Cardinalities of Fuzzy Sets and Fuzzy QuantiŻers over Residuated Lattices

Mgr. Michal HOLČAPEK

Doctoral thesis

Abstract:

During the last few years one could have already noticed a signi?cant growth of the new calculation kinds that could, in general, be called the soft com- puting. This type of computing di?ers from the traditional calculations in its primary goal, i.e. the toleration of uncertainty and inaccuracy in order to reach quick, robust and cheap results. Thus the soft computing is used to ?nd an approximate solution of an exactly formulated problem or, more frequently and most typically, to ?nd the solution of the problem that is not formulated exactly itself. The new unconventional theories, methods, techniques and technologies in computer and information science, systems analysis, decision-making and control, expert systems, data modeling, en- gineering, etc., using techniques of soft computing together with their basic tools theory of fuzzy sets and fuzzy logic, are still investigated and developed. Fuzzy set theory as well as fuzzy logic originated from ideas of L.A. Zadeh. The concept of fuzzy set together with the basic principles of fuzzy set theory was established in his seminal paper [119] published in 1965 and the basic ideas of fuzzy logic were elaborated in [121{123]. Thanks to the fact that fuzzy set theory and fuzzy logic could o?er very strong tools for description of an uncertainty in common human reasoning, an intensive development took place in these areas. It was motivated not only by requirement to de- velop these sciences themselves, but, of course, to be able to solve practical problems. One of the topic of fuzzy logic that was and still is intensively investigated is the theory of fuzzy (generalized) quanti?cations. Fuzzy quan- ti?cation is a construct that speci?es the extend of validity of a predicate, where the range of predicate validity is often expressed roughly i.e. fuzzy. Fuzzy quanti?ers are then elements of natural language which generate fuzzy quanti?cations. Fuzzy quanti?ers could be seen in mathematics, but mainly in natural language. Everyone surely knows the expressions like \many happy people", \nearly all members", \some animals", \few single women", \nearly none mistake", \about half of participants", \about ?fteen students" etc. The main goal of this thesis is to propose a new approach to fuzzy quanti?- cations. The presented approach was originally motivated by looking for the ix

Abstract:

During the last few years one could have already noticed a signi?cant growth of the new calculation kinds that could, in general, be called the soft com- puting. This type of computing di?ers from the traditional calculations in its primary goal, i.e. the toleration of uncertainty and inaccuracy in order to reach quick, robust and cheap results. Thus the soft computing is used to ?nd an approximate solution of an exactly formulated problem or, more frequently and most typically, to ?nd the solution of the problem that is not formulated exactly itself. The new unconventional theories, methods, techniques and technologies in computer and information science, systems analysis, decision-making and control, expert systems, data modeling, en- gineering, etc., using techniques of soft computing together with their basic tools theory of fuzzy sets and fuzzy logic, are still investigated and developed. Fuzzy set theory as well as fuzzy logic originated from ideas of L.A. Zadeh. The concept of fuzzy set together with the basic principles of fuzzy set theory was established in his seminal paper [119] published in 1965 and the basic ideas of fuzzy logic were elaborated in [121{123]. Thanks to the fact that fuzzy set theory and fuzzy logic could o?er very strong tools for description of an uncertainty in common human reasoning, an intensive development took place in these areas. It was motivated not only by requirement to de- velop these sciences themselves, but, of course, to be able to solve practical problems. One of the topic of fuzzy logic that was and still is intensively investigated is the theory of fuzzy (generalized) quanti?cations. Fuzzy quan- ti?cation is a construct that speci?es the extend of validity of a predicate, where the range of predicate validity is often expressed roughly i.e. fuzzy. Fuzzy quanti?ers are then elements of natural language which generate fuzzy quanti?cations. Fuzzy quanti?ers could be seen in mathematics, but mainly in natural language. Everyone surely knows the expressions like \many happy people", \nearly all members", \some animals", \few single women", \nearly none mistake", \about half of participants", \about ?fteen students" etc. The main goal of this thesis is to propose a new approach to fuzzy quanti?- cations. The presented approach was originally motivated by looking for the ix

Keywords

fuzzy sets

Language used: Czech

Date on which the thesis was submitted / produced: 20. 4. 2006

Accessible from:: 20. 4. 2006

Thesis defence

• Date of defence: 25. 5. 2006
• Supervisor: prof. RNDr. Jiří Močkoř, DrSc.

University of Ostrava

Faculty of Science

Doctoral programme / field:
Applied Mathematics / Applied Mathematics