Interval additive generators of interval t-norms and interval t-conorms

Information Sciences

Volumn 181, Issue 18, 2011, Pages 3898-3916

  Dimuro, Graaliz Pereira ^a   Bedregal, Benjamín Callejas ^b   Santiago, Regivan Hugo Nunes ^b   Reiser, Renata Hax Sander ^c  

^a FEDERAL UNIVERSITY OF RIO GRANDE   (Brazil)

^b FEDERAL UNIVERSITY OF RIO GRANDE DO NORTE   (Brazil)

^c CATHOLIC UNIVERSITY OF PELOTAS   (Brazil)

Author keywords

Interval additive generator; Interval fuzzy logic; Interval triangular conorm; Interval triangular norm; Interval valued fuzzy connectives

Indexed keywords

ADDITIVE GENERATORS; INTERVAL FUZZY LOGIC; INTERVAL-VALUED; TRIANGULAR CONORM; TRIANGULAR NORMS;

FUZZY LOGIC; FUZZY SYSTEMS;

MATHEMATICAL OPERATORS;

EID: 79959743869     PISSN: 00200255     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ins.2011.05.003     Document Type: Article

References (125)

1. B.M. Acióly, and B.C. Bedregal A quasi-metric topology compatible with inclusion monotonicity on interval space Reliable Computing 3 3 1997 305 313 (Pubitemid 127506435)
2. I. Aguiló, J. Suñer, and J. Torrens A characterization of residual implications derived from left-continuous uninorms Information Sciences 180 20 2010 3992 4005
3. R. Aliev, and A. Tserkovny Systemic approach to fuzzy logic formalization for approximate reasoning Information Sciences 181 6 2011 1045 1059
4. K.T. Atanassov, Intuitionistic fuzzy sets, Tech. Rep. 1697/84, Central Techical Library, Bulgarian Academy Science, Sofia, Bulgaria, 1983.
5. K.T. Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems 20 1 1986 87 96
6. M. Baczynsk, and B. Jayaram (U, N)-implications and their characterizations Fuzzy Sets and Systems 160 14 2009 2049 2062
7. M. Baczynski, and B. Jayaram (S, N)- and R-implications: a state-of-the-art survey Fuzzy Sets and Systems 159 14 2008 1836 1859
8. M. Baczynski, and B. Jayaram QL-implications: some properties and intersections Fuzzy Sets and Systems 161 2 2010 158 188
9. M. Baczynski, and B. Jayaram Intersections between some families of (U, N)- and RU-implications Fuzzy Sets and Systems 167 1 2011 30 44
10. B.C. Bedregal On interval fuzzy negations Fuzzy Sets and Systems 161 17 2010 2290 2313
11. B.C. Bedregal, G.P. Dimuro, R.H.N. Santiago, and R.H.S. Reiser On interval fuzzy S-implications Information Sciences 180 8 2010 1373 1389
12. B.C. Bedregal, R.H.S. Reiser, G.P. Dimuro, Xor-implications and E-implications: classes of fuzzy implications based on fuzzy xor, Electronic Notes in Theoretical Computer Science 247 (2009) 5-18 (Proceedings of the Third Workshop on Logical and Semantic Frameworks with Applications (LSFA 2008)).
13. B.C. Bedregal, R.H.N. Santiago, R.H.S. Reiser, and G.P. Dimuro Analyzing properties of fuzzy implications obtained via the interval constructor 12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, 26-29 September, Duisburg, 2006, SCAN 2006 Conference Post-Proceedings 2007 IEEE Computer Society Los Alamitos (Paper n.13)
14. B.C. Bedregal, and A. Takahashi Interval t-norms as interval representations of t-norms Proceedings of the IEEE International Conference on Fuzzy Systems, July 2005, Reno 2005 IEEE Los Alamitos 909 914 (Pubitemid 41189627)
15. B.C. Bedregal, and A. Takahashi The best interval representation of t-norms and automorphisms Fuzzy Sets and Systems 157 24 2006 3220 3230 (Pubitemid 44709759)
