dispense with the computation of extremely large axioms, thereby rendering the approach applicable albeit some completeness is lost. Moreover, we evaluate the prototype on real-world datasets.

@inproceedings{ Kr-AAAI2024,
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 38th Annual AAAI Conference on Artificial Intelligence {(AAAI 2024)}, February 20--27, 2024, Vancouver, Canada},
  doi = {https://doi.org/10.1609/aaai.v38i9.28930},
  pages = {10597--10606},
  title = {Efficient Axiomatization of OWL 2 EL Ontologies from Data by means of Formal Concept Analysis},
  year = {2024},
}

@inproceedings{ Kr-DL2024,
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 37th International Workshop on Description Logics ({DL} 2024), Bergen, Norway, June 18--21, 2024},
  publisher = {CEUR-WS.org},
  series = {{CEUR} Workshop Proceedings},
  title = {Efficient Axiomatization of OWL 2 EL Ontologies from Data by means of Formal Concept Analysis (Extended Abstract)},
  volume = {3739},
  year = {2024},
}

Ontologies based on Description Logics may contain errors, which are usually detected when reasoning produces consequences that follow from the ontology, but do not hold in the modelled application domain. In previous work, we have introduced repair approaches for \(\mathcal{EL}\) ontologies that are optimal in the sense that they preserve a maximal amount of consequences. In this paper, we will, on the one hand, review these approaches, but with an emphasis on motivation rather than on technical details. On the other hand, we will describe new results that address the problems that optimal repairs may become very large or need not even exist unless strong restrictions on the terminological part of the ontology apply. We will show how one can deal with these problems by introducing concise representations of optimal repairs.

@inproceedings{ BaKoKr-JELIA2023,
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel}},
  booktitle = {Proceedings of the 18th European Conference on Logics in Artificial Intelligence {(JELIA 2023)}, September 20--22, 2023, Dresden, Germany},
  doi = {https://doi.org/10.1007/978-3-031-43619-2_2},
  pages = {11--34},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {Optimal Repairs in the Description Logic $\mathcal{EL}$ Revisited},
  volume = {14281},
  year = {2023},
}

@inproceedings{ Kr-KR2023RPR,
  author = {Francesco {Kriegel}},
  booktitle = {Recently Published Research {(RPR)} track of the 20th International Conference on Principles of Knowledge Representation and Reasoning {(KR 2023)}, September 2--8, 2023, Rhodes, Greece},
  doi = {https://doi.org/10.5281/zenodo.8341194},
  title = {Optimal Fixed-Premise Repairs of $\mathcal{EL}$ {TBoxes} (Extended Abstract)},
  year = {2023},
}

Errors in Description Logic (DL) ontologies are often detected when a reasoner computes unwanted consequences. The question is then how to repair the ontology such that the unwanted consequences no longer follow, but as many of the other consequences as possible are preserved. The problem of computing such optimal repairs was addressed in our previous work in the setting where the data (expressed by an ABox) may contain errors, but the schema (expressed by an \(\mathcal{EL}\) TBox) is assumed to be correct. Actually, we consider a generalization of ABoxes called quantified ABoxes (qABoxes) both as input for and as result of the repair process. Using qABoxes for repair allows us to retain more information, but the disadvantage is that standard DL systems do not accept qABoxes as input. This raises the question, investigated in the present paper, whether and how one can obtain optimal repairs if one restricts the output of the repair process to being ABoxes. In general, such optimal ABox repairs need not exist. Our main contribution is that we show how to decide the existence of optimal ABox repairs in exponential time, and how to compute all such repairs in case they exist.

@inproceedings{ BaKoKrNu-ESWC2022,
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {19th Extended Semantic Web Conference, {ESWC} 2022, Hersonissos, Greece, May 29 -- June 2, 2022, Proceedings},
  doi = {https://doi.org/10.1007/978-3-031-06981-9_8},
  pages = {130--146},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {Optimal ABox Repair w.r.t.\ Static {$\mathcal{EL}$} TBoxes: from Quantified ABoxes back to ABoxes},
  volume = {13261},
  year = {2022},
}

Ontologies based on Description Logic (DL) represent general background knowledge in a terminology (TBox) and the actual data in an ABox. DL systems can then be used to compute consequences (such as answers to certain queries) from an ontology consisting of a TBox and an ABox. Since both human-made and machine-learned data sets may contain errors, which manifest themselves as unintuitive or obviously incorrect consequences, repairing DL-based ontologies in the sense of removing such unwanted consequences is an important topic in DL research. Most of the repair approaches described in the literature produce repairs that are not optimal, in the sense that they do not guarantee that only a minimal set of consequences is removed. In a series of papers, we have developed an approach for computing optimal repairs, starting with the restricted setting of an \(\mathcal{EL}\) instance store, extending this to the more general setting of a quantified ABox (where some individuals may be anonymous), and then adding a static \(\mathcal{EL}\) TBox.

Here, we extend the expressivity of the underlying DL considerably, by adding nominals, inverse roles, regular role inclusions and the bottom concept to \(\mathcal{EL}\), which yields a fragment of the well-known DL Horn-\(\mathcal{SROIQ}\). The ideas underlying our repair approach still apply to this DL, though several non-trivial extensions are needed to deal with the new constructors and axioms. The developed repair approach can also be used to treat unwanted consequences expressed by certain conjunctive queries or regular path queries, and to handle Horn-\(\mathcal{ALCOI}\) TBoxes with regular role inclusions.

@inproceedings{ BaKr-KR2022,
  author = {Franz {Baader} and Francesco {Kriegel}},
  booktitle = {Proceedings of the 19th International Conference on Principles of Knowledge Representation and Reasoning, {KR} 2022, Haifa, Israel, July 31 -- August 5, 2022},
  doi = {https://doi.org/10.24963/kr.2022/3},
  editor = {Gabriele {Kern{-}Isberner} and Gerhard {Lakemeyer} and Thomas {Meyer}},
  pages = {22--32},
  title = {Pushing Optimal ABox Repair from {$\mathcal{EL}$} Towards More Expressive Horn-DLs},
  year = {2022},
}

Ontologies based on Description Logic (DL) represent general background knowledge in a terminology (TBox) and the actual data in an ABox. Both human-made and machine-learned data sets may contain errors, which are usually detected when the DL reasoner returns unintuitive or obviously incorrect answers to queries. To eliminate such errors, classical repair approaches offer as repairs maximal subsets of the ABox not having the unwanted answers w.r.t. the TBox. It is, however, not always clear which of these classical repairs to use as the new, corrected data set. Error-tolerant semantics instead takes all repairs into account: cautious reasoning returns the answers that follow from all classical repairs whereas brave reasoning returns the answers that follow from some classical repair. It is inspired by inconsistency-tolerant reasoning and has been investigated for the DL \(\mathcal{EL}\), but in a setting where the TBox rather than the ABox is repaired. In a series of papers, we have developed a repair approach for ABoxes that improves on classical repairs in that it preserves a maximal set of consequences (i.e., answers to queries) rather than a maximal set of ABox assertions. The repairs obtained by this approach are called optimal repairs. In the present paper, we investigate error-tolerant reasoning in the DL \(\mathcal{EL}\), but we repair the ABox and use optimal repairs rather than classical repairs as the underlying set of repairs. To be more precise, we consider a static \(\mathcal{EL}\) TBox (which is assumed to be correct), represent the data by a quantified ABox (where some individuals may be anonymous), and use \(\mathcal{EL}\) concepts as queries (instance queries). We show that brave entailment of instance queries can be decided in polynomial time. Cautious entailment can be decided by a coNP procedure, but is still in P if the TBox is empty.

@inproceedings{ BaKrNu-RuleML2022,
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {Rules and Reasoning - 6th International Joint Conference, RuleML+RR 2022, Virtual, September 26-28, 2022, Proceedings},
  doi = {https://doi.org/10.1007/978-3-031-21541-4_15},
  editor = {Guido {Governatori} and Anni{-}Yasmin {Turhan}},
  pages = {227--243},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {Error-Tolerant Reasoning in the Description Logic {$\mathcal{EL}$} Based On Optimal Repairs},
  volume = {13752},
  year = {2022},
}

Reasoners can be used to derive implicit consequences from an ontology. Sometimes unwanted consequences are revealed, indicating errors or privacy-sensitive information, and the ontology needs to be appropriately repaired. The classical approach is to remove just enough axioms such that the unwanted consequences vanish. However, this is often too rough since mere axiom deletion also erases many other consequences that might actually be desired. The goal should not be to remove a minimal number of axioms but to modify the ontology such that only a minimal number of consequences is removed, including the unwanted ones. Specifically, a repair should rather be logically entailed by the input ontology, instead of being a subset. To this end, we introduce a framework for computing fixed-premise repairs of \(\mathcal{EL}\) TBoxes. In the first variant the conclusions must be generalizations of those in the input TBox, while in the second variant no such restriction is imposed. In both variants, every repair is entailed by an optimal one and, up to equivalence, the set of all optimal repairs can be computed in exponential time. A prototypical implementation is provided. In addition, we show new complexity results regarding gentle repairs.

@inproceedings{ Kr-KI2022,
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 45th German Conference on Artificial Intelligence ({KI} 2022), Virtual in Trier, Germany, September 19--23, 2022},
  doi = {https://doi.org/10.1007/978-3-031-15791-2_11},
  editor = {Ralph {Bergmann} and Lukas {Malburg} and Stephanie C. {Rodermund} and Ingo J. {Timm}},
  pages = {115--130},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {Optimal Fixed-Premise Repairs of $\mathcal{EL}$ {TBoxes}},
  volume = {13404},
  year = {2022},
}

The application of automated reasoning approaches to Description Logic (DL) ontologies may produce certain consequences that either are deemed to be wrong or should be hidden for privacy reasons. The question is then how to repair the ontology such that the unwanted consequences can no longer be deduced. An optimal repair is one where the least amount of other consequences is removed. Most of the previous approaches to ontology repair are of a syntactic nature in that they remove or weaken the axioms explicitly present in the ontology, and thus cannot achieve semantic optimality. In previous work, we have addressed the problem of computing optimal repairs of (quantified) ABoxes, where the unwanted consequences are described by concept assertions of the light-weight DL \(\mathcal{E\!L}\). In the present paper, we improve on the results achieved so far in two ways. First, we allow for the presence of terminological knowledge in the form of an \(\mathcal{E\!L}\) TBox. This TBox is assumed to be static in the sense that it cannot be changed in the repair process. Second, the construction of optimal repairs described in our previous work is best case exponential. We introduce an optimized construction that is exponential only in the worst case. First experimental results indicate that this reduces the size of the computed optimal repairs considerably.

@inproceedings{ BaKoKrNu-CADE2021,
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {Proceedings of the 28th International Conference on Automated Deduction (CADE-28), July 11--16, 2021, Virtual Event, United States},
  doi = {https://doi.org/10.1007/978-3-030-79876-5_18},
  editor = {Andr{\'e} {Platzer} and Geoff {Sutcliffe}},
  pages = {309--326},
  series = {Lecture Notes in Computer Science},
  title = {{Computing Optimal Repairs of Quantified ABoxes w.r.t. Static $\mathcal{E\!L}$ TBoxes}},
  volume = {12699},
  year = {2021},
}

@inproceedings{ BaKoKrNu-KR2021,
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {Recent Published Research (RPR) track of the 18th International Conference on Principles of Knowledge Representation and Reasoning, {KR} 2021, Virtual Event, November 3--12, 2021},
  doi = {https://doi.org/10.5281/zenodo.8341301},
  title = {{Computing Optimal Repairs of Quantified ABoxes w.r.t.\ Static $\mathcal{EL}$ TBoxes (Extended Abstract)}},
  year = {2021},
}

The \(\mathcal{EL}\) subsumption hierarchy consists of all \(\mathcal{EL}\) concept descriptions and is partially ordered by subsumption. In order to navigate within this hierarchy, one can go up to subsumers and down to subsumees. We analyze how smallest steps can be made. Specifically, we show how all upper neighbors as well as all lower neighbors of a given \(\mathcal{EL}\) concept description can be efficiently computed, where two concepts are neighbors if one subsumes the other and there is no third concept in between. We further show that the hierarchy contains very long chains: there is a sequence of concepts \(C_n\) with size linear in \(n\) such that each chain of neighbors from \(\top\) to \(C_n\) has at least \(n\)-fold exponential length. As applications, we provide a template for determining upper complexity bounds for deciding whether a concept is maximally general or maximally specific w.r.t. a property, we construct a metric on the set of all \(\mathcal{EL}\) concept descriptions, we introduce a similarity measure that fulfills the triangle inequality, and we conclude that an uninformed search for a target concept by subsequently computing neighbors or, equivalently, along an ideal refinement operator is not feasible in practical applications.

