Fuzzy Relational Systems

Foundations and Principles
Softcover Nachdruck of the original 1. Auflage 2002
ISBN: 978-1-4613-5168-9
Verlag: Springer US

Buch, Englisch, Band 20, 369 Seiten, Format (B × H): 178 mm x 254 mm, Gewicht: 721 g

Reihe: IFSR International Series in Systems Science and Systems Engineering

Since their inception, fuzzy sets and fuzzy logic became popular. The reason is that the very idea of fuzzy sets and fuzzy logic attacks an old tradition in science, namely bivalent (black-or-white, all-or-none) judg­ ment and reasoning and the thus resulting approach to formation of scientific theories and models of reality. The idea of fuzzy logic, briefly speaking, is just the opposite of this tradition: instead of full truth and falsity, our judgment and reasoning also involve intermediate truth values. Application of this idea to various fields has become known under the term fuzzy approach (or graded truth approach). Both prac­ tice (many successful engineering applications) and theory (interesting nontrivial contributions and broad interest of mathematicians, logicians, and engineers) have proven the usefulness of fuzzy approach. One of the most successful areas of fuzzy methods is the application of fuzzy relational modeling. Fuzzy relations represent formal means for modeling of rather nontrivial phenomena (reasoning, decision, control, knowledge extraction, systems analysis and design, etc. ) in the pres­ ence of a particular kind of indeterminacy called vagueness. Models and methods based on fuzzy relations are often described by logical formulas (or by natural language statements that can be translated into logical formulas). Therefore, in order to approach these models and methods in an appropriate formal way, it is desirable to have a general theory of fuzzy relational systems with basic connections to (formal) language which enables us to describe relationships in these systems.

Zielgruppe

Research

Autoren/Hrsg.

Belohlávek, Radim

Fachgebiete

• Interdisziplinäres Wissenschaften Wissenschaften: Forschung und Information Informationstheorie, Kodierungstheorie
• Mathematik | Informatik Mathematik Mathematik Allgemein Grundlagen der Mathematik
• Interdisziplinäres Wissenschaften Wissenschaften: Forschung und Information Kybernetik, Systemtheorie, Komplexe Systeme
• Mathematik | Informatik Mathematik Mathematik Allgemein Mathematische Logik
• Mathematik | Informatik EDV | Informatik Programmierung | Softwareentwicklung Algorithmen & Datenstrukturen
• Mathematik | Informatik EDV | Informatik Informatik Künstliche Intelligenz

Inhaltsverzeichnis

1. Preliminaries.- 1.1 Sets, relations, mappings.- 1.2 Ordered sets, lattices.- 1.3 Tolerances, equivalences.- 1.4 Algebras.- 2. Fuzzy Approach, Graded Truth, and Structures of Truth Values.- 2.1 Fuzzy approach: from resolving paradoxes to engineering applications.- 2.2 Graded truth and structures of truth values.- 2.3 Residuated lattices.- 2.4 Residuated lattices and t-norms.- 2.5 Bibliographical remarks.- 3. Fuzzy Structures.- 3.1 Fuzzy sets and fuzzy relations.- 3.2 Fuzzy logic: syntax and semantics.- 3.3 Similarity issues.- 3.4 Morphisms, products, substructures.- 3.5 Representation by crisp structures: cutlike semantics.- 3.6 Algebras with fuzzy equalities.- 3.7 Many-sorted case.- 3.8 Bibliographical remarks.- 4. Binary Fuzzy Relations.- 4.1 Operations and properties of binary fuzzy relations.- 4.2 Similarity: fuzzy equivalences.- 4.3 Hierarchy: fuzzy order.- 4.4 Bibliographical remarks.- 5. Object-Attribute Fuzzy Relations and Fuzzy Concept Lattices.- 5.1 Conceptual granulation: fuzzy context, fuzzy concepts, and fuzzy concept lattice.- 5.2 Similarity and logical precision.- 5.3 Fuzzy Galois connections.- 5.4 Main theorem of fuzzy concept lattices.- 5.5 Bibliographical remarks.- 6. Composition and Decomposition of Fuzzy Relations.- 6.1 Composition of fuzzy relations.- 6.2 Decomposition: fuzzy relational equations.- 6.3 Bibliographical remarks.- 7. Miscellanea.- 7.1 Fuzzy closure operators and similarity spaces.- 7.2 Applications of selected results to some problems and models.- 7.3 Fuzzy logic with truth degrees in residuated lattices.- 7.4 Bibliographical remarks.- References.