16. B.C. Bedregal, and A. Takahashi Interval valued versions of t-conorms, fuzzy negations and fuzzy implications Proceedings of the IEEE International Conference on Fuzzy Systems, Vancouver, 2006 2006 IEEE Los Alamitos 1981 1987 (Pubitemid 46949018)
17. G. Beliakov, H. Bustince, D.P. Goswami, U.K. Mukherjee, and N.R. Pal On averaging operators for Atanassov's intuitionistic fuzzy sets Information Sciences 181 6 2011 1116 1124
18. H. Bustince, E. Barrenechea, M. Pagola, and J. Fernandez Interval-valued fuzzy sets constructed from matrices: application to edge detection Fuzzy Sets and Systems 160 13 2009 1819 1840
19. H. Bustince, E. Barrenechea, M. Pagola, J. Fernandez, and J. Sanz Comment on: "Image thresholding using type II fuzzy sets". Importance of this method Pattern Recognition 43 9 2010 3188 3192
20. H. Bustince, and P. Burillo Mathematical analysis of interval-valued fuzzy relations: application to approximate reasoning Fuzzy Sets and Systems 113 2 2000 205 219
21. R. Callejas-Bedregal, and B.C. Bedregal Intervals as a domain constructor TEMA - Tendências em Matemática Aplicada e Computacional 2 1 2001 43 52 (Available in English at )
22. O. Caprani, K. Madsen, and O. Stauning Existence test for asynchronous interval iteration Reliable Computing 3 3 1997 269 275 (Pubitemid 127506432)
23. J.R. Castro, O. Castillo, P. Melin, and A. Rodríguez-Díaz A hybrid learning algorithm for a class of interval type-2 fuzzy neural networks Information Sciences 179 13 2009 2175 2193
24. C. Cornelis, G. Deschrijver, and E.E. Kerre Classification of intuitionistic fuzzy implicators: an algebraic approach H.J. Caulfield, S. Chen, H. Chen, R. Duro, V. Honavar, E.E. Kerre, M. Lu, M.G. Romay, T.K. Shih, D. Ventura, P.P. Wang, Y. Yang, Proceedings of 6th Joint Conference on Information Sciences, March 2002, North Carolina 2002 JCIS/Association for Intelligent Machinery, Inc. 105 108
25. G. Cornelis, G. Deschrijver, and E.E. Kerre Advances and challenges in interval-valued fuzzy logic Fuzzy Sets and Systems 157 5 2006 622 627 (Pubitemid 43189821)
26. G. Deschrijver Additive and multiplicative generators in interval-valued fuzzy set theory IEEE Transactions on Fuzzy Systems 15 2 2007 222 237 (Pubitemid 46563423)
27. G. Deschrijver Arithmetic operators in interval-valued fuzzy set theory Information Sciences 177 14 2007 2906 2924 (Pubitemid 46660010)
28. G. Deschrijver A representation of t-norms in interval-valued L-fuzzy set theory Fuzzy Sets and Systems 159 13 2008 1597 1618 (Pubitemid 351672311)
29. G. Deschrijver Characterizations of (weakly) archimedean t-norms in interval-valued fuzzy set theory Fuzzy Sets and Systems 160 6 2009 778 801
30. G. Deschrijver Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory Fuzzy Sets and Systems 160 21 2009 3080 3102
31. 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
32. G.P. Dimuro, B.R.C. Bedregal, R.H.S. Reiser, and R.H.N. Santiago Interval additive generators of interval T-norms W. Hodges, R. de Queiroz, Proceedings of the 15th International Workshop on Logic, Language, Information and Computation, WoLLIC 2008, Edinburgh LNAI vol. 5110 2008 Springer Berlin 123 135
33. D. Dubois, and H. Prade Random sets and fuzzy interval analysis Fuzzy Sets and Systems 42 1 1991 87 101
34. D. Dubois, and H. Prade Fuzzy Sets and Systems 1996 Academic Press New York
