Formal Concept Analysis has been originally developed as a subfield of Applied Mathematics based on the mathematization of concept and concept hierarchy. Only after more than a decade of development, the connections to the philosophical logic of human thought became clearer and even later the connections to Piaget's cognitive structuralism which Thomas Bernhard Seiler convincingly elaborated to a comprehensive theory of concepts in his recent book [Se01]. It is the main concern of this paper to show the surprisingly rich correspondences between Seiler's multifarious aspects of concepts in the human mind and the structural properties and relationships of formal concepts in Formal Concept Analysis. These correspondences make understandable, what has been experienced in a great multitude of applications, that Formal Concept Analysis may function in the sense of transdisciplinary mathematics, i.e., it allows mathematical thought to aggregate with other ways of thinking and thereby to support human thought and action.

Formal concept analysis as mathematical theory of concepts and concept hierarchies

With Linked Data, a very pragmatic approach towards achieving the vision of the Semantic Web has gained some traction in the last years. The term Linked Data refers to a set of best practices for publishing and interlinking structured data on the Web. While many standards, methods and technologies developed within by the Semantic Web community are applicable for Linked Data, there are also a number of specific characteristics of Linked Data, which have to be considered. In this article we introduce the main concepts of Linked Data. We present an overview of the Linked Data lifecycle and discuss individual approaches as well as the state-of-the-art with regard to extraction, authoring, linking, enrichment as well as quality of Linked Data. We conclude the chapter with a discussion of issues, limitations and further research and development challenges of Linked Data. This article is an updated version of a similar lecture given at Reasoning Web Summer School 2011.

Introduction to linked data and its lifecycle on the web

This chapter accompanies the foundational lecture onDescription Logics (DLs) at the 7th Reasoning Web Summer School in Galway, Ireland, 2011. It introduces basic notions and facts about this family of logics which has significantly gained in importance over the recent years as these logics constitute the formal basis for today's most expressive ontology languages, the OWL (Web Ontology Language) family.

We start out from some general remarks and examples demonstrating the modeling capabilities of description logics as well as their relation to first-order predicate logic. Then we begin our formal treatment by introducing the syntax of DL knowledge bases which comes in three parts: RBox, TBox and ABox. Thereafter, we provide the corresponding standard model-theoretic semantics and give a glimpse of the alternative way of defining the semantics via an embedding into first-order logic with equality.

We continue with an overview of the naming conventions for DLs before we delve into considerations about different notions of semantic alikeness (concept and knowledge base equivalence as well as emulation). These are crucial for investigating the expressivity of DLs and performing normalization. We move on by reviewing knowledge representation capabilities brought about by different DL features and their combinations as well as some model-theoretic properties associated thereto. Subsequently, we consider typical reasoning tasks occurring in the context of DL knowledge bases. We show how some of these tasks can be reduced to each other, and have a look at different algorithmic approaches to realize automated reasoning in DLs.

Finally, we establish connections between DLs and OWL. We show how DL knowledge bases can be expressed in OWL and, conversely, how OWL modeling features can be translated into DLs.

In our considerations, we focus on the description logic SROIQ which underlies the most recent and most expressive yet decidable version of OWL called OWL 2 DL. We concentrate on the logical aspects and omit data types as well as extralogical features from our treatise. Examples and exercises are provided throughout the chapter.

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Foundations of description logics

State-of-the-art research on automated learning of ontologies from text currently focuses on inexpressive ontologies. The acquisition of complex axioms involving logical connectives, role restrictions, and other expressive features of the Web Ontology Language OWL remains largely unexplored. In this paper, we present a method and implementation for enriching inexpressive OWL ontologies with expressive axioms which is based on a deep syntactic analysis of natural language definitions. We argue that it can serve as a core for a semi-automatic ontology engineering process supported by a methodology that integrates methods for both ontology learning and evaluation. The feasibility of our approach is demonstrated by generating complex class descriptions from Wikipedia definitions and from a fishery glossary provided by the Food and Agriculture Organization of the United Nations.

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Acquisition of OWL DL Axioms from Lexical Resources

Recent advances in data and knowledge engineering have emphasized the need for formal concept analysis (FCA) tools taking into account structured data. There are a few adaptations of the classical FCA methodology for handling contexts holding on complex data formats, e.g. graph-based or relational data. In this paper, relational concept analysis (RCA) is proposed, as an adaptation of FCA for analyzing objects described both by binary and relational attributes. The RCA process takes as input a collection of contexts and of inter-context relations, and yields a set of lattices, one per context, whose concepts are linked by relations. Moreover, a way of representing the concepts and relations extracted with RCA is proposed in the framework of a description logic. The RCA process has been implemented within the Galicia platform, offering new and efficient tools for knowledge and software engineering.

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A proposal for combining formal concept analysis and description logics for mining relational data

In Peirce’s account of the normative sciences, logical validity and truth ultimately rely upon – though they are not reduced to – ethical and esthetic insight. An examination of the relations among critical esthetics, ethics and logic in Peirce’s system suggests a possible account of the logic of creative discovery, which Peirce identified as abductive inference.

Conceptual Structures at Work

Invited Talks.- The Esthetic Grounding of Ordered Thought.- Memes as Signs in the Dynamic Logic of Semiosis: Beyond Molecular Science and Computation Theory.- Graphics and Languages for the Flexible Modular Framework.- Implicational Concept Graphs.- Types and Tokens for Logic with Diagrams.- From Data to Conceptual Structures.- Concept-Based Data Mining with Scaled Labeled Graphs.- Iceberg Query Lattices for Datalog.- 'Particles' and 'Waves' as Understood by Temporal Concept Analysis.- A Void Type for Medical Language Disambiguation.- Clustering of Conceptual Graphs with Sparse Data.- Concept Lattices and Concept Graphs.- A Cartesian Closed Category of Approximable Concept Structures.- Implications in Triadic Formal Contexts.- Exploring Relational Structures Via .- An Extension of the Theory of Information Flow to Semiconcept and Protoconcept Graphs.- Negation in Contextual Logic.- Conceptual Frameworks for Applications.- Using Brandom's Framework to Do Peirce's Normative Science: Pragmatism as the Game of Harmonizing Assertions?.- Improving the Testbed Development Process in Collaboratories.- Relation Elements for the Semantic Web.- Poset Ontologies and Concept Lattices as Semantic Hierarchies.- Concept Types and Coreference in Simple Conceptual Graphs.- An Exportable CGIF Module from the CP Environment: A Pragmatic Approach.- Reasoning with Conceptual Structures.- Using Conceptual Graphs to Represent Agent Semantic Constituents.- Improving Generalization Level in UML Models Iterative Cross Generalization in Practice.- KNAML: A Knowledge Representation Language for Distributed Reasoning.- Interoperability of Ontologies Using Conceptual Graph Theory.- A Priorean Approach to Time Ontologies.

Karl Erich Wolff

Papers

Conceptual Structures at Work

Conceptual Structures at Work: 12th International Conference on Conceptual Structures, ICCS 2004, Huntsville, AL, USA, July 19-23, 2004, Proceedings