ROBUSTNESS OF THE TRIPLE IMPLICATION INFERENCE METHOD BASED ON THE WEIGHTED LOGIC METRIC

Document Type: Research Paper

Authors

School of Science, Lanzhou University of Technology, Lanzhou 730050, Gansu, China

Abstract

This paper focuses on the robustness problem of full implication triple implication inference method for fuzzy reasoning. First of all, based on strong regular implication, the weighted logic metric for measuring distance between two fuzzy sets is proposed. Besides, under this metric, some robustness results of the triple implication method are obtained, which demonstrates that the triple implication method possesses a good behavior of robustness.

Keywords


[1] R. Belohlavek, Fuzzy Relational Systems, Foundations and Principles, Kluwer Academic
Publishers, Dordrecht, (2002), 40-116.
[2] K. Y. Cai, -equalities of fuzzy sets, Fuzzy Sets and Systems, 76(1) (1995), 97-112.
[3] K. Y. Cai, Robustness of fuzzy reasoning and -equalities of fuzzy sets, IEEE Transactions
on Fuzzy Systems, 9(5) (2001), 738-750.
[4] G. S. Cheng and Y. X. Fu, Error estimation of perturbations under CRI, IEEE Transactions
on Fuzzy Systems, 14(6) (2006), 709-715.
[5] S. S. Dai, D. W. Pei and S. M. Wang, Perturbation of fuzzy sets and fuzzy reasoning based
on normalized Minkowski distances, Fuzzy Sets and Systems, 189 (2012), 63-73.
[6] S. S. Dai, D. W. Pei and D. H. Guo, Robustness analysis of full implication inference method,
International Journal of Approximate Reasoning, 54(5) (2013), 653-666.
[7] D. Dubois, J. Lang and H. Prade, Fuzzy sets in approximate reasoning, parts 1 and 2, Fuzzy
Sets and Systems, 40(1) (1991), 143-244.
[8] P. Hajek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, (1998),
27-29.
[9] D. H. Hong and S. Y. Hwang, A note on the value similarity of fuzzy systems variables,
Fuzzy Sets and Systems, 66(3) (1994), 383-386.
[10] E. P. Klement, R. Mesiar and E. Pap, Triangular Norms, Kluwer Academic Publishers,
Dordrecht, (2000), 4-6.
[11] J. Li and J. T. Yao, Theory of integral truth degrees of formula in SMTL propositional logic,
Acta Electronica Sinica, 41(5) (2013), 878-883.
[12] Y. M. Li, D. C. Li, W. Pedrycz and J. J. Wu, An approach to measure the robustness of fuzzy
reasoning, International Journal of Intelligent Systems, 20(4) (2005), 393-413.
[13] H. W. Liu and G. J. Wang, Continuity of triple I methods based on several implications,
Computers and Mathematics with Applications, 56(8) (2008), 2079-2087.
[14] H. W. Liu and G. J. Wang, A note on the uni ed forms of triple I method, Computers and
Mathematics with Applications, 52(10) (2006), 1609-1613.
[15] H. W. Liu and G. J. Wang, Uni ed forms of fully implicational restriction methods for fuzzy
reasoning, Information Sciences, 177(3) (2007), 956-966.

[16] H. W. Liu and G. J. Wang, Triple I method based on pointwise sustaining degrees, Computers
and Mathematics with Applications, 55(11) (2008), 2680-2688.
[17] M. X. Luo and N. Yao, Triple I algorithms based on Schweizer- Sklar operators in fuzzy
reasoning, International Journal of Approximate Reasoning, 54(5) (2013), 640-652.
[18] C. P. Pappis, Value approximation of fuzzy systems variables, Fuzzy Sets and Systems, 39(1)
(1991), 111-115.
[19] D. W. Pei, Uni ed full implication algorithms of fuzzy reasoning, Information Sciences,
178(2) (2008), 520-530.
[20] G. J. Wang, Non-classical Mathematical Logic and Approximate Reasoning, second ed., Sci-
ence Press, Beijing, China, (2008), 155-165.
[21] G. J.Wang, Introduction to Mathematical Logic and Resolution Principle, second ed., Science
Press, Beijing, China, (2006), 160-162.
[22] G. J. Wang and H. Wang, Non-fuzzy versions of fuzzy reasoning in classical logics, Informa-
tion Sciences, 138 (2001), 211-236.
[23] G. J. Wang and J. Y. Duan, On robustness of the full implication triple I inference method
with respect to ner measurements, International Journal of Approximate Reasoning, 55(3)
(2014),787-796.
[24] S. W. Wang and W. X. Zheng, Real Variable Function and Functional Analysis, Higher
Education Press, Beijing, China, (2005), 4-5.
[25] G. J. Wang, The full implication triple I method of fuzzy reasoning, SCIENCE CHINA Ser.
E 29 (1999), 43-53.
[26] G. J. Wang and J. Y. Duan, Two types of fuzzy metric spaces suitable for fuzzy reasoning,
Science China Information Sciences, 44(5) (2014), 623-632.
[27] G. J. Wang and L. Fu, Uni ed forms of triple I method, Computers and Mathematics with
Applications, 49(5) (2005), 923-932.
[28] G. J. Wang, Formalized theory of general fuzzy reasoning, Information Sciences, 160(1)
(2004), 251-266.
[29] R. R. Yager, On some new classes of implication operators and their role in approximate
reasoning, Information Sciences, 167(1-4) (2004), 193-216.
[30] M. S. Ying, Perturbation of fuzzy reasoning, IEEE Transactions on Fuzzy Systems, 7(5)
(1999), 625-629.
[31] L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision
processes, IEEE Transactions on Systems Man and Cybernetics, 3(1) (1973), 28-33.