PROPERTY ANALYSIS OF TRIPLE IMPLICATION METHOD FOR APPROXIMATE REASONING ON ATANASSOVS INTUITIONISTIC FUZZY SETS

Document Type: Research Paper

Authors

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

Abstract

Firstly, two kinds of natural distances between intuitionistic fuzzy sets are generated by the classical natural distance between fuzzy sets under a unified framework of residual intuitionistic implication operators. Secondly, the continuity and approximation property of a method for solving intuitionistic fuzzy reasoning are defined. It is proved that the triple implication method for intuitionistic fuzzy modus ponens has both continuity and approximation property with respect to these two kinds of natural distances based on {\L}ukasiewicz implication, meanwhile the triple implication method for intuitionistic fuzzy modus tollens possesses unconditional continuity and conditional approximation property. Finally, some robustness results about the triple implication method of intuitionistic fuzzy reasoning are given.

Keywords


[1] K. Atanassov, Geometrical interpretation of the elements of the intuitionistic fuzzy objects,
Preprint IM-MFAIS-1-89, Sofi a, 1989. Reprinted: International Journal of Bioautomation,
20(1) (2016), S43-S54.
[2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986), 87-96.
[3] K. Atanassov, Intuitionistic fuzzy sets, International Journal of Bioautomation, 20(S1)
(2016), S27-S42.
[4] K. Atanassov, Intuitionistic Fuzzy Sets, Springer, Heidelberg, 1999.
[5] K. Atanassov, On intuitionistic fuzzy sets theory, Springer, Berlin, 8(1) (2012), 147-163.
[6] I. Bloch, Mathematical morphology on bipolar fuzzy sets: General algebraic framework, In-
ternational Journal of Approximate Reasoning, 53(7) (2012), 1031-1060.
[7] F. Chen, W. H. Xu, C. Z. Bai and X. Gao, A novel approach to guarantee good robustness
of fuzzy reasoning, Applied Soft Computing, 41(C) (2016), 224-234.
[8] R. Cignoli, I. M. L. Dottaviano and D. Mundici, Algebraic Foundations of Many-Valued
Reasoning, Kluwer Academic Publishers, Dordrecht, 2000.
[9] C. Cornelis, G. Deschrijver and E. E. Kerre, Implication in intuitionistic fuzzy and interval-
valued fuzzy set theory: construction, classifi cation, application, International Journal of
Approximate Reasoning, 35(1) (2004), 55-95.
[10] S. S. Dai, D. W. Pei and D. Guo, Robustness analysis of full implication inference method,
International Journal of Approximate Reasoning, 54(5) (2013), 653-666.
[11] 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(1) (2012), 63-73.
[12] S. Das, B. Dutta and D. Guha, Weight computation of criteria in a decision-making problem
by knowledge measure with intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set,
Soft Computing, 20(9) (2016), 3421-3442.
[13] G. Deschrijver, C. Cornelis and E. E. Kerre, Class of intuitionistic fuzzy t-norms satisfying the
residuation principle, International Journal of Uncertainty Fuzziness and Knowledge-Based
Systems, 11(6) (2003), 691-709.
[14] G. Deschrijver, C. Cornelis and E. E. Kerre, On the representation of intuitionistic fuzzy
t-norms and t-conorms, IEEE Transactions on Fuzzy Systems, 12(1) (2004), 45-61.
[15] J. Y. Duan and Y. M. Li, Robustness analysis of logic metrics on F(X), International Journal
of Approximate Reasoning, 61 (2015), 33-42.
[16] D. Dubois and H. Prade, Fuzzy Sets in approximate reasoning, Fuzzy Sets and Systems,
40(1) (1991), 143-244.
[17] D. Dubois and H. Prade, Gradualness, uncertainty and bipolarity: Making sense of fuzzy
sets, Fuzzy Sets and Systems, 192(3) (2012), 3-24.
[18] F. Esteva and L. Godo, Monoidal t-norm based logic: Towards a logic for left-continuous
t-norms, Fuzzy Sets and Systems, 124(3) (2001), 271-288.
[19] C. Franco, J. Montero and J. T. Rodriguez, A fuzzy and bipolar approach to preference
modeling with application to need and desire, Fuzzy Sets and Systems, 214(1) (2013), 20-34.
[20] P. Hajek, Mathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.
[21] X. X. He, K. Y. Qin and Y. F. Li, Robustness of fuzzy connectives and fuzzy reasoning, Fuzzy
Sets and Systems, 225(12) (2013), 93-105.
[22] 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.