@inproceedings{ Kr-DL-2021,
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 34th International Workshop on Description Logics ({DL} 2021), Hybrid Event, Bratislava, Slovakia, September 19--22, 2021},
  editor = {Martin {Homola} and Vladislav {Ryzhikov} and Renate A. {Schmidt}},
  publisher = {CEUR-WS.org},
  series = {{CEUR} Workshop Proceedings},
  title = {{Navigating the $\mathcal{EL}$ Subsumption Hierarchy}},
  volume = {2954},
  year = {2021},
}

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 \(\mathcal{E\mkern-1.618mu L}\), \(\mathcal{M}\), \(\textsf{Horn-}\mathcal{M}\), and \(\textsf{Prob-}\mathcal{E\mkern-1.618mu L}\). 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 \(\mathcal{E\mkern-1.618mu L}\) is provided. For instance, it is shown how upper and lower neighbors of \(\mathcal{E\mkern-1.618mu L}\) concept descriptions can be computed and further it is proven that the set of \(\mathcal{E\mkern-1.618mu L}\) concept descriptions forms a graded lattice with a non-elementary rank function.

@article{ Kr-KI20,
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.1007/s13218-020-00673-8},
  journal = {KI - K{\"{u}}nstliche Intelligenz},
  pages = {399--403},
  title = {{Constructing and Extending Description Logic Ontologies using Methods of Formal Concept Analysis: A Dissertation Summary}},
  volume = {34},
  year = {2020},
}

The notion of a most specific consequence with respect to some terminological box is introduced, conditions for its existence in the description logic \(\mathcal{E\mkern-1.618mu L}\) 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.

@article{ Kr-DAM20,
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.1016/j.dam.2019.01.029},
  journal = {Discrete Applied Mathematics},
  pages = {172--204},
  title = {{Most Specific Consequences in the Description Logic $\mathcal{E\mkern-1.618mu L}$}},
  volume = {273},
  year = {2020},
}

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.

@thesis{ Kriegel-Diss-2019,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  school = {Technische Universit\"{a}t Dresden},
  title = {Constructing and Extending Description Logic Ontologies using Methods of Formal Concept Analysis},
  type = {Doctoral Thesis},
  year = {2019},
}

A joining implication is a restricted form of an implication where it is explicitly specified which attributes may occur in the premise and in the conclusion, respectively. A technique for sound and complete axiomatization of joining implications valid in a given formal context is provided. In particular, a canonical base for the joining implications valid in a given formal context is proposed, which enjoys the property of being of minimal cardinality among all such bases. Background knowledge in form of a set of valid joining implications can be incorporated. Furthermore, an application to inductive learning in a Horn description logic is proposed, that is, a procedure for sound and complete axiomatization of \(\textsf{Horn-}\mathcal{M}\) concept inclusions from a given interpretation is developed. A complexity analysis shows that this procedure runs in deterministic exponential time.

@inproceedings{ Kr-ICFCA19,
  author = {Francesco {Kriegel}},
  booktitle = {15th International Conference on Formal Concept Analysis, {ICFCA} 2019, Frankfurt, Germany, June 25-28, 2019, Proceedings},
  doi = {https://doi.org/10.1007/978-3-030-21462-3_9},
  editor = {Diana {Cristea} and Florence {Le Ber} and Bar\i{}\c{s} {Sertkaya}},
  pages = {110--129},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {{Joining Implications in Formal Contexts and Inductive Learning in a Horn Description Logic}},
  volume = {11511},
  year = {2019},
}

Description logics in their standard setting only allow for representing and reasoning with crisp knowledge without any degree of uncertainty. Of course, this is a serious shortcoming for use cases where it is impossible to perfectly determine the truth of a statement. For resolving this expressivity restriction, probabilistic variants of description logics have been introduced. Their model-theoretic semantics is built upon so-called probabilistic interpretations, that is, families of directed graphs the vertices and edges of which are labeled and for which there exists a probability measure on this graph family.

Results of scientific experiments, e.g., in medicine, psychology, or biology, that are repeated several times can induce probabilistic interpretations in a natural way. In this document, we shall develop a suitable axiomatization technique for deducing terminological knowledge from the assertional data given in such probabilistic interpretations. More specifically, we consider a probabilistic variant of the description logic \(\mathcal{E\mkern-1.618mu L}^{\!\bot}\), and provide a method for constructing a set of rules, so-called concept inclusions, from probabilistic interpretations in a sound and complete manner.

@inproceedings{ Kr-JELIA19,
  author = {Francesco {Kriegel}},
  booktitle = {16th European Conference on Logics in Artificial Intelligence, {JELIA} 2019, Rende, Italy, May 7-11, 2019, Proceedings},
  doi = {https://doi.org/10.1007/978-3-030-19570-0_26},
  editor = {Francesco {Calimeri} and Nicola {Leone} and Marco {Manna}},
  pages = {399--417},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {{Learning Description Logic Axioms from Discrete Probability Distributions over Description Graphs}},
  volume = {11468},
  year = {2019},
}

The classical approach for repairing a Description Logic ontology \(\mathfrak{O}\) in the sense of removing an unwanted consequence \(\alpha\) is to delete a minimal number of axioms from \(\mathfrak{O}\) such that the resulting ontology \(\mathfrak{O}'\) does not have the consequence \(\alpha\). However, the complete deletion of axioms may be too rough, in the sense that it may also remove consequences that are actually wanted. To alleviate this problem, we propose a more gentle notion of repair in which axioms are not deleted, but only weakened. On the one hand, we investigate general properties of this gentle repair method. On the other hand, we propose and analyze concrete approaches for weakening axioms expressed in the Description Logic \(\mathcal{E\mkern-1.618mu L}\).

@inproceedings{ BaKrNuPe-KR2018,
  address = {USA},
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah} and Rafael {Pe{\~n}aloza}},
  booktitle = {Principles of Knowledge Representation and Reasoning: Proceedings of the Sixteenth International Conference, {KR} 2018, Tempe, Arizona, 30 October - 2 November 2018},
  editor = {Frank {Wolter} and Michael {Thielscher} and Francesca {Toni}},
  pages = {319--328},
  publisher = {{AAAI} Press},
  title = {{Making Repairs in Description Logics More Gentle}},
  year = {2018},
}

For the description logic \(\mathcal{E\mkern-1.618mu L}\), we consider the neighborhood relation which is induced by the subsumption order, and we show that the corresponding lattice of \(\mathcal{E\mkern-1.618mu L}\) concept descriptions is distributive, modular, graded, and metric. In particular, this implies the existence of a rank function as well as the existence of a distance function.

@inproceedings{ Kr-CLA18,
  address = {Olomouc, Czech Republic},
  author = {Francesco {Kriegel}},
  booktitle = {{Proceedings of the 14th International Conference on Concept Lattices and Their Applications ({CLA 2018})}},
  editor = {Dmitry I. {Ignatov} and Lhouari {Nourine}},
  pages = {267--278},
  publisher = {CEUR-WS.org},
  series = {{CEUR} Workshop Proceedings},
  title = {{The Distributive, Graded Lattice of $\mathcal{E\mkern-1.618mu L}$ Concept Descriptions and its Neighborhood Relation}},
  volume = {2123},
  year = {2018},
}

The Web Ontology Language (OWL) has gained serious attraction since its foundation in 2004, and it is heavily used in applications requiring representation of as well as reasoning with knowledge. It is the language of the Semantic Web, and it has a strong logical underpinning by means of so-called Description Logics (DLs). DLs are a family of conceptual languages suitable for knowledge representation and reasoning due to their strong logical foundation, and for which the decidability and complexity of common reasoning problems are widely explored. In particular, the reasoning tasks allow for the deduction of implicit knowledge from explicitly stated facts and axioms, and plenty of appropriate algorithms were developed, optimized, and implemented, e.g., tableaux algorithms and completion algorithms. In this document, we present a technique for the acquisition of terminological knowledge from social networks. More specifically, we show how OWL axioms, i.e., concept inclusions and role inclusions in DLs, can be obtained from social graphs in a sound and complete manner. A social graph is simply a directed graph, the vertices of which describe the entities, e.g., persons, events, messages, etc.; and the edges of which describe the relationships between the entities, e.g., friendship between persons, attendance of a person to an event, a person liking a message, etc. Furthermore, the vertices of social graphs are labeled, e.g., to describe properties of the entities, and also the edges are labeled to specify the concrete relationships. As an exemplary social network we consider Facebook, and show that it fits our use case.

@incollection{ Kr-FCAoSN17,
  address = {Cham},
  author = {Francesco {Kriegel}},
  booktitle = {Formal Concept Analysis of Social Networks},
  doi = {https://doi.org/10.1007/978-3-319-64167-6_5},
  editor = {Rokia {Missaoui} and Sergei O. {Kuznetsov} and Sergei {Obiedkov}},
  pages = {97--142},
  publisher = {Springer International Publishing},
  series = {Lecture Notes in Social Networks ({LNSN})},
  title = {Acquisition of Terminological Knowledge from Social Networks in Description Logic},
  year = {2017},
}

A probabilistic formal context is a triadic context the third dimension of which is a set of worlds equipped with a probability measure. After a formal definition of this notion, this document introduces probability of implications with respect to probabilistic formal contexts, and provides a construction for a base of implications the probabilities of which exceed a given lower threshold. A comparison between confidence and probability of implications is drawn, which yields the fact that both measures do not coincide. Furthermore, the results are extended towards the lightweight description logic \(\mathcal{E\mkern-1.618mu L}^{\bot}\) with probabilistic interpretations, and a method for computing a base of general concept inclusions the probabilities of which are greater than a pre-defined lower bound is proposed. Additionally, we consider so-called probabilistic attributes over probabilistic formal contexts, and provide a method for the axiomatization of implications over probabilistic attributes.

@article{ Kr-IJGS17,
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.1080/03081079.2017.1349575},
  journal = {International Journal of General Systems},
  number = {5},
  pages = {511--546},
  title = {Probabilistic Implication Bases in {FCA} and Probabilistic Bases of GCIs in $\mathcal{E\mkern-1.618mu L}^{\bot}$},
  volume = {46},
  year = {2017},
}

The canonical base of a formal context plays a distinguished role in Formal Concept Analysis, as it is the only minimal implicational base known so far that can be described explicitly. Consequently, several algorithms for the computation of this base have been proposed. However, all those algorithms work sequentially by computing only one pseudo-intent at a time – a fact that heavily impairs the practicability in real-world applications. In this paper, we shall introduce an approach that remedies this deficit by allowing the canonical base to be computed in a parallel manner with respect to arbitrary implicational background knowledge. First experimental evaluations show that for sufficiently large data sets the speed-up is proportional to the number of available CPU cores.

@article{ KrBo-IJGS17,
  author = {Francesco {Kriegel} and Daniel {Borchmann}},
  doi = {https://doi.org/10.1080/03081079.2017.1349570},
  journal = {International Journal of General Systems},
  number = {5},
  pages = {490--510},
  title = {NextClosures: Parallel Computation of the Canonical Base with Background Knowledge},
  volume = {46},
  year = {2017},
}

Description logic knowledge bases can be used to represent knowledge about a particular domain in a formal and unambiguous manner. Their practical relevance has been shown in many research areas, especially in biology and the Semantic Web. However, the tasks of constructing knowledge bases itself, often performed by human experts, is difficult, time-consuming and expensive. In particular the synthesis of terminological knowledge is a challenge every expert has to face. Because human experts cannot be omitted completely from the construction of knowledge bases, it would therefore be desirable to at least get some support from machines during this process. To this end, we shall investigate in this work an approach which shall allow us to extract terminological knowledge in the form of general concept inclusions from factual data, where the data is given in the form of vertex and edge labeled graphs. As such graphs appear naturally within the scope of the Semantic Web in the form of sets of RDF triples, the presented approach opens up another possibility to extract terminological knowledge from the Linked Open Data Cloud.

@article{ BoDiKr-JANCL16,
  author = {Daniel {Borchmann} and Felix {Distel} and Francesco {Kriegel}},
  doi = {https://doi.org/10.1080/11663081.2016.1168230},
  journal = {Journal of Applied Non-Classical Logics},
  number = {1},
  pages = {1--46},
  title = {Axiomatisation of General Concept Inclusions from Finite Interpretations},
  volume = {26},
  year = {2016},
}

The canonical base of a formal context is a minimal set of implications that is sound and complete. A recent paper has provided a new algorithm for the parallel computation of canonical bases. An important extension is the integration of expert interaction for Attribute Exploration in order to explore implicational bases of inaccessible formal contexts. This paper presents and analyzes an algorithm that allows for Parallel Attribute Exploration.

