Mathematics and Computer Science
Volume 4, Issue 6, November 2019, Pages: 130-137
Received: Nov. 12, 2019;
Accepted: Dec. 11, 2019;
Published: Dec. 24, 2019
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Yiming Tang, School of Computer and Information, Hefei University of Technology, Hefei, China; Anhui Province Key Laboratory of Affective Computing & Advanced Intelligent Machine, Hefei University of Technology, Hefei, China; Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada
Guangqing Bao, School of Computer and Information, Hefei University of Technology, Hefei, China; Anhui Province Key Laboratory of Affective Computing & Advanced Intelligent Machine, Hefei University of Technology, Hefei, China
As one of important parts of fuzzy logic, fuzzy inference plays a vital role in the fields of fuzzy control, artificial intelligence, affective computing, image processing and so forth. Two key problems of fuzzy inference are FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens). How to get the ideal solution for FMP and FMT is a difficult problem in the area of fuzzy logic. Aiming at such problem, from the idea of symmetric implicational reasoning, triple I* method and restriction theory, we put forward and investigate the α-symmetric I* restriction method, and then generalize it to the α(x,y)-symmetric I* restriction method. To begin with, the α-symmetric I* restriction principle and the α(x,y)-symmetric I* restriction principle are established. Furthermore, the equivalent condition to let a basic restriction solution exist is given. Then the unified solutions of the α-symmetric I* restriction method and the α(x,y)-symmetric I* restriction method are achieved for R-implications and (S, N)-implications. Besides, some special cases of optimal solutions are shown. Finally, the corresponding conclusions are provided when the two methods degenerate into the α-triple I* restriction method and α(x,y)-triple I* restriction method. These research results would be an important improvement for the fields of fuzzy inference, fuzzy logic and related applications.
Symmetric I* Restriction Method of Fuzzy Inference, Mathematics and Computer Science.
Vol. 4, No. 6,
2019, pp. 130-137.
J. Verstraete (2017) The spatial disaggregation problem: Simulating reasoning using a fuzzy inference system. IEEE Transactions on Fuzzy Systems 25, 627-641.
X. Y. Yang, F. S. Yu, and W. Pedrycz (2017) Long-term forecasting of time series based on linear fuzzy information granules and fuzzy inference system. International Journal of Approximate Reasoning 81, 1-27.
Y. M. Tang, X. H Hu, W. Pedrycz, and X. C. Song (2019) Possibilistic fuzzy clustering with high-density viewpoint. Neurocomputing 329, 407-423.
Y. M. Tang and F. J. Ren (2013) Universal triple I method for fuzzy reasoning and fuzzy controller. Iranian Journal of Fuzzy Systems 10, 1-24.
S. S. Dai, D. W. Pei, and S. M. Wang (2012) Perturbation of fuzzy sets and fuzzy reasoning based on normalized Minkowski distances. Fuzzy Sets and Systems 189, 63–73.
P. Hájek (1998) Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht.
Y. M. Tang and F. J. Ren (2017) Fuzzy systems based on universal triple I method and their response functions. International Journal of Information Technology & Decision Making 16, 443-471.
L. A. Zadeh (1973) Outline of a new approach to the analysis of complex systems and decision processes. IEEE Transactions on Systems, Man and Cybernetics, 3, 28-44.
B. Jayaram (2008) On the law of importation (x ∧ y) → z ≡ (x → (y → z)) in fuzzy logic. IEEE Transactions on Fuzzy Systems 16, 130-144.
M. Stepnicka and B. Jayaram (2010) On the suitability of the Bandler-Kohout subproduct as an inference mechanism. IEEE Transactions on Fuzzy Systems 18, 285-298.
G. J. Wang (1999) On the logic foundation of fuzzy reasoning. Information Sciences 117, 47-88.
S. J. Song, C. B. Feng, and C. X. Wu (2001) Theory of restriction degree of triple I method with total inference rules of fuzzy reasoning. Progress in Natural Science 11, 58-66.
