Extending Conceptualisation Modes for Generalised Formal Concept Analysis

Formal Concept Analysis (FCA) is an exploratory data analysis technique for boolean relations based on lattice theory. Its main result is the existence of a dual order isomorphism between two set lattices induced by a binary relation between a set of objects and a set of attributes. Pairs of dually isomorphic sets of objects and attributes, called formal concepts, form a concept lattice, but actually model only a conjunctive mode of conceptualisation. In this paper we augment this formalism in two ways: first we extend FCA to consider different modes of conceptualisation by changing the basic dual isomorphism in a modal-logic motivated way. This creates the three new types of concepts and lattices of extended FCA, viz., the lattice of neighbourhood of objects, that of attributes and the lattice of unrelatedness. Second, we consider incidences with values in idempotent semirings—concretely the completed max-plus or schedule algebra View the MathML source—and focus on generalising FCA to try and replicate the modes of conceptualisation mentioned above. To provide a concrete example of the use of these techniques, we analyse the performance of multi-class classifiers by conceptually analysing their confusion matrices.Spanish Government-Comisión Interministerial de Ciencia y Tecnología project 2008–06382/TEC and 2008–02473/TEC and the regional project (Comunidad Autónoma de Madrid – UC3M) CCG08-UC3M/TIC-4457Publicad

Bisimulations for Kripke models of Fuzzy Multimodal Logics

The main objective of the dissertation is to provide a detailed study of several different types of simulations and bisimulations for Kripke models of fuzzy multimodal logics. Two types of simulations (forward and backward) and five types of bisimulations (forward, backward, forward-backward, backward-forward and regular) are presented hereby. For each type of simulation and bisimulation, an algorithm is created to test the existence of the simulation or bisimulation and, if it exists, the algorithm computes the greatest one. The dissertation presents the application of bisimulations in the state reduction of fuzzy Kripke models, while preserving their semantic properties. Next, weak simulations and bisimulations were considered and the Hennessy-Milner property was examined. Finally, an algorithm was created to compute weak simulations and bisimulations for fuzzy Kripke models over locally finite algebras

Fuzzy Description Logics with General Concept Inclusions

Description logics (DLs) are used to represent knowledge of an application domain and provide standard reasoning services to infer consequences of this knowledge. However, classical DLs are not suited to represent vagueness in the description of the knowledge. We consider a combination of DLs and Fuzzy Logics to address this task. In particular, we consider the t-norm-based semantics for fuzzy DLs introduced by Hájek in 2005. Since then, many tableau algorithms have been developed for reasoning in fuzzy DLs. Another popular approach is to reduce fuzzy ontologies to classical ones and use existing highly optimized classical reasoners to deal with them. However, a systematic study of the computational complexity of the different reasoning problems is so far missing from the literature on fuzzy DLs. Recently, some of the developed tableau algorithms have been shown to be incorrect in the presence of general concept inclusion axioms (GCIs). In some fuzzy DLs, reasoning with GCIs has even turned out to be undecidable. This work provides a rigorous analysis of the boundary between decidable and undecidable reasoning problems in t-norm-based fuzzy DLs, in particular for GCIs. Existing undecidability proofs are extended to cover large classes of fuzzy DLs, and decidability is shown for most of the remaining logics considered here. Additionally, the computational complexity of reasoning in fuzzy DLs with semantics based on finite lattices is analyzed. For most decidability results, tight complexity bounds can be derived

Conceptual Factors and Fuzzy Data

With the growing number of large data sets, the necessity of complexity reduction applies today more than ever before. Moreover, some data may also be vague or uncertain. Thus, whenever we have an instrument for data analysis, the questions of how to apply complexity reduction methods and how to treat fuzzy data arise rather naturally. In this thesis, we discuss these issues for the very successful data analysis tool Formal Concept Analysis. In fact, we propose different methods for complexity reduction based on qualitative analyses, and we elaborate on various methods for handling fuzzy data. These two topics split the thesis into two parts. Data reduction is mainly dealt with in the first part of the thesis, whereas we focus on fuzzy data in the second part. Although each chapter may be read almost on its own, each one builds on and uses results from its predecessors. The main crosslink between the chapters is given by the reduction methods and fuzzy data. In particular, we will also discuss complexity reduction methods for fuzzy data, combining the two issues that motivate this thesis.Komplexitätsreduktion ist eines der wichtigsten Verfahren in der Datenanalyse. Mit ständig wachsenden Datensätzen gilt dies heute mehr denn je. In vielen Gebieten stößt man zudem auf vage und ungewisse Daten. Wann immer man ein Instrument zur Datenanalyse hat, stellen sich daher die folgenden zwei Fragen auf eine natürliche Weise: Wie kann man im Rahmen der Analyse die Variablenanzahl verkleinern, und wie kann man Fuzzy-Daten bearbeiten? In dieser Arbeit versuchen wir die eben genannten Fragen für die Formale Begriffsanalyse zu beantworten. Genauer gesagt, erarbeiten wir verschiedene Methoden zur Komplexitätsreduktion qualitativer Daten und entwickeln diverse Verfahren für die Bearbeitung von Fuzzy-Datensätzen. Basierend auf diesen beiden Themen gliedert sich die Arbeit in zwei Teile. Im ersten Teil liegt der Schwerpunkt auf der Komplexitätsreduktion, während sich der zweite Teil der Verarbeitung von Fuzzy-Daten widmet. Die verschiedenen Kapitel sind dabei durch die beiden Themen verbunden. So werden insbesondere auch Methoden für die Komplexitätsreduktion von Fuzzy-Datensätzen entwickelt