@inproceedings{ Kr-ICCS16,
  address = {Annecy, France},
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 22nd International Conference on Conceptual Structures ({ICCS 2016})},
  doi = {http://dx.doi.org/10.1007/978-3-319-40985-6_8},
  editor = {Ollivier {Haemmerl{\'{e}}} and Gem {Stapleton} and Catherine {Faron{-}Zucker}},
  pages = {91--106},
  publisher = {Springer-Verlag},
  series = {Lecture Notes in Computer Science},
  title = {Parallel Attribute Exploration},
  volume = {9717},
  year = {2016},
}

We propose an interactive repair method for the description logic \(\mathcal{EL}\) that is based on the optimal-repair framework. The obtained repair might not be optimal in the theoretical sense, i.e. more than a minimal amount of consequences might have been removed—but from a practical perspective it is superior to a theoretically optimal repair as the interaction strategy enables the users to identify further faulty consequences connected to the initially reported errors.

@inproceedings{ Kr-KRRSAC2025,
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 40th ACM/SIGAPP Symposium On Applied Computing {(SAC 2025)}, March 31 -- April 4, 2025, Sicily, Italy},
  doi = {https://doi.org/10.1145/3672608.3707750},
  pages = {1019--1026},
  publisher = {Association for Computing Machinery},
  title = {Beyond Optimal: Interactive Identification of Better-than-optimal Repairs},
  year = {2025},
}

The \(\mathcal{EL}\) family of description logics facilitates efficient polynomial-time reasoning and has been standardized as the profile OWL 2 EL of the Web Ontology Language. \(\mathcal{EL}\) can represent and reason not only with symbolic knowledge but also with concrete knowledge expressed by numbers, strings, and other concrete datatypes. Such concrete domains must be convex to avoid introducing disjunctions “through the backdoor.” However, existing concrete domains provide only limited utility. In order to overcome this issue, we introduce a novel form of concrete domains based on semi-lattices. They are convex by design and can thus be integrated into Horn-DLs such as \(\mathcal{EL}\). Moreover, they allow for FBoxes to express dependencies between concrete features. We describe four instantiations concerned with real intervals, 2D-polygons, regular languages, and graphs.

@inproceedings{ Kr-FroCoS-2025,
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 15th International Symposium on Frontiers of Combining Systems {(FroCoS 2025)}},
  note = {To appear.},
  title = {Reasoning in OWL 2 EL with Hierarchical Concrete Domains},
  year = {2025},
}

Errors in knowledge bases (KBs) written in a Description Logic (DL) are usually detected when reasoning derives an inconsistency or a consequence that does not hold in the application domain modelled by the KB. Whereas classical repair approaches produce maximal subsets of the KB not implying the inconsistency or unwanted consequence, optimal repairs maximize the consequence sets. In this paper, we extend previous results on how to compute optimal repairs from the DL \(\mathcal{EL}\) to its extension \(\mathcal{EL}^{\bot}\), which in contrast to \(\mathcal{EL}\) can express inconsistency. The problem of how to deal with inconsistency in the context of optimal repairs was addressed previously, but in a setting where the (fixed) terminological part of the KB must satisfy a restriction on cyclic dependencies. Here, we consider a setting where this restriction is not required. We also show how the notion of optimal repairs obtained this way can be used in inconsistency- and error-tolerant reasoning.

@inproceedings{ BaKrNu-FoIKS2024,
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {Proceedings of the 13th International Symposium on Foundations of Information and Knowledge Systems {(FoIKS 2024)}, April 8--11, 2024, Sheffield, United Kingdom},
  doi = {https://doi.org/10.1007/978-3-031-56940-1_1},
  pages = {3--22},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {Inconsistency- and Error-Tolerant Reasoning w.r.t.\ Optimal Repairs of $\mathcal{EL}^{\bot}$ Ontologies},
  volume = {14589},
  year = {2024},
}

We present an FCA-based axiomatization method that produces a complete \(\mathcal{EL}\) TBox (the terminological part of an OWL 2 EL ontology) from a graph dataset in at most exponential time. We describe technical details that allow for efficient implementation as well as variations that dispense with the computation of extremely large axioms, thereby rendering the approach applicable albeit some completeness is lost. Moreover, we evaluate the prototype on real-world datasets.

@inproceedings{ Kr-AAAI2024,
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 38th Annual AAAI Conference on Artificial Intelligence {(AAAI 2024)}, February 20--27, 2024, Vancouver, Canada},
  doi = {https://doi.org/10.1609/aaai.v38i9.28930},
  pages = {10597--10606},
  title = {Efficient Axiomatization of OWL 2 EL Ontologies from Data by means of Formal Concept Analysis},
  year = {2024},
}

@inproceedings{ Kr-DL2024,
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 37th International Workshop on Description Logics ({DL} 2024), Bergen, Norway, June 18--21, 2024},
  publisher = {CEUR-WS.org},
  series = {{CEUR} Workshop Proceedings},
  title = {Efficient Axiomatization of OWL 2 EL Ontologies from Data by means of Formal Concept Analysis (Extended Abstract)},
  volume = {3739},
  year = {2024},
}

Ontologies based on Description Logics may contain errors, which are usually detected when reasoning produces consequences that follow from the ontology, but do not hold in the modelled application domain. In previous work, we have introduced repair approaches for \(\mathcal{EL}\) ontologies that are optimal in the sense that they preserve a maximal amount of consequences. In this paper, we will, on the one hand, review these approaches, but with an emphasis on motivation rather than on technical details. On the other hand, we will describe new results that address the problems that optimal repairs may become very large or need not even exist unless strong restrictions on the terminological part of the ontology apply. We will show how one can deal with these problems by introducing concise representations of optimal repairs.

@inproceedings{ BaKoKr-JELIA2023,
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel}},
  booktitle = {Proceedings of the 18th European Conference on Logics in Artificial Intelligence {(JELIA 2023)}, September 20--22, 2023, Dresden, Germany},
  doi = {https://doi.org/10.1007/978-3-031-43619-2_2},
  pages = {11--34},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {Optimal Repairs in the Description Logic $\mathcal{EL}$ Revisited},
  volume = {14281},
  year = {2023},
}

@inproceedings{ BaKrNu-DL2023,
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {Proceedings of the 36th International Workshop on Description Logics ({DL} 2023), Rhodes, Greece, September 2--4, 2023},
  publisher = {CEUR-WS.org},
  series = {{CEUR} Workshop Proceedings},
  title = {Error-Tolerant Reasoning in $\mathcal{EL}$ w.r.t.\ Optimal {ABox} Repairs (Extended Abstract)},
  volume = {3515},
  year = {2023},
}

Errors in Description Logic (DL) ontologies are often detected when reasoning yields unintuitive consequences. The question is then how to repair the ontology in an optimal way, i.e., such that the unwanted consequences are removed, but a maximal set of the unobjected consequences is kept. Error-tolerant reasoning does not commit to a single optimal repair: brave reasoning asks whether the consequence is entailed by some repair and cautious reasoning whether it is entailed by all repairs. Previous research on repairing ABoxes w.r.t. TBoxes formulated in the DL \(\mathcal{EL}\) has developed methods for computing optimal repairs, and has recently also determined the complexity of error-tolerant reasoning: brave reasoning is in P and cautious reasoning is in coNP. However, in this work the unwanted consequences were restricted to being \(\mathcal{EL}\) instance assertions. In the present paper, we show that the mentioned results can be extended to a setting where also role assertions can be required to be removed. Our solution is based on a two-stage approach where first the unwanted role assertions and then the unwanted concept assertions are removed. We also investigate the complexity of error-tolerant reasoning w.r.t. classical repairs, which are maximal subsets of the ABox that do not have the unwanted consequences, and show that, in this setting, brave reasoning is NP-complete and cautious reasoning is coNP-complete.

@inproceedings{ BaKrNu-SAC2023,
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {Proceedings of the 38th ACM/SIGAPP Symposium on Applied Computing (SAC '23), March 27--31, 2023, Tallinn, Estonia},
  doi = {https://doi.org/10.1145/3555776.3577630},
  pages = {974--982},
  publisher = {Association for Computing Machinery},
  title = {Treating Role Assertions as First-class Citizens in Repair and Error-tolerant Reasoning},
  year = {2023},
}

@inproceedings{ Kr-KR2023RPR,
  author = {Francesco {Kriegel}},
  booktitle = {Recently Published Research {(RPR)} track of the 20th International Conference on Principles of Knowledge Representation and Reasoning {(KR 2023)}, September 2--8, 2023, Rhodes, Greece},
  doi = {https://doi.org/10.5281/zenodo.8341194},
  title = {Optimal Fixed-Premise Repairs of $\mathcal{EL}$ {TBoxes} (Extended Abstract)},
  year = {2023},
}

Errors in Description Logic (DL) ontologies are often detected when a reasoner computes unwanted consequences. The question is then how to repair the ontology such that the unwanted consequences no longer follow, but as many of the other consequences as possible are preserved. The problem of computing such optimal repairs was addressed in our previous work in the setting where the data (expressed by an ABox) may contain errors, but the schema (expressed by an \(\mathcal{EL}\) TBox) is assumed to be correct. Actually, we consider a generalization of ABoxes called quantified ABoxes (qABoxes) both as input for and as result of the repair process. Using qABoxes for repair allows us to retain more information, but the disadvantage is that standard DL systems do not accept qABoxes as input. This raises the question, investigated in the present paper, whether and how one can obtain optimal repairs if one restricts the output of the repair process to being ABoxes. In general, such optimal ABox repairs need not exist. Our main contribution is that we show how to decide the existence of optimal ABox repairs in exponential time, and how to compute all such repairs in case they exist.

@inproceedings{ BaKoKrNu-ESWC2022,
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {19th Extended Semantic Web Conference, {ESWC} 2022, Hersonissos, Greece, May 29 -- June 2, 2022, Proceedings},
  doi = {https://doi.org/10.1007/978-3-031-06981-9_8},
  pages = {130--146},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {Optimal ABox Repair w.r.t.\ Static {$\mathcal{EL}$} TBoxes: from Quantified ABoxes back to ABoxes},
  volume = {13261},
  year = {2022},
}

@inproceedings{ BaKoKrNu-DL2022,
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {Proceedings of the 35th International Workshop on Description Logics ({DL} 2022), Haifa, Israel, August 7--10, 2022},
  publisher = {CEUR-WS.org},
  series = {{CEUR} Workshop Proceedings},
  title = {Optimal {ABox} Repair w.r.t. Static $\mathcal{EL}$ {TBoxes:} from Quantified {ABoxes} back to {ABoxes} (Extended Abstract)},
  year = {2022},
}

Ontologies based on Description Logic (DL) represent general background knowledge in a terminology (TBox) and the actual data in an ABox. DL systems can then be used to compute consequences (such as answers to certain queries) from an ontology consisting of a TBox and an ABox. Since both human-made and machine-learned data sets may contain errors, which manifest themselves as unintuitive or obviously incorrect consequences, repairing DL-based ontologies in the sense of removing such unwanted consequences is an important topic in DL research. Most of the repair approaches described in the literature produce repairs that are not optimal, in the sense that they do not guarantee that only a minimal set of consequences is removed. In a series of papers, we have developed an approach for computing optimal repairs, starting with the restricted setting of an \(\mathcal{EL}\) instance store, extending this to the more general setting of a quantified ABox (where some individuals may be anonymous), and then adding a static \(\mathcal{EL}\) TBox.

Here, we extend the expressivity of the underlying DL considerably, by adding nominals, inverse roles, regular role inclusions and the bottom concept to \(\mathcal{EL}\), which yields a fragment of the well-known DL Horn-\(\mathcal{SROIQ}\). The ideas underlying our repair approach still apply to this DL, though several non-trivial extensions are needed to deal with the new constructors and axioms. The developed repair approach can also be used to treat unwanted consequences expressed by certain conjunctive queries or regular path queries, and to handle Horn-\(\mathcal{ALCOI}\) TBoxes with regular role inclusions.

