Garmendia, L. and Salvador , A and Montero de Juan, Francisco Javier
(2009)
*Computing a T-transitive lower approximation or opening of a proximity relation.*
Fuzzy Sets and Systems, 160
(14).
pp. 2097-2015.
ISSN 0165-0114

PDF
Restricted to Repository staff only until 2020. 138kB |

Official URL: http://www.sciencedirect.com/science/journal/01650114

## Abstract

Since transitivity is quite often violated even by decision makers that accept transitivity in their preferences as a condition for consistency, a standard approach to deal with intransitive preference elicitations is the search for a close enough transitive preference relation, assuming that such a violation is mainly due to decision maker estimation

errors. In some way, the higher the number of elicitations, the more probable is inconsistency. This is mostly the case within a fuzzy framework, even when the number of alternatives or objects to be classified is relatively small. In this paper, we propose a fast method to compute a T-indistinguishability from a reflexive and symmetric fuzzy

relation, T being any left-continuous t-norm. The computed approximation we propose will have O(n3) time complexity, where n is the number of elements under consideration,

and is expected to produce a T-transitive opening. To the authors’ knowledge, there is no other proposed algorithm that computes T-transitive lower approximations or openings while preserving the reflexivity and symmetry properties.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Fuzzy relation; Fuzzy proximity relation; T-transitive relation; fuzzy similarity; T-indistinguishability; T-transitive lower approximation; T-transitive opening |

