Fuzzy Systems

Short contents of the course

Fuzzy Systems is a field that has a strongly multidisciplinary character. Although the topic may be classified as part of mathematical logic ot set theory, and from another point of view, as part of approximation theory, in fact it was initiated by engineering research, teh basics proposed by L. Zadeh at the U. California Berkeley, and later practically used by E. mamdani at the Queen's College, London. The greatest advantage of fuzzy systems is the fact that application of the field does not need a deep mathematical knowledge or understanding, and it is much more transparent than traditional approaches. Fuzzy control and decision support, as the main application area tackles problems in engineering, medical science, fianancial and management science, and many other areas without teh necessity of doing high level mathematical calculations such as using Laplace and z-transform, etc., and in addition, the non-linear behavior is highly transparent and tunable even by non-expert appliers, while e.g. classic approximation approaches like Lagrange interpolation do not give any insight into the behavior of the approximation functions, and such approaches are very sensitive and mathematically unstable. Fuzzy rule based control and approximation is easy to handle and robust, and can be made mathematically stable even if the problem has a large number of dimensions.

This subject will go through the basics, some oparations, relations, extended fuzzy tools and will show how to handle them. Simple explanatory examples will help understanding even for the students with non CS background, and will illustrate the versatility and wide applicability of the approach.

The plan of 14 double lectures (a total of 28 academic hours).

Part One. Basics of Fuzzy Systems

1. The idea of fuzzy sets and logic (Zadeh). The Poincarre paradox. Examples for crisp (traditional) and fuzzy sets. How to represent a fuzzy set of a chosen universe. Membership degree and functions.
2. The extension of basic operations to fuzzy sets. Fuzzy complement/negation. Fuzzy union/disjunction and intersection/conjunction. Examples for operations. Axiomatic skeleton, mathematical framework.
3. Fuzzy aggregations. T-norms, co-norms, averaging aggregations. Examples for individual operations and parametric families (Sugeno, Yager, etc.). How can these operations be used? Hints to applications.
4. The idea of mathematical relations. Basic classes of relations. Examples (distances of cities, similarities of languages, etc.) Binary relations, examples
5. Fuzzy relations, operations on relations. Cylindric extension, orthogonal projection, composition and joint of relations. Hints to applications.
6. Rule based systems: Symbolic (AI) rule bases and subsymbolic approaches (CI). Mamdani's first real application, the Larsen model.
7. Fuzzy control and reasoning with fuzzy rule bases. Defuzzification methods. Simple examples.

Advanced Fuzzy Systems

1. Extensions of fuzzy sets. L-fuzzy sets by Goguen. Vector valued fuzzy sets. Simple applications. The idea of fuzzy signatures.
2. Operations on fuzzy signatures. Horizontal and vertical operations. Examples for the operations and for some simple applications (diagnostics, etc.)
3. A detailed example of using fuzzy signatures for evaluating the condition of residential and other buildings. (Based on a pool of expert evaluations of hisrtorical Budapest residential houses.)
4. Fuzzy communication. Compression of the information with help of common background knowledge and a common codebook. The Japanese example (communication between humans, the tea and coffee story, Terano)
5. Communication and collaboration of intelligent mobile robots. The tables/boxes arrangement example. Description of static and dynamic situations with fuzzy signatures. Fuzzy situational maps.
6. Fuzzy Cognitive Maps. Modeling management systems by FCM. Waste management systems, genearl company management examples (Hungarian and Lithuanian companies compared).
7. An approach to extend and to reduce components in the FCM. Example: the classic waste management system extended and then gradual reductions leading to new models.

Depending on the background knowledge of the students the above thematics can be further compressed and other areas like fuzzy rule interpolation and sparse systems, hierarchical fuzzy rule bases, and interpolation in hierarchical rule bases, etc., may be included int eh lectures, or may be given as individual projects to iterested and motivated students.

Pan profesor Laszlo T. Koczy jest jednym ze światowych liderów badań w zakresie logiki rozmytej, działu inteligencji obliczeniowej (sztucznej inteligencji). W latach 2001-03 był prezydentem International Fuzzy Systems Association, wiodącej światowej organizacji w tym zakresie. Prof. Koczy jest autorem 542 publikacji, cytowanych (według Google Scholar) 4515 razy, indeks Hirscha wynosi 34. Był visiting professorem Deakin University (Geelong), Australian National University (Canberra), Auckland University of Technology, Helsinki University, Murdoch University (Perth), University of New South Wales (Sydney), J. Kepler Universität Linz, University of Trento, Tokyo Institute of Technology, Pohang Institute of Science and Technology, Dalian Maritime University.