@inproceedings{ BaKr-KR2022,
  author = {Franz {Baader} and Francesco {Kriegel}},
  booktitle = {Proceedings of the 19th International Conference on Principles of Knowledge Representation and Reasoning, {KR} 2022, Haifa, Israel, July 31 -- August 5, 2022},
  doi = {https://doi.org/10.24963/kr.2022/3},
  editor = {Gabriele {Kern{-}Isberner} and Gerhard {Lakemeyer} and Thomas {Meyer}},
  pages = {22--32},
  title = {Pushing Optimal ABox Repair from {$\mathcal{EL}$} Towards More Expressive Horn-DLs},
  year = {2022},
}

Ontologies based on Description Logic (DL) represent general background knowledge in a terminology (TBox) and the actual data in an ABox. Both human-made and machine-learned data sets may contain errors, which are usually detected when the DL reasoner returns unintuitive or obviously incorrect answers to queries. To eliminate such errors, classical repair approaches offer as repairs maximal subsets of the ABox not having the unwanted answers w.r.t. the TBox. It is, however, not always clear which of these classical repairs to use as the new, corrected data set. Error-tolerant semantics instead takes all repairs into account: cautious reasoning returns the answers that follow from all classical repairs whereas brave reasoning returns the answers that follow from some classical repair. It is inspired by inconsistency-tolerant reasoning and has been investigated for the DL \(\mathcal{EL}\), but in a setting where the TBox rather than the ABox is repaired. In a series of papers, we have developed a repair approach for ABoxes that improves on classical repairs in that it preserves a maximal set of consequences (i.e., answers to queries) rather than a maximal set of ABox assertions. The repairs obtained by this approach are called optimal repairs. In the present paper, we investigate error-tolerant reasoning in the DL \(\mathcal{EL}\), but we repair the ABox and use optimal repairs rather than classical repairs as the underlying set of repairs. To be more precise, we consider a static \(\mathcal{EL}\) TBox (which is assumed to be correct), represent the data by a quantified ABox (where some individuals may be anonymous), and use \(\mathcal{EL}\) concepts as queries (instance queries). We show that brave entailment of instance queries can be decided in polynomial time. Cautious entailment can be decided by a coNP procedure, but is still in P if the TBox is empty.

@inproceedings{ BaKrNu-RuleML2022,
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {Rules and Reasoning - 6th International Joint Conference, RuleML+RR 2022, Virtual, September 26-28, 2022, Proceedings},
  doi = {https://doi.org/10.1007/978-3-031-21541-4_15},
  editor = {Guido {Governatori} and Anni{-}Yasmin {Turhan}},
  pages = {227--243},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {Error-Tolerant Reasoning in the Description Logic {$\mathcal{EL}$} Based On Optimal Repairs},
  volume = {13752},
  year = {2022},
}

Reasoners can be used to derive implicit consequences from an ontology. Sometimes unwanted consequences are revealed, indicating errors or privacy-sensitive information, and the ontology needs to be appropriately repaired. The classical approach is to remove just enough axioms such that the unwanted consequences vanish. However, this is often too rough since mere axiom deletion also erases many other consequences that might actually be desired. The goal should not be to remove a minimal number of axioms but to modify the ontology such that only a minimal number of consequences is removed, including the unwanted ones. Specifically, a repair should rather be logically entailed by the input ontology, instead of being a subset. To this end, we introduce a framework for computing fixed-premise repairs of \(\mathcal{EL}\) TBoxes. In the first variant the conclusions must be generalizations of those in the input TBox, while in the second variant no such restriction is imposed. In both variants, every repair is entailed by an optimal one and, up to equivalence, the set of all optimal repairs can be computed in exponential time. A prototypical implementation is provided. In addition, we show new complexity results regarding gentle repairs.

@inproceedings{ Kr-KI2022,
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 45th German Conference on Artificial Intelligence ({KI} 2022), Virtual in Trier, Germany, September 19--23, 2022},
  doi = {https://doi.org/10.1007/978-3-031-15791-2_11},
  editor = {Ralph {Bergmann} and Lukas {Malburg} and Stephanie C. {Rodermund} and Ingo J. {Timm}},
  pages = {115--130},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {Optimal Fixed-Premise Repairs of $\mathcal{EL}$ {TBoxes}},
  volume = {13404},
  year = {2022},
}

The application of automated reasoning approaches to Description Logic (DL) ontologies may produce certain consequences that either are deemed to be wrong or should be hidden for privacy reasons. The question is then how to repair the ontology such that the unwanted consequences can no longer be deduced. An optimal repair is one where the least amount of other consequences is removed. Most of the previous approaches to ontology repair are of a syntactic nature in that they remove or weaken the axioms explicitly present in the ontology, and thus cannot achieve semantic optimality. In previous work, we have addressed the problem of computing optimal repairs of (quantified) ABoxes, where the unwanted consequences are described by concept assertions of the light-weight DL \(\mathcal{E\!L}\). In the present paper, we improve on the results achieved so far in two ways. First, we allow for the presence of terminological knowledge in the form of an \(\mathcal{E\!L}\) TBox. This TBox is assumed to be static in the sense that it cannot be changed in the repair process. Second, the construction of optimal repairs described in our previous work is best case exponential. We introduce an optimized construction that is exponential only in the worst case. First experimental results indicate that this reduces the size of the computed optimal repairs considerably.

@inproceedings{ BaKoKrNu-CADE2021,
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {Proceedings of the 28th International Conference on Automated Deduction (CADE-28), July 11--16, 2021, Virtual Event, United States},
  doi = {https://doi.org/10.1007/978-3-030-79876-5_18},
  editor = {Andr{\'e} {Platzer} and Geoff {Sutcliffe}},
  pages = {309--326},
  series = {Lecture Notes in Computer Science},
  title = {{Computing Optimal Repairs of Quantified ABoxes w.r.t. Static $\mathcal{E\!L}$ TBoxes}},
  volume = {12699},
  year = {2021},
}

@inproceedings{ BaKoKrNu-KR2021,
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {Recent Published Research (RPR) track of the 18th International Conference on Principles of Knowledge Representation and Reasoning, {KR} 2021, Virtual Event, November 3--12, 2021},
  doi = {https://doi.org/10.5281/zenodo.8341301},
  title = {{Computing Optimal Repairs of Quantified ABoxes w.r.t.\ Static $\mathcal{EL}$ TBoxes (Extended Abstract)}},
  year = {2021},
}

We review our recent work on how to compute optimal repairs, optimal compliant anonymizations, and optimal safe anonymizations of ABoxes containing possibly anonymized individuals. The results can be used both to remove erroneous consequences from a knowledge base and to hide secret information before publication of the knowledge base, while keeping as much as possible of the original information.

@inproceedings{ BaKoKrNuPe-DL-2021,
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel} and Adrian {Nuradiansyah} and Rafael {Pe\~{n}aloza}},
  booktitle = {Proceedings of the 34th International Workshop on Description Logics ({DL} 2021), Hybrid Event, Bratislava, Slovakia, September 19--22, 2021},
  editor = {Martin {Homola} and Vladislav {Ryzhikov} and Renate A. {Schmidt}},
  publisher = {CEUR-WS.org},
  series = {{CEUR} Workshop Proceedings},
  title = {{Privacy-Preserving Ontology Publishing: The Case of Quantified ABoxes w.r.t.\ a Static Cycle-Restricted $\mathcal{EL}$ TBox}},
  volume = {2954},
  year = {2021},
}

In recent work, we have shown how to compute compliant anonymizations of quantified ABoxes w.r.t. \(\mathcal{E\!L}\) policies. In this setting, quantified ABoxes can be used to publish information about individuals, some of which are anonymized. The policy is given by concepts of the Description Logic (DL) \(\mathcal{E\!L}\), and compliance means that one cannot derive from the ABox that some non-anonymized individual is an instance of a policy concept. If one assumes that a possible attacker could have additional knowledge about some of the involved non-anonymized individuals, then compliance with a policy is not sufficient. One wants to ensure that the quantified ABox is safe in the sense that none of the secret instance information is revealed, even if the attacker has additional compliant knowledge. In the present paper, we show that safety can be decided in polynomial time, and that the unique optimal safe anonymization of a non-safe quantified ABox can be computed in exponential time, provided that the policy consists of a single \(\mathcal{E\!L}\) concept.

@inproceedings{ BaKrNuPe-SAC2021,
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah} and Rafael {Pe{\~n}aloza}},
  booktitle = {Proceedings of the 36th Annual ACM Symposium on Applied Computing (SAC '21), March 22--26, 2021, Virtual Event, Republic of Korea},
  doi = {https://doi.org/10.1145/3412841.3441961},
  pages = {863--872},
  publisher = {Association for Computing Machinery},
  title = {{Safety of Quantified ABoxes w.r.t.\ Singleton $\mathcal{E\!L}$ Policies}},
  year = {2021},
}

Knowledge engineers might face situations in which an unwanted consequence is derivable from an ontology. It is then desired to revise the ontology such that it no longer entails the consequence. For this purpose, we introduce a novel technique for repairing TBoxes formulated in the description logic \(\mathcal{EL}\). Specifically, we first compute a canonical model of the TBox and then transform it into a countermodel to the unwanted consequence. As formalism for the model transformation we employ transductions. We then obtain a TBox repair as the axiomatization of the logical intersection of the original TBox and the theory of the countermodel. In fact, we construct a set of countermodels, each of which induces a TBox repair. For the actual computation of the repairs we use results from Formal Concept Analysis.

@inproceedings{ HiKrNu-DL-2021,
  author = {Willi {Hieke} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {Proceedings of the 34th International Workshop on Description Logics ({DL} 2021), Hybrid Event, Bratislava, Slovakia, September 19--22, 2021},
  editor = {Martin {Homola} and Vladislav {Ryzhikov} and Renate A. {Schmidt}},
  publisher = {CEUR-WS.org},
  series = {{CEUR} Workshop Proceedings},
  title = {{Repairing $\mathcal{EL}$ TBoxes by Means of Countermodels Obtained by Model Transformation}},
  volume = {2954},
  year = {2021},
}

The \(\mathcal{EL}\) subsumption hierarchy consists of all \(\mathcal{EL}\) concept descriptions and is partially ordered by subsumption. In order to navigate within this hierarchy, one can go up to subsumers and down to subsumees. We analyze how smallest steps can be made. Specifically, we show how all upper neighbors as well as all lower neighbors of a given \(\mathcal{EL}\) concept description can be efficiently computed, where two concepts are neighbors if one subsumes the other and there is no third concept in between. We further show that the hierarchy contains very long chains: there is a sequence of concepts \(C_n\) with size linear in \(n\) such that each chain of neighbors from \(\top\) to \(C_n\) has at least \(n\)-fold exponential length. As applications, we provide a template for determining upper complexity bounds for deciding whether a concept is maximally general or maximally specific w.r.t. a property, we construct a metric on the set of all \(\mathcal{EL}\) concept descriptions, we introduce a similarity measure that fulfills the triangle inequality, and we conclude that an uninformed search for a target concept by subsequently computing neighbors or, equivalently, along an ideal refinement operator is not feasible in practical applications.

@inproceedings{ Kr-DL-2021,
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 34th International Workshop on Description Logics ({DL} 2021), Hybrid Event, Bratislava, Slovakia, September 19--22, 2021},
  editor = {Martin {Homola} and Vladislav {Ryzhikov} and Renate A. {Schmidt}},
  publisher = {CEUR-WS.org},
  series = {{CEUR} Workshop Proceedings},
  title = {{Navigating the $\mathcal{EL}$ Subsumption Hierarchy}},
  volume = {2954},
  year = {2021},
}

We adapt existing approaches for privacy-preserving publishing of linked data to a setting where the data are given as Description Logic (DL) ABoxes with possibly anonymised (formally: existentially quantified) individuals and the privacy policies are expressed using sets of concepts of the DL \(\mathcal{E\!L}\). We provide a chacterization of compliance of such ABoxes w.r.t. \(\mathcal{E\!L}\) policies, and show how optimal compliant anonymisations of ABoxes that are non-compliant can be computed. This work extends previous work on privacy-preserving ontology publishing, in which a very restricted form of ABoxes, called instance stores, had been considered, but restricts the attention to compliance. The approach developed here can easily be adapted to the problem of computing optimal repairs of quantified ABoxes.

@inproceedings{ BaKrNuPe-ISWC2020,
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah} and Rafael {Pe{\~n}aloza}},
  booktitle = {Proceedings of the 19th International Semantic Web Conference (ISWC 2020), Part {I}, Athens, Greece, November 2-6, 2020},
  doi = {https://doi.org/10.1007/978-3-030-62419-4_1},
  editor = {Jeff Z. {Pan} and Valentina A. M. {Tamma} and Claudia {d'Amato} and Krzysztof {Janowicz} and Bo {Fu} and Axel {Polleres} and Oshani {Seneviratne} and Lalana {Kagal}},
  pages = {3--20},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {{Computing Compliant Anonymisations of Quantified ABoxes w.r.t. $\mathcal{E\!L}$ Policies}},
  volume = {12506},
  year = {2020},
}

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 \(\mathcal{E\mkern-1.618mu L}\), \(\mathcal{M}\), \(\textsf{Horn-}\mathcal{M}\), and \(\textsf{Prob-}\mathcal{E\mkern-1.618mu L}\). 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 \(\mathcal{E\mkern-1.618mu L}\) is provided. For instance, it is shown how upper and lower neighbors of \(\mathcal{E\mkern-1.618mu L}\) concept descriptions can be computed and further it is proven that the set of \(\mathcal{E\mkern-1.618mu L}\) concept descriptions forms a graded lattice with a non-elementary rank function.

