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Minimal solutions of generalized fuzzy relational equations: probabilistic algorithm based on greedy approach. (English) Zbl 1335.03047
Summary: The paper deals with generalized fuzzy relational equations that are defined within a recently introduced framework of sup-preserving aggregation structures. Generalized fuzzy relational equations subsume all previously studied types of fuzzy relational equations, namely those that are based on sup-t-norm and inf-residuum compositions. The paper contributes to previous studies of generalized fuzzy relational equations by presenting a method for constructing all minimal solutions and, consequently, for determining the whole solution set for any given generalized fuzzy relational equation that is solvable and for which every solution is bounded from below by a minimal solution. Moreover, in the paper we present a simple probabilistic algorithm for finding all minimal solutions.

##### MSC:
 03E72 Theory of fuzzy sets, etc. 68T37 Reasoning under uncertainty in the context of artificial intelligence
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##### References:
 [1] Bandler, W.; Kohout, L. J., Fuzzy relational products and fuzzy implication operators, (International Workshop of Fuzzy Reasoning Theory and Applications, (1978), Queen Mary College, University of London) [2] Bandler, W.; Kohout, L. J., Fuzzy relational products as a tool for analysis and synthesis of the behaviour of complex natural and artificial systems, (Fuzzy Sets: Theory and Applications to Policy Analysis and Information Systems, (1980), Plenum Press New York), 341-367 [3] Bandler, W.; Kohout, L. J., Semantics of implication operators and fuzzy relational products, Int. J. Man-Mach. Stud., 12, 89-116, (1980) · Zbl 0435.68042 [4] Bartl, E.; Belohlavek, R., Sup-t-norm and inf-residuum are a single type of relational equations, Int. J. Gen. Syst., 40, 599-609, (2011) · Zbl 1259.03065 [5] Bartl, E.; Belohlavek, R.; Vychodil, V., Bivalent and other solutions of fuzzy relational equations via linguistic hedges, Fuzzy Sets Syst., 187, 103-112, (2012) · Zbl 1258.03080 [6] Bartl, E.; Klir, G. J., Fuzzy relational equations in general framework, Int. J. Gen. Syst., 43, 1-18, (2014) · Zbl 1286.93004 [7] Belohlavek, R., Fuzzy relational systems: foundations and principles, (2002), Kluwer/Plenum New York · Zbl 1067.03059 [8] Belohlavek, R., Sup-t-norm and inf-residuum are one type of relational product: unifying framework and consequences, Fuzzy Sets Syst., 197, 45-58, (2012) · Zbl 1266.03056 [9] Belohlavek, R.; Vychodil, V., What is a fuzzy concept lattice?, (3rd Int. Conference on Concept Lattices and Their Applications, CLA 2005, (2005)), 34-45 [10] De Baets, B., Analytical solution methods for fuzzy relation equations, (Fundamentals of Fuzzy Sets, (2000), Kluwer Dordrecht), 291-340 · Zbl 0970.03044 [11] Di Nola, A.; Sanchez, E.; Pedrycz, W.; Sessa, S., Fuzzy relation equations and their applications to knowledge engineering, (1989), Kluwer Dordrecht · Zbl 0694.94025 [12] Galatos, N.; Jipsen, P.; Kowalski, T.; Ono, H., Residuated lattices: an algebraic glimpse at substructural logics, Stud. Logic Found. Math., vol. 151, (2007), Elsevier Amsterdam · Zbl 1171.03001 [13] Goguen, J., L-fuzzy sets, J. Math. Anal. Appl., 18, 145-174, (1967) · Zbl 0145.24404 [14] Gottwald, S., Fuzzy sets and fuzzy logic. foundations of applications - from a mathematical point of view, (1993), Verlag Vieweg Darmstadt · Zbl 0782.94025 [15] Gottwald, S., A treatise on many-valued logics, (2001), Research Studies Press Baldock, Hertfordshire (UK) · Zbl 1048.03002 [16] Hájek, P., Metamathematics of fuzzy logic, Trends Log., vol. 4, (1998), Kluwer Dordrecht · Zbl 0937.03030 [17] Hromkovic, J., Algorithmics for hard problems: introduction to combinatorial optimization, randomization, approximation, and heuristics, (2002), Springer [18] Imai, H.; Kikuchi, K.; Miyakoshi, M., Unattainable solutions of a fuzzy relation equation, Fuzzy Sets Syst., 99, 193-196, (1998) · Zbl 0938.03081 [19] Klement, E.; Mesiar, R.; Pap, E., Triangular norms, (2000), Kluwer Dordrecht · Zbl 0972.03002 [20] Klir, G. J.; Yuan, B., Fuzzy sets and fuzzy logic: theory and applications, (1995), Prentice Hall Upper Saddle River (NJ) · Zbl 0915.03001 [21] Kohout, L. J.; Bandler, W., Relational-product architectures for information processing, Inf. Sci., 37, 25-37, (1985) [22] Krajči, S., A generalized concept lattice, Log. J. IGPL, 13, 543-550, (2005) · Zbl 1088.06005 [23] Lin, J.-L.; Wu, Y.-K.; Guu, S.-M., On fuzzy relational equations and the covering problem, Inf. Sci., 181, 2951-2963, (2011) · Zbl 1231.03047 [24] Markovskii, A. V., On the relation between equations with MAX-product composition and the covering problem, Fuzzy Sets Syst., 153, 261-273, (2005) · Zbl 1073.03538 [25] Pedrycz, W., Some applicational aspects of fuzzy relational equations in system analysis, Int. J. Gen. Syst., 9, 125-132, (1983) · Zbl 0521.93005 [26] Pedrycz, W., Fuzzy relational equations: bridging theory, methodology and practice, Int. J. Gen. Syst., 29, 529-554, (2000) · Zbl 0965.03066 [27] Peeva, K., Resolution of fuzzy relational equations - method, algorithm and software with applications, Inf. Sci., 234, 44-63, (2013) · Zbl 1284.03249 [28] Peeva, K.; Kyosev, Y., Fuzzy relational calculus: theory, applications and software, (2004), World Scientific Singapore · Zbl 1083.03048 [29] Sanchez, E., Resolution of composite fuzzy relation equations, Inf. Control, 30, 38-48, (1976) · Zbl 0326.02048 [30] Shieh, B.-S., Infinite fuzzy relation equations with continuous t-norms, Inf. Sci., 178, 1961-1967, (2008) · Zbl 1135.03346 [31] Turunen, E., On generalized fuzzy relation equations: necessary and sufficient conditions for the existence of solutions, Acta Univ. Carol., Math. Phys., 28, 33-37, (1987) · Zbl 0661.04004 [32] Xiong, Q.-Q.; Wang, X.-P., Some properties of sup-MIN fuzzy relational equations on infinite domains, Fuzzy Sets Syst., 151, 393-402, (2005) · Zbl 1062.03053 [33] Xiong, Q.-Q.; Wang, X.-P., Some properties of infinite fuzzy relational equations with sup-inf composition, Inf. Sci., 252, 32-41, (2013) · Zbl 1321.03073 [34] Zadeh, L., Fuzzy sets, Inf. Control, 8, 338-353, (1965) · Zbl 0139.24606 [35] Zahariev, Z., (2010)
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