Subjects: | Sciences > Mathematics > Logic, Symbolic and mathematical |

ID Code: | 16156 |

References: | C. Alsina, E. Trillas, L. Valverde, On some logical connectives for fuzzy set theory, J. Math. Ann. Appl. 93 (1983) 15–26. W. Bandler, J. Kohout, Special properties, closures and interiors of crisp and fuzzy relations, Fuzzy Sets and Systems 26 (1988) 317–331. D. Boixader, On the relationship between T-transitivity and approximate equality, Fuzzy Sets and Systems 33 (2003) 6–69. V. Cutello, J. Montero, Fuzzy rationality measures, Fuzzy Sets and Systems 62 (1994) 39–44. P. Dawyndt, H. De Meyer, B. De Baets, The complete linkage clustering algorithm revisited, Soft Comput. 9 (2005) 385–392. P. Dawyndt, H. De Meyer, B. De Baets, UPGMA clustering revisited: a weight-driven approach to transitive approximation, Internat. J. Approx. Reason. 42 (2006) 174–191. B. De Baets, H. De Meyer, Transitive approximation of fuzzy relations by alternating closures and openings, Soft Comput. 7 (2003) 210–219. B. De Baets, H. De Meyer, On the existence and construction of T-transitive closures, Inform. Sci. 152 (2003) 167–179. A. Di Nola, W. Kolodziejczyk, S. Sessa, Transitive Solutions of Relational Equations on Finite Sets and Linear Lattices, Lecture Notes in Computer Science, Vol. 521, Springer, Berlin, 1991, pp. 173–182. D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980. J. Elorza, P. Burillo, On the relation of fuzzy preorders and fuzzy consequence operators, Internat. J. Uncertainty Fuzziness Knowledge-based Systems 7(3) (1999). F. Esteva, P. Garcia, L. Godo, R.O. Rodriguez, Fuzzy approximation relations, modal structures and possibilistic logic, Mathware Soft Comput. 5 (2–3) (1998) 151–166. J. Fodor, M. Roubens, Structure of transitive valued binary relations, Math. Soc. Sci. 30 (1995) 71–94. L. Garmendia, C. Campo, S. Cubillo, A. Salvador, A method to make some fuzzy relations T-transitive, Internat. J. Intelligence Systems 14 (9) (1999) 873–882. L. Garmendia, A. Salvador, On a new method to T-transitivize fuzzy relations, in: B. Bouchon-Meunier, J. Gutierrez-Rios, L. Magdalena, R.R. Yager (Eds.), Technologies for Constructing Intelligent Systems, Vol. 2, Springer, Berlin, 2000, pp. 251–260. L. Garmendia, A. Salvador, A New Algorithm to Compute Low T-transitive Approximation of a Fuzzy Relation Preserving Symmetry. Comparisons with theT-transitive Closure, in: Lecture Notes in Artificial Intelligence, European Conf. on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, Springer, Berlin, 2005, pp. 576–586. L. Garmendia, A. Salvador, Computing a Transitive Opening of a Reflexive and Symmetric Fuzzy Relation, in: Lecture Notes in Artificial Intelligence, European Conf. on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, Springer, Berlin, 2005, pp. 587–599. H. Hashimoto, Transitivity of generalised fuzzy matrices, Fuzzy Sets and Systems 17 (1) (1985) 83–90. J. Jacas, On the generators of T-indistinguishability operators, Stochastica 12 (1988) 49–63. J. Jacas, Similarity relations. The calculation of minimal generating families, Fuzzy Sets and Systems 35 (1990) 151–162. J. Jacas, J. Recasens, Fuzzy T-transitive relations: eigenvectors and generators, Fuzzy Sets and Systems 72 (1995) 147–154. J. Jacas, J. Recasens, Decomposable indistinguishability operators, in: Proc. Sixth IFSA Congr., Sao Paulo, 1995. G.J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic. Theory and Applications, Prentice-Hall, New Jersey, 1995. B. Leclerc, Caractérisation construction et dénombrement des ultramétriques supérieures minimales, Statist. Anal. Donées 11 (1986) 26–50. H.-S. Lee, An optimal algorithm for computing the max–min transitive closure of a fuzzy similarity matrix, Fuzzy Sets and Systems 123 (2001) 129–136. J. Montero, J. Tejada, Some problems on the definition of fuzzy preference relation, Fuzzy Sets and Systems 20 (1986) 45–53. J. Montero, Arrow’s theorem under fuzzy rationality, Behavioral Sci. 32 (1987) 267–273 (now Systems Research and Behavioral Science). J. Montero, D. Gómez, J. Yánez, J. Gonález-Pachón, Rationality cores in preference representation, in: Proc. Internat. Fuzzy Systems Assoc. Conf., Bogaziai University, Istanbul, 2003, pp. 340–343. H. Naessens, H. De Meyer, B. De Baets, Algorithms for the computation of T-transitive closures, IEEE Trans. Fuzzy Systems 10 (4) (2002) 541–551. S. Ovchinnikov, Representations of transitive fuzzy relations, in: H.J. Skala, S. Termini, E. Trillas (Eds.), Aspects of Vagueness, Reidel Publishers, Dordrecht, 1984, pp. 105–118. H.B. Potoczny, On similarity relations in fuzzy relational databases, Fuzzy Sets and Systems 12 (3) (1984) 231–235. R.O. Rodriguez, F. Esteva, P. Garcia, L. Godo, On implicative closure operators in approximate reasoning, Internat. J. Approx. Reason. 33 (2003) 159–184. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, New York, 1983. E. Trillas, L. Valverde, An inquiry into indistinguishability operators, in: H.J. Skala, S. Termini y, E. Trillas (Eds.), Aspects of Vagueness, Reidel Publishers, Dordrecht, 1984, pp. 231–256. L. Valverde, On the structure of F-indistinguishability operators, Fuzzy Sets and Systems 17 (1985) 313–328. M. Wagenknecht, On transitive solutions of fuzzy equations, inequalities and lower approximation of fuzzy relations, Fuzzy Sets and Systems 75 (1995) 229–240. M.Wallace, Y. Avrithis, S. Kollias, Computationally efficient sup-T transitive closure for sparse fuzzy binary relations, Fuzzy Sets and Systems 157 (3) (2006) 341–372. X. Xiao, An algorithm for calculating fuzzy transitive closure, Fuzzy Math. 5 (4) (1985) 71–73. L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338–353. [40] L.A. Zadeh, Similarity relations and fuzzy orderings, Inform. Sci. 3 (1971) 177–200. |

Deposited On: | 11 Sep 2012 08:21 |

Last Modified: | 06 Feb 2014 10:39 |

Repository Staff Only: item control page