Barış Sertkaya

Computing the least common subsumer w.r.t. a background terminology

Abstract Methods for computing the least common subsumer (lcs) are usually restricted to rather inexpressive Description Logics (DLs) whereas existing knowledge bases are written in very expressive DLs. In order to allow the user to re-use concepts defined in such terminologies and still support the definition of new concepts by computing the lcs, we extend the notion of the lcs of concept descriptions to the notion of the lcs w.r.t. a background terminology. We will show both theoretical results on the existence of the least common subsumer in this setting, and describe a practical approach—based on a method from formal concept analysis—for computing good common subsumers, which may, however, not be the least ones. We will also describe results obtained in a first evaluation of this practical approach.

Applying Formal Concept Analysis to Description Logics

Given a finite set \(\mathcal{C} := \{ C_1, \ldots, C_n\}\) of description logic concepts, we are interested in computing the subsumption hierarchy of all least common subsumers of subsets of \(\mathcal{C}\) as well as the hierarchy of all conjunctions of subsets of \(\mathcal{C}\). These hierarchies can be used to support the bottom-up construction of description logic knowledge bases. The point is to compute the first hierarchy without having to compute the least common subsumer for all subsets of \(\mathcal{C}\), and the second hierarchy without having to check all possible pairs of such conjunctions explicitly for subsumption. We will show that methods from formal concept analysis developed for computing concept lattices can be employed for this purpose.

Computing the Least Common Subsumer w.r.t. a Background Terminology

Methods for computing the least common subsumer (lcs) are usually restricted to rather inexpressive Description Logics (DLs) whereas existing knowledge bases are written in very expressive DLs. In order to allow the user to re-use concepts defined in such terminologies and still support the definition of new concepts by computing the lcs, we extend the notion of the lcs of concept descriptions to the notion of the lcs w.r.t. a background terminology. We will both show a theoretical result on the existence of the least common subsumer in this setting, and describe a practical approach (based on a method from formal concept analysis) for computing good common subsumers, which may, however, not be the least ones.

Usability Issues in Description Logic Knowledge Base Completion

In a previous paper, we have introduced an approach for extending both the terminological and the assertional part of a Description Logic knowledge base by using information provided by the assertional part and by a domain expert. This approach, called knowledge base completion, was based on an extension of attribute exploration to the case of partial contexts. The present paper recalls this approach, and then addresses usability issues that came up during first experiments with a preliminary implementation of the completion algorithm. It turns out that these issues can be addressed by extending the exploration algorithm for partial contexts such that it can deal with implicational background knowledge.

Some Computational Problems Related to Pseudo-intents

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

On the complexity of computing generators of closed sets

We investigate the computational complexity of some decision and counting problems related to generators of closed sets fundamental in Formal Concept Analysis. We recall results from the literature about the problem of checking the existence of a generator with a specified cardinality, and about the problem of determining the number of minimal generators. Moreover, we show that the problem of counting minimum cardinality generators is #ċcoNP-complete. We also present an incremental-polynomial time algorithm from relational database theory that can be used for computing all minimal generators of an implication-closed set.

OntoComP: A Protégé Plugin for Completing OWL Ontologies

We describe OntoComP , a Protege 4 plugin that supports ontology engineers in completing OWL ontologies. More precisely, OntoComP supports an ontology engineer in checking whether an ontology contains all the relevant information about the application domain, and in extending the ontology appropriately if this is not the case. It acquires complete knowledge about the application domain efficiently by asking successive questions to the ontology engineer. By using novel techniques from Formal Concept Analysis, it ensures that, on the one hand, the interaction with the ontology engineer is kept to a minimum, and, on the other hand, the resulting ontology is complete in a certain well-defined sense.

Towards the Complexity of Recognizing Pseudo-intents

Pseudo-intents play a key role in Formal Concept Analysis. They are the premises of the implications in the Duquenne-Guigues Base, which is a minimum cardinality base for the set of implications that hold in a formal context. It has been shown that checking whether a set is a pseudo-intent is in co np . However, it is still open whether this problem is co np -hard, or it is solvable in polynomial time. In the current work we prove a first lower bound for this problem by showing that it is at least as hard as transversal hypergraph , which is the problem of identifying the minimal transversals of a given hypergraph. This is a prominent open problem in hypergraph theory that is conjectured to form a complexity class properly contained between p and co np . Our result explains why the attempts to find a polynomial algorithm for recognizing pseudo-intents have failed until now. We also formulate a decision problem, namely first pseudo-intent , and show that if this problem is not polynomial, then, unless p = np , pseudo-intents cannot be enumerated with polynomial delay in a specified lexicographic order.

Proof of the Basic Theorem on Concept Lattices in Isabelle/HOL

This paper presents a machine-checked proof of the Basic Theorem on Concept Lattices, which appears in the book “Formal Concept Analysis” by Ganter and Wille, in the Isabelle/HOL Proof Assistant. As a by-product, the underlying lattice theory by Kammueller has been extended.