Kybernetika 57 no. 4, 688-713, 2021

Symmetric implicational restriction method of fuzzy inference

Yiming Tang, Wenbin Wu, Youcheng Zhang, Witold Pedrycz, Fuji Ren and Jun LiuDOI: 10.14736/kyb-2021-4-0688


The symmetric implicational method is revealed from a different perspective based upon the restriction theory, which results in a novel fuzzy inference scheme called the symmetric implicational restriction method. Initially, the SIR-principles are put forward, which constitute optimized versions of the triple I restriction inference mechanism. Next, the existential requirements of basic solutions are given. The supremum (or infimum) of its basic solutions is achieved from some properties of fuzzy implications. The conditions are obtained for the supremum to become the maximum (or the infimum to be the minimum). Lastly, four concrete examples are provided, and it is shown that the new method is better than the triple I restriction method, because the former is able to let the inference more compact, and lead to more and superior particular inference schemes.


continuity, fuzzy entropy, fuzzy inference, compositional rule of inference


03B52, 94D05


  1. M. Baczyński and B. Jayaram: On the characterizations of (S,N)-implications. Fuzzy Sets Syst. 158 (2007), 1713-1727.   DOI:10.1016/j.fss.2007.02.010
  2. S. S. Dai: Logical foundation of symmetric implicational methods for fuzzy reasoning. J. Intell. Fuzzy Syst. 39 (2020), 1089-1095.   DOI:10.3233/JIFS-191998
  3. S. S. Dai, D. W. Pei and D. H. Guo: Robustness analysis of full implication inference method. Int. J. Approx. Reason. 54 (2013), 653-666.   DOI:10.1016/j.ijar.2012.11.007
  4. J. Fodor and M. Roubens: Fuzzy Preference Modeling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht, 1994.   CrossRef
  5. P. Hájek: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht, 1998.   CrossRef
  6. J. Hou, F. You and H. X. Li: Fuzzy systems constructed by triple I algorithm and their response ability. Prog. Nat. Sci. 15 (2005), 29-37.   DOI:10.1038/ng0105-15
  7. H. X. Li: Probability representations of fuzzy systems. Sci. China Ser. F Inf. Sci. 49 (2006), 339-363.   DOI:10.1007/s11432-006-0339-9
  8. D. C. Li and Y. M. Li: Algebraic structures of interval-valued fuzzy (S,N)-implications. Int. J. Approx. Reason. 53 (2012) 892-900.   DOI:10.1016/j.ijar.2012.03.003
  9. H. X. Li, F. You and J. Y. Peng: Fuzzy controllers based on some fuzzy implication operators and their response functions. Prog. Nat. Sci. 14 (2004), 15-20.   DOI:10.1177/8756479304201004
  10. H. W. Liu and G. J. Wang: Unified forms of fully implicational restriction methods for fuzzy reasoning. Inf. Sci. 177 (2007), 956-966.   DOI:10.1016/j.ins.2006.08.012
  11. M. X. Luo and B. Liu: Robustness of interval-valued fuzzy inference triple I algorithms based on normalized Minkowski distance. J. Log. Algebr. Methods 86 (2017), 298-307.   DOI:10.1016/j.jlamp.2016.09.006
  12. M. X. Luo and N. Yao: Triple I algorithms based on Schweizer-Sklar operators in fuzzy reasoning. Int. J. Approx. Reason. 54 (2013), 640-652.   DOI:10.1016/j.ijar.2013.01.008
  13. M. X Luo and K. Zhang: Robustness of full implication algorithms based on interval-valued fuzzy inference. Int. J. Approx. Reason. 62 (2015), 61-72.   DOI:10.1016/j.ijar.2015.05.006
  14. P. Kaur, M. Goyal and J. Lu: A comparison of bidding strategies for online auctions using fuzzy reasoning and negotiation decision functions. IEEE Trans. Fuzzy Syst. 25 (2017) 425-438.   DOI:10.1109/TFUZZ.2016.2598297
  15. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, Dordrecht, 2000.   CrossRef
  16. M. Mas, M. Monserrat, J. Torrens and E. Trillas: A survey on fuzzy implication functions. IEEE Trans. Fuzzy Syst. 15 (2007), 1107-1121.   DOI:10.1109/TFUZZ.2007.896304
  17. V. Novák, I. Perfilieva and J. Močkoř: Mathematical Principles of Fuzzy Logic. Kluwer Academic Publishes, Boston, Dordrecht, 1999.   CrossRef
  18. J. Y. Peng: Fully implicational triple I restriction algorithm for fuzzy reasoning based on some familiar implication operators. Prog. Nat. Sci. 15 (2005), 539-546.   DOI:0.1007/BF02714092
  19. S. J. Song, C. B. Feng and C. X. Wu: Theory of restriction degree of triple I method with total inference rules of fuzzy reasoning. Prog. Nat. Sci. 11 (2001), 58-66.   CrossRef
  20. S. J. Song and C. Wu: Reverse triple I method of fuzzy reasoning. Sci. China, Ser. F, Inf. Sci. 45 (2002), 344-364.   CrossRef
  21. D. W. Pei: $R_{0}$ implication: characteristics and applications. Fuzzy Set Syst. 131 (2002), 297-302.   DOI:10.1016/s0165-0114(02)00053-2
  22. D. W. Pei: On the strict logic foundation of fuzzy reasoning. Soft Comput. 8 (2004), 539-545.   DOI:10.1007/s00500-003-0312-0
  23. D. W. Pei: Formalization of implication based fuzzy reasoning method. Int. J. Approx. Reason. 53 (2012), 837-846.   DOI:10.1016/j.ijar.2012.01.007
  24. W. Pedrycz: Granular Computing: Analysis and Design of Intelligent Systems. CRC Press/Francis and Taylor, Boca Raton 2013.   CrossRef
  25. W. Pedrycz: From fuzzy data analysis and fuzzy regression to granular fuzzy data analysis. Fuzzy Set Syst. 274 (2015), 12-17.   DOI:10.1007/s10393-016-1112-1
  26. W. Pedrycz and X. M. Wang: Designing fuzzy sets with the use of the parametric principle of justifiable granularity. IEEE Trans. Fuzzy Syst. 24 (2016), 489-496.   DOI:10.1109/TFUZZ.2015.2453393
  27. Y. M. Tang and X. Z. Yang: Symmetric implicational method of fuzzy reasoning. Int. J. Approx. Reason. 54 (2013), 1034-1048.   DOI:10.1016/j.ijar.2013.04.012
  28. Y. M. Tang and W. Pedrycz: On the $\alpha$(u,v)-symmetric implicational method for R- and (S, N)-implications. Int. J. Approx. Reason. 92 (2018), 212-231.   DOI:10.1016/j.ijar.2017.10.009
  29. L. X. Wang: A Course in Fuzzy Systems and Control. Prentice-Hall, Englewood Cliffs, NJ 1997.   CrossRef
  30. G. J. Wang: On the logic foundation of fuzzy reasoning. Inform. Sci. 117 (1999), 47-88.   DOI:10.1016/S0020-0255(98)10103-2
  31. G. J. Wang and L. Fu: Unified forms of triple I method. Comput. Math. Appl. 49 (2005), 923-932.   DOI:10.1016/j.camwa.2004.01.019
  32. G. J. Wang and H. J. Zhou: Introduction to Mathematical Logic and Resolution Principle. Co-published by Science Press and Alpha International Science Ltd., 2009.   CrossRef
  33. L. A. Zadeh: Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Syst. Man Cyber. 3 (1973), 28-44.   DOI:10.1109/TSMC.1973.5408575