35. D. Dubois, H. Prade, Interval-valued fuzzy sets, possibility theory and imprecise probability, in: E. Montseny, P. Sobrevilla (Eds.), Proceedings of the Joint 4th Conference of the European Society for Fuzzy Logic and Technology and the 11th Rencontres Francophones sur la Logique Floue et ses Applications, Universidad Polytecnica de Catalunya, Barcelona, 2005, pp. 314-319.
36. D. Dubois, H. Prade, and S. Sessa Recent literature Fuzzy Sets and Systems 160 20 2009 3017 3026
37. D. Dubois, H. Prade, and S. Sessa Recent literature Fuzzy Sets and Systems 161 7 2010 1039 1046
38. W.M. Faucett Compact semigroups irreducibly connected between two idempotents Proceedings of the American Mathematical Society 6 1955 741 747
39. F. Feng, Y. Li, and V. Leoreanu-Fotea Application of level soft sets in decision making based on interval-valued fuzzy soft sets Computers & Mathematics with Applications 60 6 2010 1756 1767
40. J. Fodor, and M. Roubens Fuzzy Preference Modelling and Multicriteria Decision Support 1994 Kluwer Academic Publisher Dordrecht
41. B.V. Gasse, C. Cornelis, G. Deschrijver, and E.E. Kerre The pseudo-linear semantics of interval-valued fuzzy logics Information Sciences 179 6 2009 717 728
42. M. Gehrke, C. Walker, and E. Walker Some comments on interval valued fuzzy sets International Journal of Intelligent Systems 11 10 1996 751 759 (Pubitemid 126578776)
43. G. Gierz, K. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D. Scott Continuous Lattices and Domains 2003 Cambridge Press Cambridge
44. M.B. Gorzalczany A method of inference in approximate reasoning based on interval-valued fuzzy sets Fuzzy Sets and Systems 21 1 1987 1 17
45. I. Grattan-Guiness Fuzzy membership mapped onto interval and many-valued quantities Zeitschrift für mathematische Logik und Grundlagen der Mathematik 22 1 1976 149 160
46. O. Hadzic, and E. Pap Fixed Point Theorem in Probabilistic Metric Spaces 2001 Kluwer Academic Publishers. Dordrecht
47. T. Hickey, Q. Ju, and M. Emdem Interval arithmetic: from principles to implementation Journal of the ACM 48 5 2001 1038 1068
48. P. Hájek Metamathematics of Fuzzy Logic 1998 Kluwer Academic Publishers. Boston
49. K.-U. Jahn Intervall-wertige mengen Mathematische Nachrichten 68 1 1975 115 132
50. L. Jaulin, M. Kieffer, O. Didrit, and E. Walter Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotic 2001 Springer Heildelberg
51. B. Jayaram, and R. Mesiar On special fuzzy implications Fuzzy Sets and Systems 160 14 2009 2063 2085
52. R.B. Keafort Rigorous Global Search: Continuous problems 1996 Kluwer Academic Publishers Dordrecht
53. E.P. Klement, R. Mesiar, and E. Pap A characterization of the ordering of continuous t-norms Fuzzy Sets and Systems 86 2 1997 189 195 (Pubitemid 127700454)
54. E.P. Klement, R. Mesiar, and E. Pap Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms Fuzzy Sets and Systems 104 1 1999 3 13 (Pubitemid 129533549)
55. E.P. Klement, R. Mesiar, and E. Pap Generated triangular norms Kybernetika 36 3 2000 363 377
56. E.P. Klement, R. Mesiar, and E. Pap Triangular Norms 2000 Kluwer Academic Publisher Dordrecht
57. E.P. Klement, R. Mesiar, and E. Pap Triangular norms. Position paper II: general constructions and parameterized families Fuzzy Sets and Systems 145 3 2004 411 438
58. E.P. Klement, R. Mesiar, and E. Pap Triangular norms: basic notions and properties E.P. Klement, R. Mesiar, Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms 2005 Elsevier Amsterdam 17 60
