Daniel Borchmann

Fast algorithms for implication bases and attribute exploration using proper premises

A central task in formal concept analysis is the enumeration of a small base for the implications that hold in a formal context. The usual stem base algorithms have been proven to be costly in terms of runtime. Proper premises are an alternative to the stem base. We present a new algorithm for the fast computation of proper premises. It is based on a known link between proper premises and minimal hypergraph transversals. Two further improvements are made, which reduce the number of proper premises that are obtained multiple times and redundancies within the set of proper premises. We have evaluated our algorithms within an application related to refactoring of model variants. In this application an implicational base needs to be computed, and runtime is more crucial than minimal cardinality. In addition to the empirical tests, we provide heuristic evidence that an approach based on proper premises will also be beneficial for other applications. Finally, we show how our algorithms can be extended to an exploration algorithm that is based on proper premises.

Mining of EL-GCIs

We consider an existing approach for mining general inclusion axioms written in a lightweight Description Logic. In comparison to classical association rule mining, this approach allows more complex patterns to be obtained. Ours is the first implementation of these algorithms for learning Description Logic axioms. We use our implementation for a case study on two real world datasets. We discuss the outcome and examine what further research will be needed for this approach to be applied in a practical setting.

Axiomatisation of general concept inclusions from finite interpretations

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 that 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-labelled graphs. Because such graphs appear naturally within the scope of the Semantic Web in the form of sets of Resource Description Framework (RDF) triples, the presented approach opens up another possibility to extract terminological knowledge from the Linked Open Data Cloud.

Individuality in Social Networks

We consider individuality in bi-modal social networks, a facet that has not been considered before in the mathematical analysis of social networks. We use methods from formal concept analysis to develop a natural definition for individuality, and provide experimental evidence that this yields a meaningful approach for additional insights into the nature of social networks.

Topological entropy of formal languages

We introduce the notion of topological entropy of a formal language as the topological entropy of the minimal topological automaton accepting it. Using a characterization of this notion in terms of approximations of the Myhill–Nerode congruence relation, we are able to compute the topological entropies of certain example languages. Those examples suggest that the notion of a “simple” formal language coincides with the language having zero entropy.

The Identity Problem in Description Logic Ontologies and Its Application to View-Based Information Hiding

The work in this paper is motivated by a privacy scenario in which the identity of certain persons (represented as anonymous individuals) should be hidden. We assume that factual information about known individuals (i.e., individuals whose identity is known) and anonymous individuals is stored in an ABox and general background information is expressed in a TBox, where both the TBox and the ABox are publicly accessible. The identity problem then asks whether one can deduce from the TBox and the ABox that a given anonymous individual is equal to a known one. Since this would reveal the identity of the anonymous individual, such a situation needs to be avoided. We first observe that not all Description Logics (DLs) are able to derive any such equalities between individuals, and thus the identity problem is trivial in these DLs. We then consider DLs with nominals, number restrictions, or function dependencies, in which the identity problem is non-trivial. We show that in these DLs the identity problem has the same complexity as the instance problem. Finally, we consider an extended scenario in which users with different roles can access different parts of the TBox and ABox, and we want to check whether, by a sequence of role changes and queries asked in each role, one can deduce the identity of an anonymous individual.

On the Usability of Probably Approximately Correct Implication Bases

We revisit the notion of probably approximately correct implication bases from the literature and present a first formulation in the language of formal concept analysis, with the goal to investigate whether such bases represent a suitable substitute for exact implication bases in practical use cases. To this end, we quantitatively examine the behavior of probably approximately correct implication bases on artificial and real-world data sets and compare their precision and recall with respect to their corresponding exact implication bases. Using a small example, we also provide evidence suggesting that implications from probably approximately correct bases can still represent meaningful knowledge from a given data set.

Implications and Dependencies Between Attributes

This work is a translation of “Implikationen und Abhang-igkeiten zwischen Merkmalen” by Bernhard Ganter and Rudolf Wille, Technische Hochschule Darmstadt, Preprint-Number 1017, 1986. The manuscript has originally been published in “Die Klassifikation und ihr Umfeld”, edited by P. O. Degens, H. J. Hermes, and O. Opitz, Indeks-Verlag, Frankfurt, 1986 (rights now with Ergon-Verlag).

NextClosures: parallel computation of the canonical base with background knowledge

Abstract 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.

Exploring Faulty Data

Within formal concept analysis, attribute exploration is a powerful tool to semi-automatically check data for completeness with respect to a given domain. However, the classical formulation of attribute exploration does not take into account possible errors which are present in the initial data. To remedy this, we present in this work a generalization of attribute exploration based on the notion of confidence, that will allow for the exploration of implications which are not necessarily valid in the initial data, but instead enjoy a minimal confidence therein.