Lattice Structure of D, T, and SR Fuzzy Flip-Flops Under Max-Min Logic
Shinichi Yoshida*, and Kaoru Hirota**
*Information Science Research Center, Aoyama Gakuin University, N603, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 229-8558, Japan
**Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, G3-49, 4259 Nagatsuta, Midori-ku, Yokohama-city 226-8502, Japan
Lattice structures of fuzzy flip-flops are described. A binary flip-flop (e.g. D, T, set-type SR, or reset-type SR flip-flop) can be extended to a fuzzy flip-flop in various ways. Under max-min fuzzy logic, there are 4 types of D fuzzy flip-flops extended from a binary D flip-flop, 136 types of SR fuzzy flip-flops extended from a binary SR flip-flop, and only one T fuzzy flip-flop. There is a lattice structure among different types of fuzzy flip-flops extended from a same binary flip-flop in terms of the order of ambiguity and the order of fuzzy logical value. These results show that fuzzy flip-flops under max-min fuzzy logic construct distributive lattice structures. Moreover D and T fuzzy flip-flops constructs Boolean lattice. And there exists a order monotone between two lattices of same fuzzy flip-flop under the order of ambiguity and the order of fuzzy logical value. Proposed analysis and results have potential to establish a fuzzy sequential system design method.