Cutlike semantics for fuzzy logic and its applications

Abstract

Each fuzzy set can be represented by a nested system of ordinary sets—its a-cuts. There is an extensive literature on fuzzy sets devoted to problems of the following kind: is it possible to reduce operations with fuzzy sets to operations with their a-cuts? Is it possible to reduce properties of fuzzy relations to properties of their a-cuts? More generally, can a fuzzy concept be represented by a collection of corresponding crisp concepts? Klir and Yuan (1995) speak of cutworthiness.

We attempt to provide a general solution to this problem. The way we proceed is thus: a structure for fuzzy predicate logic can be represented by a nested system of crisp structures. The system of crisp structures can be used to define semantics of fuzzy predicate logic in an alternative way by using the nested structure and Boolean connectives only. Answers to the above questions are then obtained by simple application of the obtained general results; we present some examples (extension principle, properties of binary fuzzy relations, fuzzy automata).

Keywords:

• Fuzzy set
• Alpha cut
• Predicate fuzzy logic
• Semantics

Acknowledgements

Supported partially by grants no. 201/02/P076 of the Grant Agency of the Czech Republic and no. B1137301 of the Grant Agency of the Academy of Sciences of the Czech Republic.

Notes

Radim Bělohlávek is a head of the Department of Computer Science, Palacký University of Olomouc (Czech Republic). He obtained a PhD in Computer Science from the Technical University of Ostrava in 1998 and a PhD in Mathematics from the Palacký University of Olomouc in 2001. His interests are in pure and applied logic, mathematical methods for processing of indeterminate information, foundations of computing and mathematics. He is the author of one monograph and about 40 journal papers in these areas.