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 qualitative insight that implications from probably approximately correct bases can still represent meaningful knowledge from a given data set.Comment: 17 pages, 8 figures; typos added, corrected x-label on graph

Tuplix Calculus

We introduce a calculus for tuplices, which are expressions that generalize matrices and vectors. Tuplices have an underlying data type for quantities that are taken from a zero-totalized field. We start with the core tuplix calculus CTC for entries and tests, which are combined using conjunctive composition. We define a standard model and prove that CTC is relatively complete with respect to it. The core calculus is extended with operators for choice, information hiding, scalar multiplication, clearing and encapsulation. We provide two examples of applications; one on incremental financial budgeting, and one on modular financial budget design.Comment: 22 page

Most specific consequences in the description logic EL

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

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

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

On the complexity of enumerating pseudo-intents

AbstractWe investigate whether the pseudo-intents of a given formal context can efficiently be enumerated. We show that they cannot be enumerated in a specified lexicographic order with polynomial delay unless P=NP. Furthermore we show that if the restriction on the order of enumeration is removed, then the problem becomes at least as hard as enumerating minimal transversals of a given hypergraph. We introduce the notion of minimal pseudo-intents and show that recognizing minimal pseudo-intents is polynomial. Despite their less complicated nature, surprisingly it turns out that minimal pseudo-intents cannot be enumerated in output-polynomial time unless P=NP

FCA2VEC: Embedding Techniques for Formal Concept Analysis

Embedding large and high dimensional data into low dimensional vector spaces is a necessary task to computationally cope with contemporary data sets. Superseding latent semantic analysis recent approaches like word2vec or node2vec are well established tools in this realm. In the present paper we add to this line of research by introducing fca2vec, a family of embedding techniques for formal concept analysis (FCA). Our investigation contributes to two distinct lines of research. First, we enable the application of FCA notions to large data sets. In particular, we demonstrate how the cover relation of a concept lattice can be retrieved from a computational feasible embedding. Secondly, we show an enhancement for the classical node2vec approach in low dimension. For both directions the overall constraint of FCA of explainable results is preserved. We evaluate our novel procedures by computing fca2vec on different data sets like, wiki44 (a dense part of the Wikidata knowledge graph), the Mushroom data set and a publication network derived from the FCA community.Comment: 25 page