@article{ Kr-KI20,
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.1007/s13218-020-00673-8},
  journal = {KI - K{\"{u}}nstliche Intelligenz},
  pages = {399--403},
  title = {{Constructing and Extending Description Logic Ontologies using Methods of Formal Concept Analysis: A Dissertation Summary}},
  volume = {34},
  year = {2020},
}

The notion of a most specific consequence with respect to some terminological box is introduced, conditions for its existence in the description logic \(\mathcal{E\mkern-1.618mu L}\) 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.

@article{ Kr-DAM20,
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.1016/j.dam.2019.01.029},
  journal = {Discrete Applied Mathematics},
  pages = {172--204},
  title = {{Most Specific Consequences in the Description Logic $\mathcal{E\mkern-1.618mu L}$}},
  volume = {273},
  year = {2020},
}

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 \(\mathcal{EL}\), which corresponds to the setting where the ontology is an \(\mathcal{EL}\) 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 \(\mathcal{EL}\) concept can be computed. In addition, we investigate the complexity of the optimality problem.

@inproceedings{ BaKrNu-JELIA19,
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  booktitle = {16th European Conference on Logics in Artificial Intelligence, {JELIA} 2019, Rende, Italy, May 7-11, 2019, Proceedings},
  doi = {https://doi.org/10.1007/978-3-030-19570-0_21},
  editor = {Francesco {Calimeri} and Nicola {Leone} and Marco {Manna}},
  pages = {323--338},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {{Privacy-Preserving Ontology Publishing for $\mathcal{EL}$ Instance Stores}},
  volume = {11468},
  year = {2019},
}

A joining implication is a restricted form of an implication where it is explicitly specified which attributes may occur in the premise and in the conclusion, respectively. A technique for sound and complete axiomatization of joining implications valid in a given formal context is provided. In particular, a canonical base for the joining implications valid in a given formal context is proposed, which enjoys the property of being of minimal cardinality among all such bases. Background knowledge in form of a set of valid joining implications can be incorporated. Furthermore, an application to inductive learning in a Horn description logic is proposed, that is, a procedure for sound and complete axiomatization of \(\textsf{Horn-}\mathcal{M}\) concept inclusions from a given interpretation is developed. A complexity analysis shows that this procedure runs in deterministic exponential time.

@inproceedings{ Kr-ICFCA19,
  author = {Francesco {Kriegel}},
  booktitle = {15th International Conference on Formal Concept Analysis, {ICFCA} 2019, Frankfurt, Germany, June 25-28, 2019, Proceedings},
  doi = {https://doi.org/10.1007/978-3-030-21462-3_9},
  editor = {Diana {Cristea} and Florence {Le Ber} and Bar\i{}\c{s} {Sertkaya}},
  pages = {110--129},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {{Joining Implications in Formal Contexts and Inductive Learning in a Horn Description Logic}},
  volume = {11511},
  year = {2019},
}

Description logics in their standard setting only allow for representing and reasoning with crisp knowledge without any degree of uncertainty. Of course, this is a serious shortcoming for use cases where it is impossible to perfectly determine the truth of a statement. For resolving this expressivity restriction, probabilistic variants of description logics have been introduced. Their model-theoretic semantics is built upon so-called probabilistic interpretations, that is, families of directed graphs the vertices and edges of which are labeled and for which there exists a probability measure on this graph family.

Results of scientific experiments, e.g., in medicine, psychology, or biology, that are repeated several times can induce probabilistic interpretations in a natural way. In this document, we shall develop a suitable axiomatization technique for deducing terminological knowledge from the assertional data given in such probabilistic interpretations. More specifically, we consider a probabilistic variant of the description logic \(\mathcal{E\mkern-1.618mu L}^{\!\bot}\), and provide a method for constructing a set of rules, so-called concept inclusions, from probabilistic interpretations in a sound and complete manner.

@inproceedings{ Kr-JELIA19,
  author = {Francesco {Kriegel}},
  booktitle = {16th European Conference on Logics in Artificial Intelligence, {JELIA} 2019, Rende, Italy, May 7-11, 2019, Proceedings},
  doi = {https://doi.org/10.1007/978-3-030-19570-0_26},
  editor = {Francesco {Calimeri} and Nicola {Leone} and Marco {Manna}},
  pages = {399--417},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {{Learning Description Logic Axioms from Discrete Probability Distributions over Description Graphs}},
  volume = {11468},
  year = {2019},
}

@inproceedings{ BaKrNuPe-DL2018,
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah} and Rafael {Pe{\~n}aloza}},
  booktitle = {Proceedings of the 31st International Workshop on Description Logics, Tempe, Arizona, October 27-29, 2018},
  editor = {Magdalena {Ortiz} and Thomas {Schneider}},
  publisher = {CEUR-WS.org},
  series = {{CEUR} Workshop Proceedings},
  title = {{Making Repairs in Description Logics More Gentle (Extended Abstract)}},
  volume = {2211},
  year = {2018},
}

The classical approach for repairing a Description Logic ontology \(\mathfrak{O}\) in the sense of removing an unwanted consequence \(\alpha\) is to delete a minimal number of axioms from \(\mathfrak{O}\) such that the resulting ontology \(\mathfrak{O}'\) does not have the consequence \(\alpha\). However, the complete deletion of axioms may be too rough, in the sense that it may also remove consequences that are actually wanted. To alleviate this problem, we propose a more gentle notion of repair in which axioms are not deleted, but only weakened. On the one hand, we investigate general properties of this gentle repair method. On the other hand, we propose and analyze concrete approaches for weakening axioms expressed in the Description Logic \(\mathcal{E\mkern-1.618mu L}\).

@inproceedings{ BaKrNuPe-KR2018,
  address = {USA},
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah} and Rafael {Pe{\~n}aloza}},
  booktitle = {Principles of Knowledge Representation and Reasoning: Proceedings of the Sixteenth International Conference, {KR} 2018, Tempe, Arizona, 30 October - 2 November 2018},
  editor = {Frank {Wolter} and Michael {Thielscher} and Francesca {Toni}},
  pages = {319--328},
  publisher = {{AAAI} Press},
  title = {{Making Repairs in Description Logics More Gentle}},
  year = {2018},
}

For a probabilistic extension of the description logic \(\mathcal{E\mkern-1.618mu L}^{\bot}\), we consider the task of automatic acquisition of terminological knowledge from a given probabilistic interpretation. Basically, such a probabilistic interpretation is a family of directed graphs the vertices and edges of which are labeled, and where a discrete probability measure on this graph family is present. The goal is to derive so-called concept inclusions which are expressible in the considered probabilistic description logic and which hold true in the given probabilistic interpretation. A procedure for an appropriate axiomatization of such graph families is proposed and its soundness and completeness is justified.

@inproceedings{ Kr-KI18,
  address = {Berlin, Germany},
  author = {Francesco {Kriegel}},
  booktitle = {{{KI} 2018: Advances in Artificial Intelligence - 41st German Conference on AI, Berlin, Germany, September 24-28, 2018, Proceedings}},
  doi = {https://doi.org/10.1007/978-3-030-00111-7_5},
  editor = {Frank {Trollmann} and Anni-Yasmin {Turhan}},
  pages = {46--53},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  title = {{Acquisition of Terminological Knowledge in Probabilistic Description Logic}},
  volume = {11117},
  year = {2018},
}

For the description logic \(\mathcal{E\mkern-1.618mu L}\), we consider the neighborhood relation which is induced by the subsumption order, and we show that the corresponding lattice of \(\mathcal{E\mkern-1.618mu L}\) concept descriptions is distributive, modular, graded, and metric. In particular, this implies the existence of a rank function as well as the existence of a distance function.

@inproceedings{ Kr-CLA18,
  address = {Olomouc, Czech Republic},
  author = {Francesco {Kriegel}},
  booktitle = {{Proceedings of the 14th International Conference on Concept Lattices and Their Applications ({CLA 2018})}},
  editor = {Dmitry I. {Ignatov} and Lhouari {Nourine}},
  pages = {267--278},
  publisher = {CEUR-WS.org},
  series = {{CEUR} Workshop Proceedings},
  title = {{The Distributive, Graded Lattice of $\mathcal{E\mkern-1.618mu L}$ Concept Descriptions and its Neighborhood Relation}},
  volume = {2123},
  year = {2018},
}

The Web Ontology Language (OWL) has gained serious attraction since its foundation in 2004, and it is heavily used in applications requiring representation of as well as reasoning with knowledge. It is the language of the Semantic Web, and it has a strong logical underpinning by means of so-called Description Logics (DLs). DLs are a family of conceptual languages suitable for knowledge representation and reasoning due to their strong logical foundation, and for which the decidability and complexity of common reasoning problems are widely explored. In particular, the reasoning tasks allow for the deduction of implicit knowledge from explicitly stated facts and axioms, and plenty of appropriate algorithms were developed, optimized, and implemented, e.g., tableaux algorithms and completion algorithms. In this document, we present a technique for the acquisition of terminological knowledge from social networks. More specifically, we show how OWL axioms, i.e., concept inclusions and role inclusions in DLs, can be obtained from social graphs in a sound and complete manner. A social graph is simply a directed graph, the vertices of which describe the entities, e.g., persons, events, messages, etc.; and the edges of which describe the relationships between the entities, e.g., friendship between persons, attendance of a person to an event, a person liking a message, etc. Furthermore, the vertices of social graphs are labeled, e.g., to describe properties of the entities, and also the edges are labeled to specify the concrete relationships. As an exemplary social network we consider Facebook, and show that it fits our use case.

@incollection{ Kr-FCAoSN17,
  address = {Cham},
  author = {Francesco {Kriegel}},
  booktitle = {Formal Concept Analysis of Social Networks},
  doi = {https://doi.org/10.1007/978-3-319-64167-6_5},
  editor = {Rokia {Missaoui} and Sergei O. {Kuznetsov} and Sergei {Obiedkov}},
  pages = {97--142},
  publisher = {Springer International Publishing},
  series = {Lecture Notes in Social Networks ({LNSN})},
  title = {Acquisition of Terminological Knowledge from Social Networks in Description Logic},
  year = {2017},
}

Entropy is a measure for the uninformativeness or randomness of a data set, i.e., the higher the entropy is, the lower is the amount of information. In the field of propositional logic it has proven to constitute a suitable measure to be maximized when dealing with models of probabilistic propositional theories. More specifically, it was shown that the model of a probabilistic propositional theory with maximal entropy allows for the deduction of other formulae which are somehow expected by humans, i.e., allows for some kind of common sense reasoning. In order to pull the technique of maximum entropy entailment to the field of Formal Concept Analysis, we define the notion of entropy of a formal context with respect to the frequency of its object intents, and then define maximum entropy entailment for quantified implication sets, i.e., for sets of partial implications where each implication has an assigned degree of confidence. Furthermore, then this entailment technique is utilized to define so-called maximum entropy implicational bases (ME-bases), and a first general example of such a ME-base is provided.

@inproceedings{ Kr-ICFCA17b,
  address = {Rennes, France},
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 14th International Conference on Formal Concept Analysis ({ICFCA} 2017)},
  doi = {https://doi.org/10.1007/978-3-319-59271-8_10},
  editor = {Karell {Bertet} and Daniel {Borchmann} and Peggy {Cellier} and S\'{e}bastien {Ferr\'{e}}},
  pages = {155--167},
  publisher = {Springer Verlag},
  series = {Lecture Notes in Computer Science ({LNCS})},
  title = {First Notes on Maximum Entropy Entailment for Quantified Implications},
  volume = {10308},
  year = {2017},
}

We consider the task of acquisition of terminological knowledge from given assertional data. However, when evaluating data of real-world applications we often encounter situations where it is impractical to deduce only crisp knowledge, due to the presence of exceptions or errors. It is rather appropriate to allow for degrees of uncertainty within the derived knowledge. Consequently, suitable methods for knowledge acquisition in a probabilistic framework should be developed. In particular, we consider data which is given as a probabilistic formal context, i.e., as a triadic incidence relation between objects, attributes, and worlds, which is furthermore equipped with a probability measure on the set of worlds. We define the notion of a probabilistic attribute as a probabilistically quantified set of attributes, and define the notion of validity of implications over probabilistic attributes in a probabilistic formal context. Finally, a technique for the axiomatization of such implications from probabilistic formal contexts is developed. This is done is a sound and complete manner, i.e., all derived implications are valid, and all valid implications are deducible from the derived implications. In case of finiteness of the input data to be analyzed, the constructed axiomatization is finite, too, and can be computed in finite time.

