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Vol: 51(65) No: 1 / March 2006 Fuzzy J-K Flip-Flops Based on Various Classic and Non-Associative Norms Rita Lovassy Institute of Microelectronics and Technology, Budapest Tech, Hungary, Budapest, Hungary, e-mail: lovassy.rita@kvk.bmf.hu Laszlo T. Koczy Institute of Information Technology, Mechanical and Electrical Engineering, Széchenyi István University, Győr, Hungary, Győr, Hungary, e-mail: koczy@sze.hu Keywords: fuzzy logic, non-associative operations, fuzzy flip-flop. Abstract JK flip-flops are the elementary digital circuits providing sequential features/memory functions. The definitive equation of these units is well known in both the minimal disjunctive and the minimal conjunctive forms. As fuzzy connectives do not possess all axiomatic properties of their Boolean counterparts, if each operation in these equations is substituted by a corresponding fuzzy operator, two non-equivalent definitions are obtained, which were named reset type and set type fuzzy flip-flops (F3) by Hirota et al. when first introducing the concept of F3. There are many alternative possibilities to extend the idea of digital flip-flop, first of all based on classic (Zadeh-type), algebraic and other connectives. This paper gives an overview of the behavior of some of the most famous F3-s by presenting the definitions and examining the graphs of the inner state Q for several typical state value combinations. It is clearly shown that set and reset type definitions do not match in any case. In the next part a pair of non-associative operators is introduced, referring to an earlier definition by Fodor and Kóczy and the properties of this F3 are also discussed. The surprising fact is presented that the graphs of the two different flip-flops are identical. It is also proven that the two definitive forms are identical in the mathematical sense. References [1] B. Bouchon-Meunier, V. Kreinovich and H. T. Nguyen, “Non-associative operationsâ€, Proc. Second International Conference on Intelligent Technologies (InTech 2001), Bangkok, Thailand, 2001. [2] J. C. Fodor and L. T. Kóczy, “Some remarks on fuzzy flip-flopsâ€, In: L. T. Kóczy, K. Hirota, eds., Proc. Joint Hungarian-Japanese Symposium on Fuzzy Systems and Applications, Budapest, pp. 60 – 63, 1991. [3] J. C. Fodor and M. Roubens, Fuzzy Preference Modeling and Multicriteria Decision Support, Kluwer Academic Publishers, Dordrecht, 272 p., 1994. [4] I. R. Goodma, “A decision aid for nodes in Command& Control Systems based on cognitive probability logicâ€, Proceedings of the 1999 Command & Control Research and Technology Symposium, US Naval War College, Newport, RI, pp. 898 – 941, 1991. [5] K. Hirota and K. Ozawa, “Concept of fuzzy flip-flopâ€, Preprints of 2nd IFSA Congress, Tokyo, pp. 556 – 559, 1987. [6] K. Hirota and K. Ozawa, “Fuzzy flip-flop as a basis of fuzzy memory modulesâ€, In: M. M. Gupta et al., eds., Fuzzy Computing. Theory, Hardware and Applications, North Holland, Amsterdam, pp. 173 – 183, 1998. [7] G. J. Klir and T. A. Folger, Fuzzy sets, uncertainty, and information, Prentice Hall, Upper Saddle River, NJ, pp. 37 – 63, 1988. [8] G. J. Klir and B. Yuan, Fuzzy sets and fuzzy logic: Theory and applications, Prentice Hall, Upper Saddle River, NJ, pp. 50 – 88, 1995. [9] R. Lovassy and L. T. Kóczy, “Comparison of elementary fuzzy sequential digital units based on various popular T-norms and Co-normsâ€, Proc. 3rd Romanian-Hungarian Joint Symposium on Applied Computational Intelligence, Timisoara, pp. 164 – 181, 2006. [10] K. Ozawa, K. Hirota, L. T. Kóczy and K. Ohmori, “Algebraic fuzzy flip-flop circuitsâ€, Fuzzy Sets and Systems, vol. 39, no. 2, pp. 215 – 226, 1991. [11] K. Ozawa, K. Hirota and L. T. Kóczy, “Fuzzy flip-flopâ€, In: M. J. Patyra, D. M. Mlynek, eds., Fuzzy Logic. Implementation and Applications, Wiley, Chichester, pp. 197 – 236, 1996. [12] L. A. Zadeh, “Fuzzy setsâ€, Information and Control, vol. 8, pp. 338 – 353, 1965. [13] H. H. Zimmerman and P. Zysno, “Latent connectives in human decision makingâ€, Fuzzy Sets and Systems, vol. 4, pp. 37 – 51, 1980. |