59. E.P. Klement, and M. Navara A survey on different triangular norm-based fuzzy logics Fuzzy Sets and Systems 101 2 1999 241 251 (Pubitemid 129496790)
60. G.J. Klir, and B. Yuan Fuzzy Sets and Fuzzy Logics: Theory and Applications 1995 Prentice Hall Upper Saddle River
61. V. Kreinovich, A. Lakeyev, J. Rohn, and P. Kahl Computational Complexity and Feasibility of Data Processing and Interval Computations 1997 Kluwer Dordrecht
62. J. Leventides, and A. Bounas An approach to the selection of fuzzy connectives in terms of their additive generators Fuzzy Sets and Systems 126 2 2002 219 224 (Pubitemid 34116146)
63. L. Lianzhen States on finite monoidal t-norm based algebras Information Sciences 181 7 2011 1369 1383
64. F.-J. Lin, S.-Y. Chen, P.-H. Chou, and P.-H. Shieh Interval type-2 fuzzy neural network control for x-y-theta motion control stage using linear ultrasonic motors Neurocomputing 72 4-6 2009 1138 1151
65. C.-H. Ling Representation of associative functions Publicationes Mathematicae Debrecen 12 1965 189 212
66. W.A. Lodwick Preface Reliable Computing 10 4 2004 247 248
67. R. Martínez, O. Castillo, and L.T. Aguilar Optimization of interval type-2 fuzzy logic controllers for a perturbed autonomous wheeled mobile robot using genetic algorithms Information Sciences 179 13 2009 2158 2174
68. M. Mas, M. Monserrat, and J. Torrens Two types of implications derived from uninorms Fuzzy Sets and Systems 158 3 2007 2612 2626 (Pubitemid 47445751)
69. M. Mas, M. Monserrat, and J. Torrens A characterization of (U, N), RU, QL and D-implications derived from uninorms satisfying the law of importation Fuzzy Sets and Systems 161 10 2010 1369 1387
70. S. Massanet, and J. Torrens On a new class of fuzzy implications: h-implications and generalizations Information Sciences 181 11 2011 2111 2127
71. G. Mayorand, and J. Monreal Additive generators of discrete conjunctive aggregation operations IEEE Transactions on Fuzzy Systems 15 6 2007 1046 1052 (Pubitemid 350268637)
72. K. Meer On the complexity of some problems in interval arithmetic B. Rovan, P. Vojtás, Mathematical Foundations of Computer Science 2003 Proceedings of the 28th International Symposium, MFCS 2003, Bratislava, Slovakia, August 25-29, 2003 vol. 2747 2003 Springer Berlin 582 591
73. J.M. Mendel Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions 2001 Prentice Hall Upper Saddle River
74. J.M. Mendel Advances in type-2 fuzzy sets and systems Information Sciences 177 1 2007 84 110 (Pubitemid 44566500)
75. J.M. Mendel On answering the question where do i start in order to solve a new problem involving interval type-2 fuzzy sets? Information Sciences 179 19 2009 3418 3431
76. O. Mendoza, P. Melín, and O. Castillo Interval type-2 fuzzy logic and modular neural networks for face recognition applications Applied Soft Computing 9 4 2009 1377 1387
77. A. Mesiarová Generators of triangular norms E.P. Klement, R. Mesiar, Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms 2005 Elsevier Amsterdam 95 111
78. A. Mesiarová H-transformation of t-norms Information Sciences 176 11 2006 1531 1545
79. A. Mesiarová-Zemánková Ranks of additive generators Fuzzy Sets and Systems 160 14 2009 2032 2048
80. S. Miller, and R. John An interval type-2 fuzzy multiple echelon supply chain model Knowledge-Based Systems 23 4 2010 363 368
