Symmetric I* Restriction Method of Fuzzy Inference

Yiming Tang; Guangqing Bao

doi:doi:10.11648/j.mcs.20190406.14

| Peer-Reviewed

Symmetric I* Restriction Method of Fuzzy Inference

Yiming Tang, Guangqing Bao

Published in Mathematics and Computer Science (Volume 4, Issue 6)

Received: 12 November 2019 Accepted: 11 December 2019 Published: 24 December 2019

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

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.

Published in	Mathematics and Computer Science (Volume 4, Issue 6)
DOI	10.11648/j.mcs.20190406.14
Page(s)	130-137
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Fuzzy Inference, Fuzzy Implication, Triple I Method, Symmetric Implicational Method

References

[1]	J. Verstraete (2017) The spatial disaggregation problem: Simulating reasoning using a fuzzy inference system. IEEE Transactions on Fuzzy Systems 25, 627-641.
[2]	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.
[3]	Y. M. Tang, X. H Hu, W. Pedrycz, and X. C. Song (2019) Possibilistic fuzzy clustering with high-density viewpoint. Neurocomputing 329, 407-423.
[4]	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.
[5]	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.
[6]	P. Hájek (1998) Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht.
[7]	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.
[8]	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.
[9]	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.
[10]	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.
[11]	G. J. Wang (1999) On the logic foundation of fuzzy reasoning. Information Sciences 117, 47-88.
[12]	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.
[13]	S. J. Song and C. Wu (2002) Reverse triple I method of fuzzy reasoning. Science in China, Ser. F, Information Sciences 45, 344–364.
[14]	G. J. Wang and L. Fu (2005) Unified forms of triple I method. Computers & Mathematics with Applications 49, 923-932.
[15]	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.
[16]	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.
[17]	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.
[18]	Y. M. Tang and F. J. Ren (2015) Variable differently implicational algorithm of fuzzy inference. Journal of Intelligent and Fuzzy Systems 28, 1885–1897.
[19]	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.
[20]	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.
[21]	Y. M. Tang and W. Pedrycz (2019) On continuity of the entropy-based differently implicational algorithm. Kybernetika 55, 307-336.
[22]	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.
[23]	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.
[24]	D. W. Pei (2001) Two triple methods for problem and their reductivity. Fuzzy Systems and Mathematics 15, 1-7.
[25]	Y. M. Tang and X. Z. Yang (2013) Symmetric implicational method of fuzzy reasoning. International Journal of Approximate Reasoning 54, 1034-1048.
[26]	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.
[27]	E. P. Klement, R. Mesiar, and E. Pap (2000) Triangular Norms. Kluwer Academic Publishers, Dordrecht.
[28]	M. Mas, M. Monserrat, J. Torrens, et al. (2007) A survey on fuzzy implication functions. IEEE Transactions on Fuzzy Systems 15, 1107–1121.
[29]	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.
[30]	V. Novak, I. Perfifilieva, and J. Mockor (1999) Mathematical Principles of Fuzzy Logic. Kluwer Academic Publishers, Boston, Dordrecht.
[31]	M. Baczynski and B. Jayaram (2007) On the characterizations of (S,N)-implications. Fuzzy Sets and Systems 158, 1713–1727.
[32]	J. Fodor and M. Roubens (1994) Fuzzy Preference Modeling and Multicriteria Decision Support, Kluwer Academic Publishers, Dordrecht.
[33]	W. Pedrycz (2013) Granular Computing: Analysis and Design of Intelligent Systems. CRC Press/Francis & Taylor, Boca Raton, FL, USA.
[34]	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.

Cite This Article

Plain Text BibTeX RIS

APA Style

Yiming Tang, Guangqing Bao. (2019). Symmetric I* Restriction Method of Fuzzy Inference. Mathematics and Computer Science, 4(6), 130-137. https://doi.org/10.11648/j.mcs.20190406.14

Copy | Download

ACS Style

Yiming Tang; Guangqing Bao. Symmetric I* Restriction Method of Fuzzy Inference. Math. Comput. Sci. 2019, 4(6), 130-137. doi: 10.11648/j.mcs.20190406.14

Copy | Download

AMA Style

Yiming Tang, Guangqing Bao. Symmetric I* Restriction Method of Fuzzy Inference. Math Comput Sci. 2019;4(6):130-137. doi: 10.11648/j.mcs.20190406.14

Copy | Download

@article{10.11648/j.mcs.20190406.14,
  author = {Yiming Tang and Guangqing Bao},
  title = {Symmetric I* Restriction Method of Fuzzy Inference},
  journal = {Mathematics and Computer Science},
  volume = {4},
  number = {6},
  pages = {130-137},
  doi = {10.11648/j.mcs.20190406.14},
  url = {https://doi.org/10.11648/j.mcs.20190406.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20190406.14},
  abstract = {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.},
 year = {2019}
}

Copy | Download

TY - JOUR
T1 - Symmetric I* Restriction Method of Fuzzy Inference
AU - Yiming Tang
AU - Guangqing Bao
Y1 - 2019/12/24
PY - 2019
N1 - https://doi.org/10.11648/j.mcs.20190406.14
DO - 10.11648/j.mcs.20190406.14
T2 - Mathematics and Computer Science
JF - Mathematics and Computer Science
JO - Mathematics and Computer Science
SP - 130
EP - 137
PB - Science Publishing Group
SN - 2575-6028
UR - https://doi.org/10.11648/j.mcs.20190406.14
AB - 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.
VL - 4
IS - 6
ER -

Copy | Download

Author Information

Yiming Tang

School of Computer and Information, Hefei University of Technology, Hefei, China
Guangqing Bao

School of Computer and Information, Hefei University of Technology, Hefei, China

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Yiming Tang, Guangqing Bao. (2019). Symmetric I* Restriction Method of Fuzzy Inference. Mathematics and Computer Science, 4(6), 130-137. https://doi.org/10.11648/j.mcs.20190406.14

Copy | Download

ACS Style

Yiming Tang; Guangqing Bao. Symmetric I* Restriction Method of Fuzzy Inference. Math. Comput. Sci. 2019, 4(6), 130-137. doi: 10.11648/j.mcs.20190406.14

Copy | Download

AMA Style

Yiming Tang, Guangqing Bao. Symmetric I* Restriction Method of Fuzzy Inference. Math Comput Sci. 2019;4(6):130-137. doi: 10.11648/j.mcs.20190406.14

Copy | Download

@article{10.11648/j.mcs.20190406.14,
  author = {Yiming Tang and Guangqing Bao},
  title = {Symmetric I* Restriction Method of Fuzzy Inference},
  journal = {Mathematics and Computer Science},
  volume = {4},
  number = {6},
  pages = {130-137},
  doi = {10.11648/j.mcs.20190406.14},
  url = {https://doi.org/10.11648/j.mcs.20190406.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20190406.14},
  abstract = {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.},
 year = {2019}
}

Copy | Download