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Research

Algebraic theory of structures of truth values for fuzzy logic

Classical logic is based on the lattice structure of the simplest non-degenerated lattice L = {0,1}. Fuzzy logics are based mostly on the lattice L = [0,1] equipped with some special functions and the algebraic structure of such tuples (e.g., BL-algebra, MTL-algebra, etc.) heavily influence the corresponding fuzzy logic. Deeper knowledge of algebraic structures of truth degrees is thus an important step towards the description of the relevant fuzzy logics as well as to the possible fields of applications. Related areas of fuzzy approximation and fuzzy modeling also requires deeper understanding of the fuzzy numbers and fuzzy measures (or fuzzy densities) theories.

Developing new evolutionary algorithms for global optimization in fuzzy models

The global optimization problem with box constraints follows the form: minimize f(x) subject to x ε D

Formal theories of fuzzy logic

Mathematical fuzzy logic is a special formal theory of many-valued logic generalizing classical mathematical logic that focuses on the development of tools for modeling vagueness phenomenon. We distinguish fuzzy logic in narrow (FLn) and in broader sense (FLb). Fuzzy logic has been initiated at the end of sixties and beginning of seventies by L. A. Zadeh and J. A. Goguen. Nowadays, it is a well developed theory thanks to the results of an international group of mathematicians (e.g., P. Hájek, D. Mundici, S. Gottwald, F. Esteva, L. Godo, A. diNola, F. Montagna, V. Novák, P. Cintula, L. Běhounek, and many others).

Linguistic representation of knowledge and reasoning based on it

Fuzzy logic is known by many applications in various areas. A common denominator of them in most cases is the ability of FL to capture, in a certain sense, semantics of a part of natural language semantics. A crucial role is played by the, so called evaluative linguistic expressions and conditional statements containing them. The former are used in control, managerial decision-making, identification, etc. Typical examples of evaluative expressions are “very small, more or less medium, roughly big, extremely deep, more or less brief, about twenty five, approximately 1000, not small, not very big, roughly small or medium, quite roughly medium and/but not big”, etc. The expressions containing words “small, medium, big” can be taken as canonical.

Theory and methods of fuzzy approximation

Fuzzy approximation is a special phenomenon which explains how complicated functions can be described by a set of fuzzy IF-THEN rules. In many cases, the way of designing the concrete set of these rules is left unexplained, or rather exhaustive technique based on neural networks or genetic algorithms is used. On the other hand, there is a series of theoretical papers unified by the key words “universal approximation” where the problem of representation of a continuous function by a so called fuzzy system (which can be regarded as an interpretation of fuzzy IF-THEN rules) is considered.
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