[23] C. C. Lee, Fuzzy logic in control systems: Fuzzy logic controller, IEEE Transactions on
Systems, Man, and Cybernetics, 20(2) (1990), 404-418.
[24] D. C. Li, Y. M. Li and Y. J. Xie, Robustness of interval-valued fuzzy inference, Information
Sciences, 181(20) (2011), 4754-4764.
[25] D. F. Li and C. T. Cheng, New similarity measures of intuitionistic fuzzy sets and application
to pattern recognitions, Pattern Recognition Letters, 23(1-3) (2002), 221-225.
[26] Y. F. Li, K. Y. Qin, X. X. He and D. Meng, Robustness of fuzzy connectives and fuzzy
reasoning with respect to general divergence measures, Fuzzy Sets and Systems, 294 (2016),
63-78.
[27] H. W. Liu and G. J. Wang, Continuity of triple I methods based on several implications,
Mathematical and Computer Modelling, 56(8) (2008), 2079-2087.
[28] H. W. Liu and G. J. Wang, Multi-criteria decision-making methods based on intuitionistic
fuzzy sets, European Journal of Operational Research, 179(1) (2007), 220-233.
[29] H. W. Liu and G. J. Wang, Uni fied forms of fully implicational restriction methods for fuzzy
reasoning, Information Sciences, 177(3) (2007), 956-966.
[30] H. W. Liu, New similarity measures between intuitionistic fuzzy sets and between elements,
Mathematical and Computer Modelling, 42(1) (2005), 61-70.
[31] R. Lowen and W. Peeters, Distances between fuzzy sets representing grey level images, Fuzzy
Sets and Systems, 99(2) (1998), 135-149.
[32] E. H. Mamdani and B. R. Gaines, Fuzzy Reasoning and its Applications, Academic Press,
London, 18(11) (1981), 1925-1935.
[33] D. W. Pei, Formalization of implication based fuzzy reasoning method, International Journal
of Approximate Reasoning, 53(5) (2012), 837-846.
[34] D. W. Pei, On equivalent forms of fuzzy logic systems NM and IMTL, Fuzzy Sets and Sys-
tems, 138(1) (2003), 187-195.
[35] D. W. Pei, On the strict logic foundation of fuzzy reasoning, Soft Computing, 8(8) (2004),
539-545.
[36] D. W. Pei, The full implication triple I algorithms and their consistency in fuzzy reasoning,
Journal of Mathematical Research and Exposition, 24(2) (2004), 359-368 (in Chinese).
[37] D. W. Pei, Uni ed full implication algorithms of fuzzy reasoning, Information Sciences,
178(2) (2008), 520-530.
[38] R. H. S. Reiser and B. Bedregal, Robustness on intuitionistic fuzzy connectives, Tema, 15(2)
(2014), 133-149.
[39] R. Srinivasan and S. S. Begum, Properties on Intuitionistic Fuzzy Sets of Third Type, Fuzzy
Information Processing Society, IEEE, 12(2) (2017), 189-195.
[40] Y. M. Tang and X. Z. Yang, Symmetric implicational method of fuzzy reasoning, International
Journal of Approximate Reasoning, 54(8) (2015), 1034-1048.
[41] G. J. Wang and J. Y. Duan, On robustness of the full implication triple I inference method
with respect to fi ner measurements, International Journal of Approximate Reasoning, 55(3)
(2014), 787-796.
[42] G. J. Wang and L. Fu, Uni fied forms of triple I method, Computers and Mathematics with
Applications, 49(5-6) (2005), 923-932.
[43] G. J. Wang and H. Wang, Non-fuzzy versions of fuzzy reasoning in classical logics, Informa-
tion Sciences, 138(1) (2001), 211-236.
[44] G. J. Wang, Formalized theory of general fuzzy reasoning, Information Sciences, 160(1)
(2004), 251-266.
[45] G. J. Wang, Full implicational triple I method for fuzzy reasoning, Science in China, Series
E, 29(1) (1999), 43-53.
[46] G. J. Wang, Introduction to Mathematical Logic and Resolution Principle, Science Press,
Beijing, (in Chinese), 2006.
[47] G. J. Wang, Non-classical Mathematical Logic and Approximate Reasoning, Science Press,
Beijing, (in Chinese), 2008.
[48] G. J. Wang, On the logic foundation of fuzzy reasoning, Information Sciences, 117(1-2)
(1999), 47-88.

9] L. X. Wang, A Course in Fuzzy Systems and Control, Upper Saddle River, Prentice Hall
PTR, NJ, 1997.
[50] W. H. Xu, Z. K. Xie, J. Y. Yang and Y. E. You-Pei, Continuity and approximation properties
of two classes of algorithms for fuzzy inference, Journal of Software, 15(10) (2004), (in
chinese), 1485-1492.
[51] L. A. Zadeh, Fuzzy sets, information and control, Information and Control, 8(3) (1965),
338-353.
[52] L. A. Zadeh, Generalized theory of uncertainty (GTU)-principal concepts and ideas, Com-
putational Statistics and Data Analysis, 51(1) (2006), 15-46.
[53] L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision
processes, IEEE Transactions on Systems, Man, and Cybernetics, smc-3(1) (1973), 28-44.
[54] L. A. Zadeh, Position Paper: Toward extended fuzzy logic-A fi rst step, Fuzzy Sets and Sys-
tems, 160(21) (2009), 3175-3181.
[55] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning,
Learning Systems and Intelligent Robots. Springer US, 8(3) (1974), 199-249.
[56] M. C. Zheng, Z. K. Shi and Y. Liu, Triple I method of approximate reasoning on Atanassov's
intuitionistic fuzzy sets, International Journal of Approximate Reasoning, 55(6) (2014), 1369-
1382.
[57] M. C. Zheng, Z. K. Shi and Y. Liu, Triple I method of intuitionistic fuzzy reasoning based
on residual implicator, Information Sciences, 43(6) (2013), (in chinese), 810-820.