81. S. Mitra, and S.K. Pal Fuzzy sets in pattern recognition and machine intelligence Fuzzy Sets and Systems 156 3 2005 381 386 (Pubitemid 41475496)
82. E.P. Monreal, R. Mesiar, E. Pap, Additive generators of t-norms which are not necessarily continuous, in: Proceedings of the Fourth European Congress on Intelligent Techniques and Soft Computing - EUFIT'96, vol. 1, ELITE-Foundation, Aachen, 1996, pp. 60-73.
83. R.E. Moore Methods and Applications of Interval Analysis 1979 SIAM Philadelphia
84. R.E. Moore, R.B. Kearfott, and M.J. Cloud Introduction to Interval Analysis 2009 SIAM Philadelphia
85. R.E. Moore, and W. Lodwick Interval analysis and fuzzy set theory Fuzzy Sets and Systems 135 1 2003 5 9
86. P.S. Mostert, and A.L. Shields On the structure of semigroups on a compact manifold with boundary Annals of Mathematics 65 1 1957 117 143
87. H.T. Nguyen, V. Kreinovich, and Q. Zuo Interval-valued degrees of belief: applications of interval computations to expert systems and intelligent control International Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems 5 3 1997 317 358 (Pubitemid 127530908)
88. H.T. Nguyen, and E.A. Walker A First Course in Fuzzy Logic 2006 Chapman & Hall, CRC Boca Raton
89. Y. Ouyang On the construction of boundary weak triangular norms through additive generators Nonlinear Analysis 66 1 2007 125 130 (Pubitemid 44692981)
90. Y. Ouyang, J. Fang, and Z. Zhao A generalization of additive generator of triangular norms International Journal of Approximate Reasoning 49 2 2008 417 421
91. R.H.S. Reiser, B.C. Bedregal, R.N. Santiago, and G.P. Dimuro Analyzing the relationship between interval-valued D-implications and interval-valued QL-implications TEMA - Tendências em Matemática Aplicada e Computacional 11 1 2010 89 100 (Available in English at )
92. R.H.S. Reiser, G.P. Dimuro, B.C. Bedregal, and R.H.N. Santiago Interval valued QL-implications D. Leivant, R. Queiroz, Proceedings of the 14th International Workshop on Logic, Language, Information and Computation, WoLLIC 2007, Rio de Janeiro LNCS vol. 4576 2007 Springer Berlin 307 321
93. 2 and their distributivity over uninorms Fuzzy Sets and Systems 160 6 2009 832 852
94. E. Sainio, E. Turunen, and R. Mesiar A characterization of fuzzy implications generated by generalized quantifiers Fuzzy Sets and Systems 159 4 2008 491 499 (Pubitemid 350298582)
95. R. Sambuc, Fonctions φ-floues. Application l'aide au diagnostic en pathologie thyroidienne, Ph.D. Thesis, Université Marseille, Marseille, 1975.
96. S. Saminger-Platz The dominance relation in some families of continuous archimedean t-norms and copulas Fuzzy Sets and Systems 160 14 2009 2017 2031
97. R.H.N. Santiago, B.C. Bedregal, and B.M. Acióly Formal aspects of correctness and optimality in interval computations Formal Aspects of Computing 18 2 2006 231 243 (Pubitemid 43787486)
98. B. Schweizer, and A. Sklar Statistical metric spaces Pacific Journal of Mathematics 10 1 1960 313 334
99. B. Schweizer, and A. Sklar Associative functions and statistical triangle inequalities Publicationes Mathematicae Debrecen 8 1961 168 186
100. B. Schweizer, and A. Sklar Associative functions and abstract semigroups Publicationes Mathematicae Debrecen 10 1963 69 81
101. B. Schweizer, and A. Sklar Probabilistic Metric Spaces 1983 North-Holland New York
