Bernhard Ganter – Radim Belohlavek – Sebastian Rudolph – M. Eugenia Cornejo
Bernhard Ganter is a Professor Emeritus of Mathematics (Algebraic Theory of Structures) at TU Dresden, Germany. In his scientific work, Ganter initially dealt with general algebra, order theory, discrete mathematics and connections between these areas. He received his doctorate in 1974. Four years later he became an associate professor in Rudolf Wille’s group at TU Darmstadt, which founded and developed the field of Formal Concept Analysis in the early 1980s.
In 1993, Ganter was then appointed to a position as full professor at TU Dresden, which he held until his retirement, eventually as the Dean of Science.
Ganter is co-author and co-editor of several fundamental publications on Formal Concept Analysis, including the standard monographies “Formal Concept Analysis – Mathematical Foundations” and “Conceptual Exploration”.
In 2008, together with Volker Nollau, he founded “Erlebnisland Mathematik” (“Mathematics Adventure Land”), a popular permanent exhibition located in the Dresden Technology Collections Museum. Many of the exhibits shown there were designed by him.
Title: Contextual Attribute Logics
Abstract: Mathematical formal logic is a comprehensive and profound theory that is valuable and significant also for Formal Concept Analysis. However, it is not to be understood as “concept logic” in the philosophical sense. Its language and technical terms are geared towards its main field of application, mathematics. That is why Rudolf Wille began early on to formulate a logic of formal concepts under the name “Contextual Logic”.
A first building block for this is Contextual Attribute Logic. However, the initial publication on this topic remained largely unnoticed because it did not show much that was mathematically new. In fact, a reader could quite rightly say that this attribute logic is merely reformulated Propositional Logic (and more generally that Contextual Logic can be translated into Predicate Logic).
However, this very argument was also put forward for years to Formal Concept Analysis itself. The fact that a reformulation, guided by an interpretative goal, and the associated restructuring can release completely new forces, was only understood when Formal Concept Analysis was taken up by the academic world in thousands of scientific publications. A suitably coordinated logical language is part of such a restructuring.
In our talk, we will present Contextual Attribute Logic again, going a few steps further than the previously published contributions. As always, our goal is to provide a framework that is suitable for holding together the divergent developments within Formal Concept Analysis and contributes to a more unified theory.
Radim Belohlavek is professor of computer science at Palacky University Olomouc. He holds PhD in computer science and in mathematics, and obtained a DSc in informatics and cybernetics from the Academy of Sciences of the Czech Republic. Earlier in his career he was research fellow and in 2007-2009 tenured professor at the State University of New York. His academic interests are in the areas of logic, discrete mathematics, artificial intelligence, data analysis, and uncertainty and information. R. Belohlavek authored or coauthored four books published by Kluwer, Springer, the MIT Press, and the Oxford University Press, and over 200 papers in conference proceedings and journals including J. Computer and System Sciences, Annals of Pure and Applied Logic, Archive for Math. Logic, Discrete Applied Mathematics, Neural Computation, J. Logic and Computation, J. Mathematical Analysis and Applications, Fundamenta Informaticae, Order, Algebra Universalis, IEEE Trans. on Systems, Man, and Cybernetics, Int. J. Approximate Reasoning, and Fuzzy Sets and Systems. He served as editor-in-chief of the International Journal of General Systems and serves on editorial boards of other journals including J. Computer and System Sciences, Fundamenta Informaticae, Annals of Mathematics and Artificial Intelligence, and Fuzzy Sets and Systems. R. Belohlavek is married and has four children. His hobbies include literature and sports.
Title: Interactions between the psychology of concepts and conceptual exploration of data
Abstract: Both the psychology of concepts and conceptual exploration of data have developed a rich body of knowledge (repertoire of theories and tools). Naturally, these two areas have different aims. We argue that despite their focus on different phenomena, proper interactions between the psychology of concepts and conceptual exploration of data may benefit both areas. For this purpose, we provide examples of recent work involving human categorization, typicality, similarity, and factor analysis.
