On the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations

Fuzzy Sets and Systems

Volumn 161, Issue 24, 2010, Pages 3081-3113

  Ignjatović, Jelena ^a   Ćirić, Miroslav ^a   Bogdanović, Stojan ^a  

^a UNIVERSITY OF NI   (Serbia)

Author keywords

Complete residuated lattices; Fuzzy equivalences; Fuzzy quasi orders; Fuzzy relation equations; Fuzzy relation inequalities; Fuzzy relations; Post fixed points; Regular fuzzy relations; Residuals of fuzzy relations

Indexed keywords

FIXED POINTS; FUZZY EQUIVALENCE; FUZZY QUASI-ORDERS; FUZZY RELATION EQUATIONS; FUZZY RELATIONS; RESIDUATED LATTICES;

ALGORITHMS; AUTOMATA THEORY; ELECTRIC NETWORK ANALYSIS; FUZZY SYSTEMS;

LINEAR SYSTEMS;

EID: 78049233670     PISSN: 01650114     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.fss.2010.08.002     Document Type: Article

References (107)

1. W. Bandler, and L.J. Kohout Fuzzy relational products as a tool for analysis and synthesis of the behaviour of complex natural and artificial systems S.K. Wang, P.P. Chang, Fuzzy Sets: Theory and Application to Policy Analysis and Information Systems 1980 Plenum Press New York 341 367
2. N.C. Basak, and A. Gupta On quotient machines of a fuzzy automaton and the minimal machine Fuzzy Sets and Systems 125 2002 223 229
3. R. Bělohlávek Fuzzy Relational Systems: Foundations and Principles 2002 Kluwer New York
4. R. Bělohlávek, and V. Vychodil Fuzzy Equational Logic 2005 Springer Berlin, Heidelberg
5. M. Bezema, R. Nieuwenhuis, and E. Rodriguez-Carbonell Exponential behaviour of the Butkovic-Zimmermann algorithm for solving two-sided linear systems in max-algebra Discrete Applied Mathematics 156 2008 3506 3509
6. G. Birkhoff Lattice Theory third ed. 1973 American Mathematical Society Providence, RI
7. T.S. Blyth Lattices and Ordered Algebraic Structures 2005 Springer London
8. U. Bodenhofer, B. De Baets, and J. Fodor A compendium of fuzzy weak orders: representations and constructions Fuzzy Sets and Systems 158 2007 811 829
9. S.P. Borgatti, and M.G. Everett The class of all regular equivalences: algebraic structure and computation Social Networks 11 1989 65 88
10. S.P. Borgatti, and M.G. Everett Regular blockmodels of multiway, multimode matrices Social Networks 14 1992 91 120
11. S.P. Borgatti, and M.G. Everett Two algorithms for computing regular equivalences Social Networks 15 1993 361 376
12. J. Boyd Social Semigroups 1991 George Mason University Press Fairfax, VA
13. J.P. Boyd, and M.G. Everett Relations, residuals, regular interiors, and relative regular equivalence Social Networks 21 1999 147 165
14. U. Brandes, T. Erlebach (Eds.), Network Analysis: Methodological Foundations, Lecture Notes in Computer Science, vol. 3418, Springer, 2005.
15. T. Brihaye Words and bisimulations of dynamical systems Discrete Mathematics and Theoretical Computer Science 9 2 2007 11 32
16. P. Buchholz Bisimulation relations for weighted automata Theoretical Computer Science 393 2008 109 123
17. P. Butkovic, and K. Zimmermann A strongly polynomial algorithm for solving two-sided linear systems in max-algebra Discrete Applied Mathematics 154 2006 437 446
18. C. Câmpeanu, N. Sântean, and S. Yu Mergible states in large NFA Theoretical Computer Science 330 2005 23 34
19. K. Cechlarova On max-min linear inequalities and coalitional resource games with sharable resources Linear Algebra and its Applications 433 2010 127 135
20. J.-M. Champarnaud, F. Coulon, NFA reduction algorithms by means of regular inequalities, in: Z. Ésik, Z. Fülöp (Eds.), DLT 2003, Lecture Notes in Computer Science, vol. 2710, 2003, pp. 194-205.
21. J.-M. Champarnaud, and F. Coulon NFA reduction algorithms by means of regular inequalities Theoretical Computer Science 327 2004 241 253
22. W. Cheng, and Z. Mo Minimization algorithm of fuzzy finite automata Fuzzy Sets and Systems 141 2004 439 448
23. M. Ćirić, M. Droste, J. Ignjatović, and H. Vogler Determinization of weighted finite automata over strong bimonoids Information Sciences 180 2010 3497 3520