@inproceedings{ Kr-ICFCA17a,
  address = {Rennes, France},
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 14th International Conference on Formal Concept Analysis ({ICFCA} 2017)},
  doi = {https://doi.org/10.1007/978-3-319-59271-8_11},
  editor = {Karell {Bertet} and Daniel {Borchmann} and Peggy {Cellier} and S\'{e}bastien {Ferr\'{e}}},
  pages = {168--183},
  publisher = {Springer Verlag},
  series = {Lecture Notes in Computer Science ({LNCS})},
  title = {Implications over Probabilistic Attributes},
  volume = {10308},
  year = {2017},
}

A probabilistic formal context is a triadic context the third dimension of which is a set of worlds equipped with a probability measure. After a formal definition of this notion, this document introduces probability of implications with respect to probabilistic formal contexts, and provides a construction for a base of implications the probabilities of which exceed a given lower threshold. A comparison between confidence and probability of implications is drawn, which yields the fact that both measures do not coincide. Furthermore, the results are extended towards the lightweight description logic \(\mathcal{E\mkern-1.618mu L}^{\bot}\) with probabilistic interpretations, and a method for computing a base of general concept inclusions the probabilities of which are greater than a pre-defined lower bound is proposed. Additionally, we consider so-called probabilistic attributes over probabilistic formal contexts, and provide a method for the axiomatization of implications over probabilistic attributes.

@article{ Kr-IJGS17,
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.1080/03081079.2017.1349575},
  journal = {International Journal of General Systems},
  number = {5},
  pages = {511--546},
  title = {Probabilistic Implication Bases in {FCA} and Probabilistic Bases of GCIs in $\mathcal{E\mkern-1.618mu L}^{\bot}$},
  volume = {46},
  year = {2017},
}

The canonical base of a formal context plays a distinguished role in Formal Concept Analysis, as it is the only minimal implicational base known so far that can be described explicitly. Consequently, several algorithms for the computation of this base have been proposed. However, all those algorithms work sequentially by computing only one pseudo-intent at a time – a fact that heavily impairs the practicability in real-world applications. In this paper, we shall introduce an approach that remedies this deficit by allowing the canonical base to be computed in a parallel manner with respect to arbitrary implicational background knowledge. First experimental evaluations show that for sufficiently large data sets the speed-up is proportional to the number of available CPU cores.

@article{ KrBo-IJGS17,
  author = {Francesco {Kriegel} and Daniel {Borchmann}},
  doi = {https://doi.org/10.1080/03081079.2017.1349570},
  journal = {International Journal of General Systems},
  number = {5},
  pages = {490--510},
  title = {NextClosures: Parallel Computation of the Canonical Base with Background Knowledge},
  volume = {46},
  year = {2017},
}

Description logic knowledge bases can be used to represent knowledge about a particular domain in a formal and unambiguous manner. Their practical relevance has been shown in many research areas, especially in biology and the Semantic Web. However, the tasks of constructing knowledge bases itself, often performed by human experts, is difficult, time-consuming and expensive. In particular the synthesis of terminological knowledge is a challenge every expert has to face. Because human experts cannot be omitted completely from the construction of knowledge bases, it would therefore be desirable to at least get some support from machines during this process. To this end, we shall investigate in this work an approach which shall allow us to extract terminological knowledge in the form of general concept inclusions from factual data, where the data is given in the form of vertex and edge labeled graphs. As such graphs appear naturally within the scope of the Semantic Web in the form of sets of RDF triples, the presented approach opens up another possibility to extract terminological knowledge from the Linked Open Data Cloud.

@article{ BoDiKr-JANCL16,
  author = {Daniel {Borchmann} and Felix {Distel} and Francesco {Kriegel}},
  doi = {https://doi.org/10.1080/11663081.2016.1168230},
  journal = {Journal of Applied Non-Classical Logics},
  number = {1},
  pages = {1--46},
  title = {Axiomatisation of General Concept Inclusions from Finite Interpretations},
  volume = {26},
  year = {2016},
}

We propose applications that utilize the infimum and the supremum of closure operators that are induced by structures occuring in the field of Description Logics. More specifically, we consider the closure operators induced by interpretations as well as closure operators induced by TBoxes, and show how we can learn GCIs from streams of interpretations, and how an error-tolerant axiomatization of GCIs from an interpretation guided by a hand-crafted TBox can be achieved.

@inproceedings{ Kr-FCA4AI16,
  address = {The Hague, The Netherlands},
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 5th International Workshop "What can {FCA} do for Artificial Intelligence?" ({FCA4AI} 2016) co-located with the European Conference on Artificial Intelligence ({ECAI} 2016)},
  editor = {Sergei {Kuznetsov} and Amedeo {Napoli} and Sebastian {Rudolph}},
  pages = {9--16},
  publisher = {CEUR-WS.org},
  series = {{CEUR} Workshop Proceedings},
  title = {Axiomatization of General Concept Inclusions from Streams of Interpretations with Optional Error Tolerance},
  volume = {1703},
  year = {2016},
}

In a former paper, the algorithm NextClosures for computing the set of all formal concepts as well as the canonical base for a given formal context has been introduced. Here, this algorithm shall be generalized to a setting where the data-set is described by means of a closure operator in a complete lattice, and furthermore it shall be extended with the possibility to handle constraints that are given in form of a second closure operator. As a special case, constraints may be predefined as implicational background knowledge. Additionally, we show how the algorithm can be modified in order to do parallel Attribute Exploration for unconstrained closure operators, as well as give a reason for the impossibility of (parallel) Attribute Exploration for constrained closure operators if the constraint is not compatible with the data-set.

@inproceedings{ Kr-CLA16,
  address = {Moscow, Russia},
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 13th International Conference on Concept Lattices and Their Applications ({CLA 2016})},
  editor = {Marianne {Huchard} and Sergei {Kuznetsov}},
  pages = {231--243},
  publisher = {CEUR-WS.org},
  series = {{CEUR} Workshop Proceedings},
  title = {NextClosures with Constraints},
  volume = {1624},
  year = {2016},
}

The canonical base of a formal context is a minimal set of implications that is sound and complete. A recent paper has provided a new algorithm for the parallel computation of canonical bases. An important extension is the integration of expert interaction for Attribute Exploration in order to explore implicational bases of inaccessible formal contexts. This paper presents and analyzes an algorithm that allows for Parallel Attribute Exploration.

@inproceedings{ Kr-ICCS16,
  address = {Annecy, France},
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 22nd International Conference on Conceptual Structures ({ICCS 2016})},
  doi = {http://dx.doi.org/10.1007/978-3-319-40985-6_8},
  editor = {Ollivier {Haemmerl{\'{e}}} and Gem {Stapleton} and Catherine {Faron{-}Zucker}},
  pages = {91--106},
  publisher = {Springer-Verlag},
  series = {Lecture Notes in Computer Science},
  title = {Parallel Attribute Exploration},
  volume = {9717},
  year = {2016},
}

Probabilistic interpretations consist of a set of interpretations with a shared domain and a measure assigning a probability to each interpretation. Such structures can be obtained as results of repeated experiments, e.g., in biology, psychology, medicine, etc. A translation between probabilistic and crisp description logics is introduced, and then utilized to reduce the construction of a base of general concept inclusions of a probabilistic interpretation to the crisp case for which a method for the axiomatization of a base of GCIs is well-known.

@inproceedings{ Kr-KI15,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 38th German Conference on Artificial Intelligence ({KI 2015})},
  doi = {http://dx.doi.org/10.1007/978-3-319-24489-1_10},
  editor = {Steffen {H{\"o}lldobler} and Sebastian {Rudolph} and Markus {Kr{\"o}tzsch}},
  pages = {124--136},
  publisher = {Springer Verlag},
  series = {Lecture Notes in Artificial Intelligence ({LNAI})},
  title = {Axiomatization of General Concept Inclusions in Probabilistic Description Logics},
  volume = {9324},
  year = {2015},
}

A description graph is a directed graph that has labeled vertices and edges. This document proposes a method for extracting a knowledge base from a description graph. The technique is presented for the description logic \(\mathcal{A\mkern-1.618mu L\mkern-1.618mu E\mkern-1.618mu Q\mkern-1.618mu R}(\mathsf{Self})\) which allows for conjunctions, primitive negation, existential restrictions, value restrictions, qualified number restrictions, existential self restrictions, and complex role inclusion axioms, but also sublogics may be chosen to express the axioms in the knowledge base. The extracted knowledge base entails all statements that can be expressed in the chosen description logic and are encoded in the input graph.

@inproceedings{ Kr-SNAFCA15,
  address = {Nerja, Spain},
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the International Workshop on Social Network Analysis using Formal Concept Analysis ({SNAFCA 2015}) in conjunction with the 13th International Conference on Formal Concept Analysis ({ICFCA} 2015)},
  editor = {Sergei O. {Kuznetsov} and Rokia {Missaoui} and Sergei A. {Obiedkov}},
  publisher = {CEUR-WS.org},
  series = {CEUR Workshop Proceedings},
  title = {Extracting $\mathcal{A\mkern-1.618mu L\mkern-1.618mu E\mkern-1.618mu Q\mkern-1.618mu R}(\mathsf{Self})$-Knowledge Bases from Graphs},
  volume = {1534},
  year = {2015},
}

Formal Concept Analysis and its methods for computing minimal implicational bases have been successfully applied to axiomatise minimal \(\mathcal{E\mkern-1.618mu L}\)-TBoxes from models, so called bases of GCIs. However, no technique for an adjustment of an existing \(\mathcal{E\mkern-1.618mu L}\)-TBox w.r.t. a new model is available, i.e., on a model change the complete TBox has to be recomputed. This document proposes a method for the computation of a minimal extension of a TBox w.r.t. a new model. The method is then utilised to formulate an incremental learning algorithm that requires a stream of interpretations, and an expert to guide the learning process, respectively, as input.

@inproceedings{ Kr-DL15,
  address = {Athens, Greece},
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 28th International Workshop on Description Logics ({DL 2015})},
  editor = {Diego {Calvanese} and Boris {Konev}},
  pages = {452--464},
  publisher = {CEUR-WS.org},
  series = {CEUR Workshop Proceedings},
  title = {Incremental Learning of TBoxes from Interpretation Sequences with Methods of Formal Concept Analysis},
  volume = {1350},
  year = {2015},
}

A probabilistic formal context is a triadic context whose third dimension is a set of worlds equipped with a probability measure. After a formal definition of this notion, this document introduces probability of implications, and provides a construction for a base of implications whose probability satisfy a given lower threshold. A comparison between confidence and probability of implications is drawn, which yields the fact that both measures do not coincide, and cannot be compared. Furthermore, the results are extended towards the light-weight description logic \(\mathcal{E\mkern-1.618mu L}^{\bot}\) with probabilistic interpretations, and a method for computing a base of general concept inclusions whose probability fulfill a certain lower bound is proposed.

@inproceedings{ Kr-CLA15,
  address = {Clermont-Ferrand, France},
  author = {Francesco {Kriegel}},
  booktitle = {Proceedings of the 12th International Conference on Concept Lattices and their Applications ({CLA 2015})},
  editor = {Sadok {Ben Yahia} and Jan {Konecny}},
  pages = {193--204},
  publisher = {CEUR-WS.org},
  series = {CEUR Workshop Proceedings},
  title = {Probabilistic Implicational Bases in FCA and Probabilistic Bases of GCIs in $\mathcal{E\mkern-1.618mu L}^{\bot}$},
  volume = {1466},
  year = {2015},
}

The canonical base of a formal context plays a distinguished role in formal concept analysis. This is because it is the only minimal base so far that can be described explicitly. For the computation of this base several algorithms have been proposed. However, all those algorithms work sequentially, by computing only one pseudo-intent at a time - a fact which heavily impairs the practicability of using the canonical base in real-world applications. In this paper we shall introduce an approach that remedies this deficit by allowing the canonical base to be computed in a parallel manner. First experimental evaluations show that for sufficiently large data-sets the speedup is proportional to the number of available CPUs.

@inproceedings{ KrBo-CLA15,
  address = {Clermont-Ferrand, France},
  author = {Francesco {Kriegel} and Daniel {Borchmann}},
  booktitle = {Proceedings of the 12th International Conference on Concept Lattices and their Applications ({CLA 2015})},
  editor = {Sadok {Ben Yahia} and Jan {Konecny}},
  note = {Best Paper Award.},
  pages = {182--192},
  publisher = {CEUR-WS.org},
  series = {CEUR Workshop Proceedings},
  title = {NextClosures: Parallel Computation of the Canonical Base},
  volume = {1466},
  year = {2015},
}

Suppose a formal context \(\mathbb{K}=(G,M,I)\) is given, whose concept lattice \(\mathfrak{B}(\mathbb{K})\) with an attribute-additive concept diagram is already known, and an attribute column \(\mathbb{C}=(G,\{n\},J)\) shall be inserted to or removed from it. This paper introduces and proves an incremental update algorithm for both tasks.

