n-Dimensional Fuzzy Implications: Analytical, Algebraic and Applicational Approaches

• Rosana Medina Zanotelli Universidade Federal de Pelotas (UFPel) http://orcid.org/0000-0001-8271-897X
• Renata H. Sander Reiser Universidade Federal de Pelotas (UFPel) http://orcid.org/0000-0001-9934-3115
• Benjamín René C. Bedregal Universidade Federal do Rio Grande do Norte (UFRN) http://orcid.org/0000-0002-6757-7934

DOI: https://doi.org/10.5753/weit.2021.18938

Resumo

The study of n-dimensional fuzzy logic contributes to overcome the insufficiency of traditional FL in modeling imperfect and imprecise information coming from different experts. Based on representability, we extend results from fuzzy connectives to n-dimensional approach. This research on n-dimensional fuzzy implications (n-DI) pass through the next steps: (i) analytical studies; (ii) algebraic aspects; (iii) n-dimensional approach of fuzzy implication classes represented by fuzzy connectives as (S, N)-implications and QL-implications; (iv) studies of n-dimensional R-implications (n-DRI); (v) constructive method obtaining n-DRI based on n-dimensional aggregation operators and (vi) an introductory study considering an n-DI in modeling approximate reasoning.

Palavras-chave: n-dimensional fuzzy logic, n-dimensional fuzzy implication, n-dimensional R-implication, approximate reasoning

Referências

Atanassov, K. and Gargov, G. (1998). Elements of intuitionistic fuzzy logic. part i. Fuzzy Sets and Systems, 95(1):39–52.

Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1):87–96.

Bedregal, B., Beliakov, G., Bustince, H., Calvo, T., Fernández, J., and Mesiar, R. (2011). A characterization theorem for t-representable n-dimensional triangular norms. In Eurofuse 2011, pages 103–112. Springer.

Bedregal, B., Beliakov, G., Bustince, H., Calvo, T., Mesiar, R., and Paternain, D. (2012). A class of fuzzy multisets with a fixed number of memberships. Information Sciences, 189:1–17.

Bedregal, B., Mezzomo, I., and Reiser, R. H. S. (2018). n-dimensional fuzzy negations. IEEE Transactions on Fuzzy Systems, 26(6):3660–3672.

Bedregal, B., Reiser, R., Bustince, H., Lopez-Molina, C., and Torra, V. (2014). Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms. Information Sciences, 255:82–99.

De Miguel, L., Sesma-Sara, M., Elkano, M., Asiain, M., and Bustince, H. (2017). An algorithm for group decision making using n-dimensional fuzzy sets, admissible orders and owa operators. Information Fusion, 37:126–131.

Deschrijver, G. and Kerre, E. E. (2005). Implicators based on binary aggregation operators in interval-valued fuzzy set theory. Fuzzy Sets and Systems, 153(2):229–248.

Dubois, D. and Prade, H. (2005). Interval-valued fuzzy sets, possibility theory and imprecise probability. In EUSFLAT Conf., pages 314–319.

Lima, A. A. d. (2019). Conjuntos Fuzzy Multidimensionais. PhD thesis, UFRN, Dep. of Infor. and Applied Mathematics, Grad. Prog. in Systems and Computing, Natal, RN, Brazil.

Mezzomo, I., Bedregal, B., and Milfont, T. (2018a). Equilibrium point of representable moore continuous n-dimensional interval fuzzy negations. In North American Fuzzy Information Processing Society Annual Conf., pages 265–277. Springer.

Mezzomo, I., Bedregal, B., and Milfont, T. (2018b). Moore continuous n-dimensional interval fuzzy negations. In 2018 IEEE Int. Conf. on Fuzzy Systems (FUZZ-IEEE), pages 1–6. IEEE.

Mezzomo, I., Bedregal, B., Milfont, T., da Cruz Asmus, T., and Bustince, H. (2019). n-dimensional interval uninorms. In 2019 IEEE Int. Conf. on Fuzzy Systems (FUZZ-IEEE), pages 1–6. IEEE.

Mezzomo, I., Bedregal, B., and Reiser, R. (2017). Natural n-dimensional fuzzy negations for n-dimensional t-norms and t-conorms. In 2017 IEEE Int. Conf. on Fuzzy Systems (FUZZ-IEEE), pages 1–6.

Mezzomo, I., Bedregal, B. C., Reiser, R. H., Bustince, H., and Paternain, D. (2016). On n-dimensional strict fuzzy negations. In Fuzzy Systems (FUZZ-IEEE), 2016 IEEE Int. Conf. on, pages 301–307. IEEE.

Mizumoto, M. and Tanaka, K. (1976). Some properties of fuzzy sets of type 2. Information and Control, 31(4):312–340.

Sambuc, R. (1975). Fonctions and Floues: Application a l’aide au Diagnostic en Pathologie Thyroidienne. Faculté de Médecine de Marseille.

Shang, Y.-g., Yuan, X.-h., and Lee, E. S. (2010). The n-dimensional fuzzy sets and zadeh fuzzy sets based on the finite valued fuzzy sets. Computers & Mathematics with Applications, 60(3):442–463.

Silva, I., Bedregal, B., and Bustince, H. (2015). Weighted average operators generated by n-dimensional overlaps and an application in decision making. In 16th World Congress of the International Fuzzy Systems Association,(Eindhoven, The Netherlands), pages 1473–1478.

Torra, V. (2010). Hesitant fuzzy sets. Intl Journal of Int Systems, 25(6):529–539.

Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning—i. Information sciences, 8(3):199–249.

Zanotelli, R., Reiser, R., and Bedregal, B. (2018). Towards the study of main properties of n-dimensional ql-implicators. In SBMAC, pages 1–8. Federal University of Ceará.

Zanotelli, R. M. (2020). n-Dimensional Fuzzy Implications: Analytical, Algebraic and Applicational Approaches. PhD thesis, UFPEL, CDTec, Pelotas, RS, Brazil.

Zanotelli, R. M., Reiser, R., and Bedregal, B. R. C. (2020). n-dimensional (S, N)-implications. Int. J. Approx. Reason., 126:1–26.

Zanotelli, R. M., Reiser, R., and Bedregal, B. R. C. (2021). On the residuation principle of ndimensional r-implications. Int. J. Soft Computing, 1:1–30. under revision.