24. M. Ćirić, J. Ignjatović, and S. Bogdanović Fuzzy equivalence relations and their equivalence classes Fuzzy Sets and Systems 158 2007 1295 1313
25. M. Ćirić, J. Ignjatović, and S. Bogdanović Uniform fuzzy relations and fuzzy functions Fuzzy Sets and Systems 160 2009 1054 1081
26. M. Ćirić, A. Stamenković, J. Ignjatović, T. Petković, Factorization of fuzzy automata, in: E. Csuhaj-Varju, Z. Ésik (Eds.), FCT 2007, Lecture Notes in Computer Science, vol. 4639, 2007, pp. 213-225.
27. M. Ćirić, A. Stamenković, J. Ignjatović, and T. Petković Fuzzy relation equations and reduction of fuzzy automata Journal of Computer and System Sciences 76 2010 609 633
28. R.A. Cuninghame-Green, and P. Butkovic The equation A ⊗ x = B ⊗ y over (max; +) Theoretical Computer Science 293 2003 3 12
29. G.B. Davis, and K.M. Carley Clearing the FOG: fuzzy, overlapping groups for social networks Social Networks 30 2008 201 212
30. B. De Baets Analytical solution methods for fuzzy relational equations D. Dubois, H. Prade, Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series vol. 1 2000 Kluwer Academic Publishers 291 340
31. A. Di Nola, E. Sanchez, W. Pedrycz, and S. Sessa Fuzzy Relation Equations and their Application to Knowledge Engineering 1989 Kluwer Academic Press Dordrecht
32. A. Dovier, C. Piazza, and A. Policriti An efficient algorithm for computing bisimulation equivalence Theoretical Computer Science 311 2004 221 256
33. D. Dubois, H. Prade (Eds.), Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series, vol. 1, Kluwer Academic Publishers, 2000.
34. M.G. Everett, and S.P. Borgatti Regular equivalences: general theory Journal of Mathematical Sociology 18 1994 29 52
35. T.F. Fan, C.J. Liau, T.Y. Lin, Positional analysis in fuzzy social networks, in: Proceedings of the Third IEEE International Conference on Granular Computing, 2007, pp. 423-428.
36. T.F. Fan, C.J. Liau, and T.S. Lin A theoretical investigation of regular equivalences for fuzzy graphs International Journal of Approximate Reasoning 49 2008 678 688
37. R. Gentilini, C. Piazza, and A. Policriti From bisimulation to simulation: coarsest partition problems Journal of Automated Reasoning 31 2003 73 103
38. J.A. Goguen L-fuzzy sets Journal of Mathematical Analysis and Applications 18 1967 145 174
39. R.A. Hanneman, and M. Riddle Introduction to Social Network Methods 2005 University of California Riverside
40. T.A. Henzinger, P.W. Kopke, A. Puri, and P. Varaiya What's decidable about hybrid automata? Journal of Computer and System Sciences 57 1998 94 124
41. J. Högberg, A. Maletti, and J. May Backward and forward bisimulation minimisation of tree automata J. Holub, J. Ždárek, IAA07, Lecture Notes in Computer Science vol. 4783 2007 Springer Heidelberg 109 121
42. J. Högberg, A. Maletti, and J. May Backward and forward bisimulation minimisation of tree automata Theoretical Computer Science 410 2009 3539 3552
43. U. Höhle Commutative, residuated ℓ -monoids U. Höhle, E.P. Klement, Non-Classical Logics and Their Applications to Fuzzy Subsets 1995 Kluwer Academic Publishers Boston, Dordrecht 53 106
44. J. Ignjatović, and M. Ćirić Formal power series and regular operations on fuzzy languages Information Sciences 180 2010 1104 1120
45. J. Ignjatović, M. Ćirić, and S. Bogdanović Determinization of fuzzy automata with membership values in complete residuated lattices Information Sciences 178 2008 164 180
46. J. Ignjatović, M. Ćirić, and S. Bogdanović Fuzzy homomorphisms of algebras Fuzzy Sets and Systems 160 2009 2345 2365
47. J. Ignjatović, M. Ćirić, S. Bogdanović, and T. Petković Myhill-Nerode type theory for fuzzy languages and automata Fuzzy Sets and Systems 161 2010 1288 1324
48. L. Ilie, S. Yu, Algorithms for computing small NFAs, in: K. Diks, et al. (Eds.), MFCS 2002, Lecture Notes in Computer Science, vol. 2420, 2002, pp. 328-340.
49. L. Ilie, and S. Yu Reducing NFAs by invariant equivalences Theoretical Computer Science 306 2003 373 390
50. L. Ilie, G. Navarro, S. Yu, On NFA reductions, in: J. Karhumäki, et al. (Eds.), Theory is Forever, Lecture Notes in Computer Science, vol. 3113, 2004, pp. 112-124.