@article{ Kr-ICFCA14,
  address = {Cluj-Napoca, Romania},
  author = {Francesco {Kriegel}},
  journal = {Studia Universitatis Babe{\c{s}}-Bolyai Informatica},
  note = {Supplemental proceedings of the 12th International Conference on Formal Concept Analysis (ICFCA 2014), Cluj-Napoca, Romania},
  pages = {45--61},
  title = {Incremental Computation of Concept Diagrams},
  volume = {59},
  year = {2014},
}

The \(\mathcal{EL}\) family of description logics facilitates efficient polynomial-time reasoning and has been standardized as the profile OWL 2 EL of the Web Ontology Language. \(\mathcal{EL}\) can represent and reason not only with symbolic knowledge but also with concrete knowledge expressed by numbers, strings, and other concrete datatypes. Such concrete domains must be convex to avoid introducing disjunctions “through the backdoor.” However, existing concrete domains provide only limited utility. In order to overcome this issue, we introduce a novel form of concrete domains based on semi-lattices. They are convex by design and can thus be integrated into Horn-DLs such as \(\mathcal{EL}\). Moreover, they allow for FBoxes to express dependencies between concrete features. We describe four instantiations concerned with real intervals, 2D-polygons, regular languages, and graphs.

@techreport{ Kr-LTCS-25-04,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.25368/2025.127},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  number = {25-04},
  title = {Reasoning in OWL 2 EL with Hierarchical Concrete Domains (Extended Version)},
  type = {LTCS-Report},
  year = {2025},
}

Errors in knowledge bases (KBs) written in a Description Logic (DL) are usually detected when reasoning derives an inconsistency or a consequence that does not hold in the application domain modelled by the KB. Whereas classical repair approaches produce maximal subsets of the KB not implying the inconsistency or unwanted consequence, optimal repairs maximize the consequence sets. In this paper, we extend previous results on how to compute optimal repairs from the DL \(\mathcal{EL}\) to its extension \(\mathcal{EL}^{\bot}\), which in contrast to \(\mathcal{EL}\) can express inconsistency. The problem of how to deal with inconsistency in the context of optimal repairs was addressed previously, but in a setting where the (fixed) terminological part of the KB must satisfy a restriction on cyclic dependencies. Here, we consider a setting where this restriction is not required. We also show how the notion of optimal repairs obtained this way can be used in inconsistency- and error-tolerant reasoning.

@techreport{ BaKrNu-LTCS-24-02,
  address = {Dresden, Germany},
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  doi = {https://doi.org/10.25368/2024.6},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  number = {24-02},
  title = {Inconsistency- and Error-Tolerant Reasoning w.r.t.\ Optimal Repairs of $\mathcal{EL}^{\bot}$ Ontologies (Extended Version)},
  type = {LTCS-Report},
  year = {2024},
}

We propose an interactive repair method for the description logic \(\mathcal{EL}\) that is based on the optimal-repair framework. The obtained repair might not be optimal in the theoretical sense, i.e. more than a minimal amount of consequences might have been removed—but from a practical perspective it is superior to a theoretically optimal repair as the interaction strategy enables the users to identify further faulty consequences connected to the initially reported errors.

@techreport{ Kr-LTCS-24-05,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.25368/2024.319},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  number = {24-05},
  title = {Beyond Optimal: Interactive Identification of Better-than-optimal Repairs (Extended Version)},
  type = {LTCS-Report},
  year = {2024},
}

We present an FCA-based axiomatization method that produces a complete \(\mathcal{EL}\) TBox (the terminological part of an OWL 2 EL ontology) from a graph dataset in at most exponential time. We describe technical details that allow for efficient implementation as well as variations that dispense with the computation of extremely large axioms, thereby rendering the approach applicable albeit some completeness is lost. Moreover, we evaluate the prototype on real-world datasets.

@techreport{ Kr-LTCS-23-01,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.25368/2023.214},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  number = {23-01},
  title = {Efficient Axiomatization of OWL 2 EL Ontologies from Data by means of Formal Concept Analysis (Extended Version)},
  type = {LTCS-Report},
  year = {2023},
}

Ontologies based on Description Logics may contain errors, which are usually detected when reasoning produces consequences that follow from the ontology, but do not hold in the modelled application domain. In previous work, we have introduced repair approaches for \(\mathcal{EL}\) ontologies that are optimal in the sense that they preserve a maximal amount of consequences. In this paper, we will, on the one hand, review these approaches, but with an emphasis on motivation rather than on technical details. On the other hand, we will describe new results that address the problems that optimal repairs may become very large or need not even exist unless strong restrictions on the terminological part of the ontology apply. We will show how one can deal with these problems by introducing concise representations of optimal repairs.

@techreport{ BaKoKr-LTCS-23-03,
  address = {Dresden, Germany},
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel}},
  doi = {https://doi.org/10.25368/2023.121},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  number = {23-03},
  title = {Optimal Repairs in the Description Logic $\mathcal{EL}$ Revisited (Extended Version)},
  type = {LTCS-Report},
  year = {2023},
}

Errors in Description Logic (DL) ontologies are often detected when a reasoner computes unwanted consequences. The question is then how to repair the ontology such that the unwanted consequences no longer follow, but as many of the other consequences as possible are preserved. The problem of computing such optimal repairs was addressed in our previous work in the setting where the data (expressed by an ABox) may contain errors, but the schema (expressed by an \(\mathcal{EL}\) TBox) is assumed to be correct. Actually, we consider a generalization of ABoxes called quantified ABoxes (qABoxes) both as input for and as result of the repair process. Using qABoxes for repair allows us to retain more information, but the disadvantage is that standard DL systems do not accept qABoxes as input. This raises the question, investigated in the present paper, whether and how one can obtain optimal repairs if one restricts the output of the repair process to being ABoxes. In general, such optimal ABox repairs need not exist. Our main contribution is that we show how to decide the existence of optimal ABox repairs in exponential time, and how to compute all such repairs in case they exist.

@techreport{ BaKoKrNu-LTCS-22-01,
  address = {Dresden, Germany},
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  doi = {https://doi.org/10.25368/2022.65},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  number = {22-01},
  title = {Optimal ABox Repair w.r.t.\ Static {$\mathcal{EL}$} TBoxes: from Quantified ABoxes back to ABoxes (Extended Version)},
  type = {LTCS-Report},
  year = {2022},
}

Ontologies based on Description Logic (DL) represent general background knowledge in a terminology (TBox) and the actual data in an ABox. DL systems can then be used to compute consequences (such as answers to certain queries) from an ontology consisting of a TBox and an ABox. Since both human-made and machine-learned data sets may contain errors, which manifest themselves as unintuitive or obviously incorrect consequences, repairing DL-based ontologies in the sense of removing such unwanted consequences is an important topic in DL research. Most of the repair approaches described in the literature produce repairs that are not optimal, in the sense that they do not guarantee that only a minimal set of consequences is removed. In a series of papers, we have developed an approach for computing optimal repairs, starting with the restricted setting of an \(\mathcal{EL}\) instance store, extending this to the more general setting of a quantified ABox (where some individuals may be anonymous), and then adding a static \(\mathcal{EL}\) TBox.

Here, we extend the expressivity of the underlying DL considerably, by adding nominals, inverse roles, regular role inclusions and the bottom concept to \(\mathcal{EL}\), which yields a fragment of the well-known DL Horn-\(\mathcal{SROIQ}\). The ideas underlying our repair approach still apply to this DL, though several non-trivial extensions are needed to deal with the new constructors and axioms. The developed repair approach can also be used to treat unwanted consequences expressed by certain conjunctive queries or regular path queries, and to handle Horn-\(\mathcal{ALCOI}\) TBoxes with regular role inclusions.

@techreport{ BaKr-LTCS-22-02,
  address = {Dresden, Germany},
  author = {Franz {Baader} and Francesco {Kriegel}},
  doi = {https://doi.org/10.25368/2022.131},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  number = {22-02},
  title = {Pushing Optimal ABox Repair from {$\mathcal{EL}$} Towards More Expressive Horn-DLs (Extended Version)},
  type = {LTCS-Report},
  year = {2022},
}

Reasoners can be used to derive implicit consequences from an ontology. Sometimes unwanted consequences are revealed, indicating errors or privacy-sensitive information, and the ontology needs to be appropriately repaired. The classical approach is to remove just enough axioms such that the unwanted consequences vanish. However, this is often too rough since mere axiom deletion also erases many other consequences that might actually be desired. The goal should not be to remove a minimal number of axioms but to modify the ontology such that only a minimal number of consequences is removed, including the unwanted ones. Specifically, a repair should rather be logically entailed by the input ontology, instead of being a subset. To this end, we introduce a framework for computing fixed-premise repairs of \(\mathcal{EL}\) TBoxes. In the first variant the conclusions must be generalizations of those in the input TBox, while in the second variant no such restriction is imposed. In both variants, every repair is entailed by an optimal one and, up to equivalence, the set of all optimal repairs can be computed in exponential time. A prototypical implementation is provided. In addition, we show new complexity results regarding gentle repairs.

@techreport{ Kr-LTCS-22-04,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.25368/2022.321},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  number = {22-04},
  title = {Optimal Fixed-Premise Repairs of {$\mathcal{EL}$} TBoxes (Extended Version)},
  type = {LTCS-Report},
  year = {2022},
}

The application of automated reasoning approaches to Description Logic (DL) ontologies may produce certain consequences that either are deemed to be wrong or should be hidden for privacy reasons. The question is then how to repair the ontology such that the unwanted consequences can no longer be deduced. An optimal repair is one where the least amount of other consequences is removed. Most of the previous approaches to ontology repair are of a syntactic nature in that they remove or weaken the axioms explicitly present in the ontology, and thus cannot achieve semantic optimality. In previous work, we have addressed the problem of computing optimal repairs of (quantified) ABoxes, where the unwanted consequences are described by concept assertions of the light-weight DL \(\mathcal{EL}\). In the present paper, we improve on the results achieved so far in two ways. First, we allow for the presence of terminological knowledge in the form of an \(\mathcal{EL}\) TBox. This TBox is assumed to be static in the sense that it cannot be changed in the repair process. Second, the construction of optimal repairs described in our previous work is best case exponential. We introduce an optimized construction that is exponential only in the worst case. First experimental results indicate that this reduces the size of the computed optimal repairs considerably.

@techreport{ BaKoKrNu-LTCS-21-01,
  address = {Dresden, Germany},
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  doi = {https://doi.org/10.25368/2022.64},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  number = {21-01},
  title = {Computing Optimal Repairs of Quantified ABoxes w.r.t. Static {$\mathcal{EL}$} TBoxes (Extended Version)},
  type = {LTCS-Report},
  year = {2021},
}

We review our recent work on how to compute optimal repairs, optimal compliant anonymizations, and optimal safe anonymizations of ABoxes containing possibly anonymized individuals. The results can be used both to remove erroneous consequences from a knowledge base and to hide secret information before publication of the knowledge base, while keeping as much as possible of the original information.

@techreport{ BaKoKrNuPe-LTCS-21-04,
  address = {Dresden, Germany},
  author = {Franz {Baader} and Patrick {Koopmann} and Francesco {Kriegel} and Adrian {Nuradiansyah} and Rafael {Pe\~{n}aloza}},
  doi = {https://doi.org/10.25368/2022.268},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  number = {21-04},
  title = {{Privacy-Preserving Ontology Publishing: The Case of Quantified ABoxes w.r.t.\ a Static Cycle-Restricted $\mathcal{EL}$ TBox (Extended Version)}},
  type = {LTCS-Report},
  year = {2021},
}

We adapt existing approaches for privacy-preserving publishing of linked data to a setting where the data are given as Description Logic (DL) ABoxes with possibly anonymised (formally: existentially quantified) individuals and the privacy policies are expressed using sets of concepts of the DL \(\mathcal{E\!L}\). We provide a chacterization of compliance of such ABoxes w.r.t. \(\mathcal{E\!L}\) policies, and show how optimal compliant anonymisations of ABoxes that are non-compliant can be computed. This work extends previous work on privacy-preserving ontology publishing, in which a very restricted form of ABoxes, called instance stores, had been considered, but restricts the attention to compliance. The approach developed here can easily be adapted to the problem of computing optimal repairs of quantified ABoxes.

@techreport{ BaKrNuPe-LTCS-20-08,
  address = {Dresden, Germany},
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah} and Rafael {Pe\~{n}aloza}},
  doi = {https://doi.org/10.25368/2022.263},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  number = {20-08},
  title = {Computing Compliant Anonymisations of Quantified ABoxes w.r.t. $\mathcal{E\!L}$ Policies (Extended Version)},
  type = {LTCS-Report},
  year = {2020},
}

In recent work, we have shown how to compute compliant anonymizations of quantified ABoxes w.r.t. \(\mathcal{E\!L}\) policies. In this setting, quantified ABoxes can be used to publish information about individuals, some of which are anonymized. The policy is given by concepts of the Description Logic (DL) \(\mathcal{E\!L}\), and compliance means that one cannot derive from the ABox that some non-anonymized individual is an instance of a policy concept. If one assumes that a possible attacker could have additional knowledge about some of the involved non-anonymized individuals, then compliance with a policy is not sufficient. One wants to ensure that the quantified ABox is safe in the sense that none of the secret instance information is revealed, even if the attacker has additional compliant knowledge. In the present paper, we show that safety can be decided in polynomial time, and that the unique optimal safe anonymization of a non-safe quantified ABox can be computed in exponential time, provided that the policy consists of a single \(\mathcal{E\!L}\) concept.

