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Fuzzy relational equations with min-biimplication composition. (English) Zbl 1254.03101

Summary: We discuss fuzzy relational equations with min-biimplication composition where the biimplication is the biresiduation operation with respect to the Łukasiewicz t-norm. It is shown that determining whether a finite system of fuzzy relational equations with min-biimplication composition has a solution is NP-complete. Moreover, a system of such equations can be fully characterized by a system of integer linear inequalities and consequently its solution set can be expressed in the terms of the minimal solutions of this system of integer linear inequalities.

MSC:

03E72 Theory of fuzzy sets, etc.
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)

Software:

JBool
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References:

[1] Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds) (2009) Handbook of satisfiability. IOS Press, Amsterdam · Zbl 1183.68568
[2] Chandra A. K., Markowsky G. (1978) On the number of prime implicants. Discrete Mathematics 24: 7–11 · Zbl 0392.03038
[3] Crama Y., Hammer P. (2011) Boolean functions: Theory, algorithms, and applications. Cambridge University Press, Cambridge · Zbl 1237.06001
[4] De Baets B. (2000) Analytical solution methods for fuzzy relational equations. In: Dubois D., Prade H. (eds) Fundamentals of fuzzy sets, The handbooks of fuzzy sets series (Vol. 1). Kluwer, Dordrecht, pp 291–340 · Zbl 0970.03044
[5] Di Nola A., Pedrycz W., Sessa S. (1988) Fuzzy relation equations with equality and difference composition operations. Fuzzy Sets and Systems 25: 205–215 · Zbl 0645.04004
[6] Di Nola A., Pedrycz W., Sessa S. (1990) Modus ponens for fuzzy data realized via equations with equality operators. International Journal of Intelligent Systems 5: 1–14 · Zbl 0703.03005
[7] Di Nola A., Sessa S., Pedrycz W. (1990) On some finite fuzzy relation equations. Information Sciences 50: 93–109 · Zbl 0688.90002
[8] Di Nola A., Sessa S., Pedrycz W., Sanchez E. (1989) Fuzzy relation equations and their applications to knowledge engineering. Kluwer, Dordrecht · Zbl 0694.94025
[9] Johnson D. S., Yannakakis M., Papadimitriou C. H. (1988) On generating all maximal independent sets. Information Processing Letters 27: 119–123 · Zbl 0654.68086
[10] Klawonn F., Gebhardt J., Kruse R. (1995) Fuzzy control on the basis of equality relations with an example from idle speed control. IEEE Transactions on Fuzzy Systems 3: 336–349
[11] Klawonn F., Kruse R. (1993) Equality relations as a basis for fuzzy control. Fuzzy Sets and Systems 54: 147–156 · Zbl 0785.93059
[12] Klement E. P., Mesiar R., Pap E. (2000) Triangular norms. Kluwer, Dordrecht
[13] Li, P. (2009). Fuzzy relational equations: Resolution and optimization, Ph.D. Dissertation, North Carolina State University.
[14] Li P., Fang S. -C. (2008) On the resolution and optimization of a system of fuzzy relational equations with sup-T composition. Fuzzy Optimization and Decision Making 7: 169–214 · Zbl 1169.90493
[15] Li P., Fang S. -C. (2009) A survey on fuzzy relational equations, Part I: Classification and solvability. Fuzzy Optimization and Decision Making 8: 179–229 · Zbl 1180.03051
[16] Liu W. J., Wang X. P. (1989) Biimplication operator and its application on fuzzy relation equation. BUSEFAL 41: 31–40
[17] Moser B. (2006) On the T-transitivity of kernels. Fuzzy Sets and Systems 157: 1787–1796 · Zbl 1100.68095
[18] Moser B. (2006) On representing and generating kernels by fuzzy equivalence relations. Journal of Machine Learning Research 7: 2603–2620 · Zbl 1222.68269
[19] Palopoli L., Pirri F., Pizzuti C. (1999) Algorithms for selective enumeration of prime implicants. Artificial Intelligence 111: 41–72 · Zbl 0996.68181
[20] Peeva K., Kyosev Y. (2004) Fuzzy relational calculus: Theory, applications and software. World Scientific, New Jersey · Zbl 1083.03048
[21] Perfilieva I., Novák V. (2007) System of fuzzy relation equations as a continuous model of IF-THEN rules. Information Sciences 177: 3218–3227 · Zbl 1124.03029
[22] Recasens J. (2010) Indistinguishability operators: Modelling fuzzy equalities and fuzzy equivalence relations. Springer, Heidelberg · Zbl 1215.03065
[23] Sanchez E. (1976) Resolution of composite fuzzy relation equation. Information and Control 30: 38–48 · Zbl 0326.02048
[24] Sanchez E. (1977) Solutions in composite fuzzy relation equations: Application to medical diagnosis in Brouwerian logic. In: Gupta M. M., Saridis G. N., Gaines B. R. (eds) Fuzzy automata and decision processes. North-Holland, Amsterdam, pp 221–234
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