102. Y. Shi, B.V. Gasse, D. Ruan, and E.E. Kerre On the first place antitonicity in QL-implications Fuzzy Sets and Systems 159 22 2008 2988 3013
103. J.T. Starczewski Efficient triangular type-2 fuzzy logic systems International Journal of Approximate Reasoning 50 5 2009 799 811
104. J.T. Starczewski Extended triangular norms Information Sciences 179 6 2009 742 757
105. H.R. Tizhoosh Image thresholding using type II fuzzy sets Pattern Recognition 38 12 2005 2363 2372 (Pubitemid 41336705)
106. I.B. Turksen Interval valued fuzzy sets based on normal forms Fuzzy Sets and Systems 20 2 1986 191 210
107. P. Viceník Additive generators of non-continuous triangular norms S.E. Rodabaugh, E.P. Klement, Topological an Algebraic Structures in Fuzzy Sets 2003 Kluwer Dordrecht 441 454
108. P. Viceník Additive generators of associative functions Fuzzy Sets and Systems 153 2 2005 137 160 (Pubitemid 40644710)
109. P. Viceník Additive generators of border-continuous triangular norms Fuzzy Sets and Systems 159 13 2008 1631 1645
110. P. Viceník Intersections of ranges of additive generators of associative functions Tatra Mountains Mathematical Publications 40 2008 117 131
111. P. Viceník On a class of generated triangular norms and their isomorphisms Fuzzy Sets and Systems 161 10 2010 1448 1458
112. Y.-M. Wang, J.-B. Yang, D.-L. Xu, and K.-S. Chin On the combination and normalization of interval-valued belief structures Information Sciences 177 5 2007 1230 1247 (Pubitemid 44908286)
113. D. Wu, and J.M. Mendel Uncertainty measures for interval type-2 fuzzy sets Information Sciences 177 23 2007 5378 5393 (Pubitemid 47409992)
114. D. Wu, and J.M. Mendel A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets Information Sciences 179 8 2009 1169 1192
115. R.R. Yager Level sets and the extension principle for interval valued fuzzy sets and its application to uncertainty measures Information Sciences 178 18 2008 3565 3576
116. S. Yamak, O. Kazanci, and B. Dawaz Applications of interval valued t-norms (t-conorms) to fuzzy n-ary sub-hypergroups Information Sciences 178 20 2008 3957 3972
117. X. Yang, T.Y. Lin, J. Yang, Y. Li, and D. Yu Combination of interval-valued fuzzy set and soft set Computers & Mathematics with Applications 58 3 2009 521 527
118. F. You, H. Ying, Interval type-2 boolean fuzzy systems are universal approximators, in: Fuzzy Information Processing Society (NAFIPS), 2010 Annual Meeting of the North American, 2010, pp. 1-4.
119. L.A. Zadeh Fuzzy sets Information and Control 8 3 1965 338 353
120. L.A. Zadeh The concept of a linguistic variable and its application to approximate reasoning - I Information Sciences 8 3 1975 199 249
121. L.A. Zadeh Toward a generalized theory of uncertainty (GTU) - an outline Information Sciences 172 1-2 2005 1 40 (Pubitemid 40612866)
122. L.A. Zadeh Is there a need for fuzzy logic? Information Sciences 178 13 2008 2751 2779
123. M.H.F. Zarandi, M. Zarinbal, and M. Izadi Systematic image processing for diagnosing brain tumors: a type-II fuzzy expert system approach Applied Soft Computing 11 1 2011 285 294
124. W. Zeng, and P. Guo Normalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship Information Sciences 178 5 2008 1334 1342 (Pubitemid 350266638)
125. Q. Zuo, Description of strictly monotonic interval and/or operations, in: Proceedings of the International Workshop on Applications of Interval Computations, APIC'95, UTEP, El Paso, 1995, pp. 232-235.