@techreport{ BaKrNuPe-LTCS-20-09,
  address = {Dresden, Germany},
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah} and Rafael {Pe\~{n}aloza}},
  doi = {https://doi.org/10.25368/2022.264},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  number = {20-09},
  title = {Computing Safe Anonymisations of Quantified ABoxes w.r.t.\ $\mathcal{E\!L}$ Policies (Extended Version)},
  type = {LTCS-Report},
  year = {2020},
}

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 \(\mathcal{EL}\), which corresponds to the setting where the ontology is an \(\mathcal{EL}\) 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 \(\mathcal{EL}\) concept can be computed. In addition, we investigate the complexity of the optimality problem.

@techreport{ BaKrNu-LTCS-19-01,
  address = {Dresden, Germany},
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah}},
  doi = {https://doi.org/10.25368/2022.250},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  note = {\url{https://tu-dresden.de/inf/lat/reports#BaKrNu-LTCS-19-01}},
  number = {19-01},
  title = {{Privacy-Preserving Ontology Publishing for $\mathcal{EL}$ Instance Stores (Extended Version)}},
  type = {LTCS-Report},
  year = {2019},
}

A joining implication is a restricted form of an implication where it is explicitly specified which attributes may occur in the premise and in the conclusion, respectively. A technique for sound and complete axiomatization of joining implications valid in a given formal context is provided. In particular, a canonical base for the joining implications valid in a given formal context is proposed, which enjoys the property of being of minimal cardinality among all such bases. Background knowledge in form of a set of valid joining implications can be incorporated. Furthermore, an application to inductive learning in a Horn description logic is proposed, that is, a procedure for sound and complete axiomatization of \(\textsf{Horn-}\mathcal{M}\) concept inclusions from a given interpretation is developed. A complexity analysis shows that this procedure runs in deterministic exponential time.

@techreport{ Kr-LTCS-19-02,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.25368/2022.251},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  note = {\url{https://tu-dresden.de/inf/lat/reports#Kr-LTCS-19-02}},
  number = {19-02},
  title = {{Joining Implications in Formal Contexts and Inductive Learning in a Horn Description Logic (Extended Version)}},
  type = {LTCS-Report},
  year = {2019},
}

The classical approach for repairing a Description Logic ontology \(\mathfrak{O}\) in the sense of removing an unwanted consequence \(\alpha\) is to delete a minimal number of axioms from \(\mathfrak{O}\) such that the resulting ontology \(\mathfrak{O}'\) does not have the consequence \(\alpha\). However, the complete deletion of axioms may be too rough, in the sense that it may also remove consequences that are actually wanted. To alleviate this problem, we propose a more gentle way of repair in which axioms are not necessarily deleted, but only weakened. On the one hand, we investigate general properties of this gentle repair method. On the other hand, we propose and analyze concrete approaches for weakening axioms expressed in the Description Logic \(\mathcal{E\mkern-1.618mu L}\).

@techreport{ BaKrNuPe-LTCS-18-01,
  address = {Dresden, Germany},
  author = {Franz {Baader} and Francesco {Kriegel} and Adrian {Nuradiansyah} and Rafael {Pe\~{n}aloza}},
  doi = {https://doi.org/10.25368/2022.238},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  note = {\url{https://tu-dresden.de/inf/lat/reports#BaKrNuPe-LTCS-18-01}},
  number = {18-01},
  title = {{Repairing Description Logic Ontologies by Weakening Axioms}},
  type = {LTCS-Report},
  year = {2018},
}

For a probabilistic extension of the description logic \(\mathcal{E\mkern-1.618mu L}^{\bot}\), we consider the task of automatic acquisition of terminological knowledge from a given probabilistic interpretation. Basically, such a probabilistic interpretation is a family of directed graphs the vertices and edges of which are labeled, and where a discrete probability measure on this graph family is present. The goal is to derive so-called concept inclusions which are expressible in the considered probabilistic description logic and which hold true in the given probabilistic interpretation. A procedure for an appropriate axiomatization of such graph families is proposed and its soundness and completeness is justified.

@techreport{ Kr-LTCS-18-03,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.25368/2022.239},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  note = {\url{https://tu-dresden.de/inf/lat/reports#Kr-LTCS-18-03}},
  number = {18-03},
  title = {{Terminological Knowledge Acquisition in Probabilistic Description Logic}},
  type = {LTCS-Report},
  year = {2018},
}

For the description logic \(\mathcal{E\mkern-1.618mu L}\), we consider the neighborhood relation which is induced by the subsumption order, and we show that the corresponding lattice of \(\mathcal{E\mkern-1.618mu L}\) concept descriptions is distributive, modular, graded, and metric. In particular, this implies the existence of a rank function as well as the existence of a distance function.

@techreport{ Kr-LTCS-18-10,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.25368/2022.245},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  note = {\url{https://tu-dresden.de/inf/lat/reports#Kr-LTCS-18-10}},
  number = {18-10},
  title = {{The Distributive, Graded Lattice of $\mathcal{E\mkern-1.618mu L}$ Concept Descriptions and its Neighborhood Relation (Extended Version)}},
  type = {LTCS-Report},
  year = {2018},
}

The notion of a most specific consequence with respect to some terminological box is introduced, conditions for its existence in the description logic \(\mathcal{E\mkern-1.618mu L}\) 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.

@techreport{ Kr-LTCS-18-11,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.25368/2022.246},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  note = {\url{https://tu-dresden.de/inf/lat/reports#Kr-LTCS-18-11}, accepted for publication in {Discrete Applied Mathematics}},
  number = {18-11},
  title = {{Most Specific Consequences in the Description Logic $\mathcal{E\mkern-1.618mu L}$}},
  type = {LTCS-Report},
  year = {2018},
}

Description logics in their standard setting only allow for representing and reasoning with crisp knowledge without any degree of uncertainty. Of course, this is a serious shortcoming for use cases where it is impossible to perfectly determine the truth of a statement. For resolving this expressivity restriction, probabilistic variants of description logics have been introduced. Their model-theoretic semantics is built upon so-called probabilistic interpretations, that is, families of directed graphs the vertices and edges of which are labeled and for which there exists a probability measure on this graph family.

Results of scientific experiments, e.g., in medicine, psychology, or biology, that are repeated several times can induce probabilistic interpretations in a natural way. In this document, we shall develop a suitable axiomatization technique for deducing terminological knowledge from the assertional data given in such probabilistic interpretations. More specifically, we consider a probabilistic variant of the description logic \(\mathcal{E\mkern-1.618mu L}^{\!\bot}\), and provide a method for constructing a set of rules, so-called concept inclusions, from probabilistic interpretations in a sound and complete manner.

@techreport{ Kr-LTCS-18-12,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.25368/2022.247},
  institution = {Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  note = {\url{https://tu-dresden.de/inf/lat/reports#Kr-LTCS-18-12}},
  number = {18-12},
  title = {{Learning Description Logic Axioms from Discrete Probability Distributions over Description Graphs (Extended Version)}},
  type = {LTCS-Report},
  year = {2018},
}

It is well-known that the canonical implicational base of all implications valid w.r.t. a closure operator can be obtained from the set of all pseudo-closures. NextClosures is a parallel algorithm to compute all closures and pseudo-closures of a closure operator in a graded lattice, e.g., in a powerset. Furthermore, the closures and pseudo-closures can be constrained, and partially known closure operators can be explored.

@techreport{ Kr-LTCS-15-01,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  institution = {Chair for Automata Theory, Institute for Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  note = {\url{https://tu-dresden.de/inf/lat/reports#Kr-LTCS-15-01}},
  number = {15-01},
  title = {{NextClosures -- Parallel Exploration of Constrained Closure Operators}},
  type = {LTCS-Report},
  year = {2015},
}

Model-based most specific concept descriptions are a useful means to compactly represent all knowledge about a certain individual of an interpretation that is expressible in the underlying description logic. Existing approaches only cover their construction in the case of \(\mathcal{E\mkern-1.618mu L}\) and \(\mathcal{F\mkern-1.618mu L\mkern-1.618mu E}\) w.r.t. greatest fixpoint semantics, and the case of \(\mathcal{E\mkern-1.618mu L}\) w.r.t. a role-depth bound, respectively. This document extends the results towards the more expressive description logic \(\mathcal{A\mkern-1.618mu L\mkern-1.618mu E\mkern-1.618mu Q}^{\geq}\mkern-1.618mu\mathcal{N}^{\leq}\mkern-1.618mu(\mathsf{Self})\) w.r.t. role-depth bounds and also gives a method for the computation of least common subsumers.

@techreport{ Kr-LTCS-15-02,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  institution = {Chair for Automata Theory, Institute for Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  note = {\url{https://tu-dresden.de/inf/lat/reports#Kr-LTCS-15-02}},
  number = {15-02},
  title = {{Model-Based Most Specific Concept Descriptions and Least Common Subsumers in $\mathcal{A\mkern-1.618mu L\mkern-1.618mu E\mkern-1.618mu Q}^{\geq}\mkern-1.618mu\mathcal{N}^{\leq}\mkern-1.618mu(\mathsf{Self})$}},
  type = {LTCS-Report},
  year = {2015},
}

Description logic knowledge bases can be used to represent knowledge about a particular domain in a formal and unambiguous manner. Their practical relevance has been shown in many research areas, especially in biology and the semantic web. However, the tasks of constructing knowledge bases itself, often performed by human experts, is difficult, time-consuming and expensive. In particular the synthesis of terminological knowledge is a challenge every expert has to face. Because human experts cannot be omitted completely from the construction of knowledge bases, it would therefore be desirable to at least get some support from machines during this process. To this end, we shall investigate in this work an approach which shall allow us to extract terminological knowledge in the form of general concept inclusions from factual data, where the data is given in the form of vertex and edge labeled graphs. As such graphs appear naturally within the scope of the Semantic Web in the form of sets of RDF triples, the presented approach opens up the possibility to extract terminological knowledge from the Linked Open Data Cloud. We shall also present first experimental results showing that our approach has the potential to be useful for practical applications.

@techreport{ BoDiKr-LTCS-15-13,
  address = {Dresden, Germany},
  author = {Daniel {Borchmann} and Felix {Distel} and Francesco {Kriegel}},
  doi = {https://doi.org/10.25368/2022.219},
  institution = {Chair for Automata Theory, Institute for Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  note = {\url{https://tu-dresden.de/inf/lat/reports#BoDiKr-LTCS-15-13}},
  number = {15-13},
  title = {{Axiomatization of General Concept Inclusions from Finite Interpretations}},
  type = {LTCS-Report},
  year = {2015},
}

Probabilistic interpretations consist of a set of interpretations with a shared domain and a measure assigning a probability to each interpretation. Such structures can be obtained as results of repeated experiments, e.g., in biology, psychology, medicine, etc. A translation between probabilistic and crisp description logics is introduced and then utilised to reduce the construction of a base of general concept inclusions of a probabilistic interpretation to the crisp case for which a method for the axiomatisation of a base of GCIs is well-known.

@techreport{ Kr-LTCS-15-14,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  doi = {https://doi.org/10.25368/2022.220},
  institution = {Chair for Automata Theory, Institute for Theoretical Computer Science, Technische Universit{\"a}t Dresden},
  note = {\url{https://tu-dresden.de/inf/lat/reports#Kr-LTCS-15-14}},
  number = {15-14},
  title = {{Learning General Concept Inclusions in Probabilistic Description Logics}},
  type = {LTCS-Report},
  year = {2015},
}

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.

@thesis{ Kriegel-Diss-2019,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  school = {Technische Universit\"{a}t Dresden},
  title = {Constructing and Extending Description Logic Ontologies using Methods of Formal Concept Analysis},
  type = {Doctoral Thesis},
  year = {2019},
}

Draft and proof of an algorithm computing incremental changes within a labeled layouted concept lattice upon insertion or removal of an attribute column in the underlying formal context. Furthermore some implementational details and mathematical background knowledge are presented.

@thesis{ Kriegel-Diploma-2013,
  address = {Dresden, Germany},
  author = {Francesco {Kriegel}},
  school = {Technische Universit\"{a}t Dresden},
  title = {Visualization of Conceptual Data with Methods of Formal Concept Analysis},
  type = {Diploma Thesis},
  year = {2013},
}