S. J. Song and C. Wu (2002) Reverse triple I method of fuzzy reasoning. Science in China, Ser. F, Information Sciences 45, 344–364.
G. J. Wang and L. Fu (2005) Unified forms of triple I method. Computers & Mathematics with Applications 49, 923-932.
M. C. Zheng, Z. K. Shi, and Y. Liu (2014) Triple I method of approximate reasoning on Atanassov’s intuitionistic fuzzy sets, International Journal of Approximate Reasoning 55, 1369-1382.
Y. M. Tang and X. P. Liu (2010) Differently implicational universal triple I method of (1, 2, 2) type. Computers & Mathematics with Applications 59, 1965-1984.
Y. M. Tang, F. J. Ren, and Y. X. Chen (2012) Differently implicational α-universal triple I restriction method of (1, 2, 2) type. Journal of Systems Engineering and Electronics 23, 560-573.
Y. M. Tang and F. J. Ren (2015) Variable differently implicational algorithm of fuzzy inference. Journal of Intelligent and Fuzzy Systems 28, 1885–1897.
Y. M. Tang, X. Z Yang, X. P. Liu, and J. Yang (2015) Double fuzzy implications-based restriction inference algorithm. Iranian Journal of Fuzzy Systems 12, 17-40.
Y. M. Tang and F. J. Ren (2016) Variable differently implicational inference for R- and S-implications. International Journal of Information Technology & Decision Making 15, 1235-1264.
Y. M. Tang and W. Pedrycz (2019) On continuity of the entropy-based differently implicational algorithm. Kybernetika 55, 307-336.
M. X. Luo and X. L. Zhou (2015) Robustness of reverse triple I algorithms based on interval-valued fuzzy inference. International Journal of Approximate Reasoning 66, 16-26.
M. X. Luo and B. Liu (2017) Robustness of interval-valued fuzzy inference triple I algorithms based on normalized Minkowski distance. Journal of Logical and Algebraic Methods in Programming 86, 298-307.
D. W. Pei (2001) Two triple methods for problem and their reductivity. Fuzzy Systems and Mathematics 15, 1-7.
Y. M. Tang and X. Z. Yang (2013) Symmetric implicational method of fuzzy reasoning. International Journal of Approximate Reasoning 54, 1034-1048.
Y. M. Tang and W. Pedrycz (2018) On the α(u,v)-symmetric implicational method for R- and (S, N)-implications. International Journal of Approximate Reasoning 92, 212-231.
E. P. Klement, R. Mesiar, and E. Pap (2000) Triangular Norms. Kluwer Academic Publishers, Dordrecht.
M. Mas, M. Monserrat, J. Torrens, et al. (2007) A survey on fuzzy implication functions. IEEE Transactions on Fuzzy Systems 15, 1107–1121.
G. J. Wang and H. J. Zhou (2009) Introduction to Mathematical Logic and Resolution Principle. Co-published by Science Press and Alpha International Science Ltd, Oxford.
V. Novak, I. Perfifilieva, and J. Mockor (1999) Mathematical Principles of Fuzzy Logic. Kluwer Academic Publishers, Boston, Dordrecht.
M. Baczynski and B. Jayaram (2007) On the characterizations of (S,N)-implications. Fuzzy Sets and Systems 158, 1713–1727.
J. Fodor and M. Roubens (1994) Fuzzy Preference Modeling and Multicriteria Decision Support, Kluwer Academic Publishers, Dordrecht.
W. Pedrycz (2013) Granular Computing: Analysis and Design of Intelligent Systems. CRC Press/Francis & Taylor, Boca Raton, FL, USA.
W. Pedrycz and X. M. Wang (2016) Designing fuzzy sets with the use of the parametric principle of justifiable granularity. IEEE Transactions on Fuzzy Systems 24, 489-496.