Sebastian Rudolph is full professor for computational logic and head of the AI Institute at TU Dresden. His scientific interests revolve around artificial intelligence, with a particular focus on logical formalisms and methods for knowledge representation and reasoning, ranging from theoretical foundations (such as semantic and computational properties) to practical deployment (including ontological modeling and interactive knowledge acquisition). In the course of an ongoing ERC Consolidator Grant, his team currently investigates general principles of decidability in logic-based knowledge representation.
Title: How to Agree to Disagree: Managing Conceptual Diversity using Standpoint Logic
Abstract: The importance of taking individual, potentially conflicting perspectives into account when dealing with knowledge has been widely recognised. Many existing ontology management approaches fully merge knowledge perspectives, which may require weakening in order to maintain consistency; others represent the distinct views in an entirely detached way. This talk presents an alternative, referred to as Standpoint Logic, a simple, yet versatile multi-modal logic “add-on” for existing KR languages intended for the integrated representation of domain knowledge relative to diverse standpoints, which can be hierarchically organised, combined, and put in relation with each other.
Starting from the generic framework of First-Order Standpoint Logic (FOSL), we first present the fragment of so-called sentential formulas, for which we provide a polytime translation into the standpoint-free version. This result yields decidability and favourable complexities for several decidable fragments of first-order logic, including the very expressive description logic SROIQbs underlying the OWL 2 DL ontology language. By virtue of this, existing highly optimised OWL reasoners can be used to provide practical reasoning support for ontology languages extended by standpoint modelling.
Shifting our focus to tractable lightweight formalisms of enhanced scalability, we present Standpoint EL+, a standpoint extension of the popular description logic EL. Satisfiability in this logic can be checked in polynomial time thanks to a satisfiability-checking deduction calculus that allows for a implementation by means of elementary logic programming.
M. Eugenia Cornejo is Assistant Professor at the Department of Mathematics, University of Cádiz, Spain. Her research areas of interest are fuzzy sets, fuzzy logic, logic programming, relational data analysis, fuzzy relation equations and algebraic structures for soft computing. She co-leads two national projects, a working group of the COST Action (DigForASP) and she has participated as a researcher in different national projects and research contracts with companies. She has published more than 150 papers in conference proceedings and scientific journals, including the Journal of Computational and Applied Mathematics, Fuzzy Sets and Systems, Information Sciences, International Journal of Intelligent Systems, International Journal of Approximate Reasoning Fuzzy Sets and Systems, and Transactions on Fuzzy Systems.
Title: Multi-adjoint Formal Concept Analysis. Theory and Applications
Abstract: Due to the inherent imperfection in real databases the fuzzy extension of formal concept analysis (FCA) was a necessity. Multi-adjoint formal concept analysis, introduced by Medina and Ojeda-Aciego, arose as a fuzzy framework of formal concept analysis focused on introducing a general approach capable of conveniently embed the different fuzzy extensions of concept lattices given in the literature, such as the original framework given by Burusco and Fuentes-González, and the so-called approaches given by Belohlavek, Pollandt, Georgescu, Popescu and Krajci. For instance, a general algebraic structure was considered in which the operators could be non-commutative and non-associative, allowing a more flexible modeling of the examples/dataset. Another interesting feature is that it is possible to consider different adjoint triples or pairs with respect to a given triple of posets, which allows to offer the possibility of considering different degrees of preference on the attributes/objects of a database. These preference degrees can be interpreted as the values of a membership function modeling a preference on the attributes and/or objects, following the semantics proposed by Zadeh, in which the values represent the intensity of preference in favor of a specific attribute/object.
Attribute reduction and size reduction of the concept lattice, and the computation of reducts are significant challenges in this theory, which is fundamental for detecting redundant information and selecting the most relevant variables in the (real) dataset. Different approaches have been introduced with remarkable results.
Attribute implications theory is another fundamental challenge in FCA, where recent advances have also been obtained and they have been applied to real cases with good results.
This presentation provides a wide view on multi-adjoint formal concept analysis including theoretical definitions, properties, and important tasks where this theory has successfully been applied, such as the commented above attribute reduction and attribute implications theory. In addition, different applications will be shown.