51. L. Ilie, R. Solis-Oba, S. Yu, Reducing the size of NFAs by using equivalences and preorders, in: A. Apostolico, M. Crochemore, K. Park (Eds.), CPM 2005, Lecture Notes in Computer Science, vol. 3537, 2005, pp. 310-321.
52. P.C. Kannellakis, and S.A. Smolka CCS expressions, finite state processes, and three problems of equivalence Information and Computation 86 1990 43 68
53. F. Klawonn Fuzzy points, fuzzy relations and fuzzy functions V. Novák, I. Perfilieva, Discovering the World with Fuzzy Logic, Advances in Soft Computing 2000 Physica-Verlag Heidelberg 431 453
54. G.J. Klir, and B. Yuan Fuzzy Sets and Fuzzy Logic, Theory and Application 1995 Prentice-Hall Englewood Cliffs, NJ
55. S. Konstantinidis, N. Santean, and S. Yu Fuzzification of rational and recognizable sets Fundamenta Informaticae 76 2007 413 447
56. H. Lei, and Y.M. Li Minimization of states in automata theory based on finite lattice-ordered monoids Information Sciences 177 2007 1413 1421
57. J. Lerner Role assignments U. Brandes, T. Erlebach, Network Analysis: Methodological Foundations, Lecture Notes in Computer Science vol. 3418 2005 Springer 216 252
58. P. Li, and S.-C. Fang A survey on fuzzy relational equations, part I: classification and solvability Fuzzy Optimization and Decision Making 8 2009 179 229
59. T.-J. Li, Y. Leung, and W.-X. Zhang Generalized fuzzy rough approximation operators based on fuzzy coverings International Journal of Approximate Reasoning 48 2008 836 856
60. Y.M. Li, and W. Pedrycz Fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids Fuzzy Sets and Systems 156 2005 68 92
61. F. Lorrain, and H.C. White Structural equivalence of individuals in social networks Journal of Mathematical Sociology 1 1971 49 80
62. N. Lynch, and F. Vaandrager Forward and backward simulations: Part I. Untimed systems Information and Computation 121 1995 214 233
63. D.S. Malik, J.N. Mordeson, and M.K. Sen Minimization of fuzzy finite automata Information Sciences 113 1999 323 330
64. M. Marx, and M. Masuch Regular equivalence and dynamic logic Social Networks 25 2003 51 65
65. P. Mika Social Networks and the Semantic Web 2007 Springer
66. R. Milner A calculus of communicating systems G. Goos, J. Hartmanis, Lecture Notes in Computer Science vol. 92 1980 Springer
67. R. Milner Communication and Concurrency 1989 Prentice-Hall International
68. R. Milner Communicating and Mobile Systems: The π -Calculus 1999 Cambridge University Press Cambridge
69. J.N. Mordeson, and D.S. Malik Fuzzy Automata and Languages: Theory and Applications 2002 Chapman & Hall, CRC Boca Raton, London
70. P.S. Nair, S. Sarasamma, Data mining through fuzzy social network analysis, in: Proceedings of the 26th Annual Meeting of the North American Fuzzy Information Processing Society, 2007, pp. 251-255.
71. M.E.J. Newman Analysis of weighted networks Physical Review E 70 2004 056131:1 9
72. R. Paige, and R.E. Tarjan Three partition refinement algorithms SIAM Journal of Computing 16 1987 973 989
73. D. Park Concurrency and automata on infinite sequences P. Deussen, Proceedings of the 5th GI Conference, Lecture Notes in Computer Science vol. 104 1981 Springer-Verlag Karlsruhe, Germany 167 183
74. P. Pattison The analysis of semigroups of multirelational systems Journal of Mathematical Psychology 25 1982 87 117
75. P.E. Pattison Algebraic Models for Social Networks 1993 Cambridge University Press Cambridge
76. W. Pedrycz, and F. Gomide Fuzzy Systems Engineering: Toward Human-Centric Computing 2007 Wiley-IEEE Press
77. K. Peeva Finite L-fuzzy acceptors regular L-fuzzy grammars and syntactic pattern recognition International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 12 2004 89 104
78. K. Peeva Finite L-fuzzy machines Fuzzy Sets and Systems 141 2004 415 437
79. K. Peeva, and Y. Kyosev Fuzzy relational calculus: theory, applications and software (with CD-ROM) Advances in Fuzzy Systems - Applications and Theory vol. 22 2004 World Scientific
80. K. Peeva, and Y. Kyosev Algorithm for solving max-product fuzzy relational equations Soft Computing 11 2007 593 605
81. K. Peeva, and Z. Zahariev Computing behavior of finite fuzzy machines - algorithm and its application to reduction and minimization Information Sciences 178 2008 4152 4165
82. I. Perfilieva Fuzzy function as an approximate solution to a system of fuzzy relation equations Fuzzy Sets and Systems 147 2004 363 383
83. I. Perfilieva, and S. Gottwald Fuzzy function as a solution to a system of fuzzy relation equations International Journal of General Systems 32 2003 361 372
84. I. Perfilieva, and V. Novák System of fuzzy relation equations as a continuous model of IF-THEN rules Information Sciences 177 2007 3218 3227
85. T. Petković Congruences and homomorphisms of fuzzy automata Fuzzy Sets and Systems 157 2006 444 458
86. D.-W. Qiu Automata theory based on completed residuated lattice-valued logic (I) Science in China, Series F 44 6 2001 419 429
87. D.-W. Qiu Automata theory based on completed residuated lattice-valued logic (II) Science in China, Series F 45 6 2002 442 452
88. D.-W. Qiu Pumping lemma in automata theory based on complete residuated lattice-valued logic: a note Fuzzy Sets and Systems 157 2006 2128 2138
89. X.-B. Qu, and X.-P. Wang Some properties of infinite fuzzy relational equations on complete Brouwerian lattices Fuzzy Sets and Systems 158 2007 1327 1339
90. F. Ranzato, and F. Tapparo Generalizing the Paige-Tarjan algorithm by abstract interpretation Information and Computation 206 2008 620 651
91. M. Roggenbach, and M. Majster-Cederbaum Towards a unified view of bisimulation: a comparative study Theoretical Computer Science 238 2000 81 130
92. S. Roman Lattices and Ordered Sets 2008 Springer New York
93. E. Sanchez, Equations de relations floues, Thse de Doctorat, Faculté de Médecine de Marseille, 1974.
94. E. Sanchez Resolution of composite fuzzy relation equations Information and Control 30 1976 38 48
95. E. Sanchez Solutions in composite fuzzy relation equations: application to medical diagnosis in Brouwerian logic M.M. Gupta, G.N. Saridis, B.R. Gaines, Fuzzy Automata and Decision Processes 1977 North-Holland Amsterdam 221 234
96. E. Sanchez Resolution of eigen fuzzy sets equations Fuzzy Sets and Systems 1 1978 69 74
97. D. Sangiorgi On the origins of bisimulation and coinduction ACM Transactions on Programming Languages and Systems 31 4 2009 111 151
98. A. Stamenković, M. Ćirić, J. Ignjatović, Reduction of fuzzy automata by means of fuzzy quasi-orders, Information Sciences, submitted for publication.
99. H.G. Xing, and D.W. Qiu Pumping lemma in context-free grammar theory based on complete residuated lattice-valued logic Fuzzy Sets and Systems 160 2009 1141 1151
100. H.-G. Xing, and D.-W. Qiu Automata theory based on complete residuated lattice-valued logic: a categorical approach Fuzzy Sets and Systems 160 2009 2416 2428
101. H.-G. Xing, D.-W. Qiu, and F.-C. Liu Automata theory based on complete residuated lattice-valued logic: pushdown automata Fuzzy Sets and Systems 160 2009 1125 1140
102. H.-G. Xing, D.-W. Qiu, F.C. Liu, and Z.J. Fan Equivalence in automata theory based on complete residuated lattice-valued logic Fuzzy Sets and Systems 158 2007 1407 1422
103. X.-P. Wang, and C. Xia The solution sets of infinite fuzzy relational equations with sup-conjunctor composition on complete distributive lattices Fuzzy Sets and Systems 160 2009 2989 3006
104. S. Wasserman, and K. Faust Social Network Analysis: Methods and Applications 1994 Cambridge University Press Cambridge
105. D.R. White, and K.P. Reitz Graph and semigroup homomorphisms on networks and relations Social Networks 5 1983 143 234
106. L.H. Wu, and D.W. Qiu Automata theory based on complete residuated lattice-valued logic: reduction and minimization Fuzzy Sets and Systems 161 2010 1635 1656
107. Y.-K. Wu, and S.-M. Guu An efficient procedure for solving a fuzzy relational equation with max-Archimedean t-norm composition IEEE Transactions on Fuzzy Systems 16 2008 73 84