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Program for generating fuzzy logical operations and its use in mathematical proofs. (English) Zbl 1265.28041
Summary: Fuzzy logic is one of the tools for management of uncertainty; it works with more than two values, usually with a continuous scale, the real interval $$[0,1]$$. Implementation restrictions in applications force us to use in fact a finite scale (finite chain) of truth degrees. In this paper, we study logical operations on finite chains, in particular conjunctions. We describe a computer program generating all finitely-valued fuzzy conjunctions ($$t$$-norms). It allows also to select these $$t$$-norms according to various criteria. Using this program, we formulated several conjectures which we verified by theoretical proofs, thus obtaining new mathematical theorems. We found out several properties of $$t$$-norms that are quite surprising. As a consequence, we give arguments why there is no “satisfactory” finitely-valued conjunction. Such an operation is desirable, e. g., for search in large databases. We present an example demonstrating both the motivation and the difficulties encountered in using many-valued conjunctions. As a by-product, we found some consequences showing that the characterization of diagonals of finitely-valued conjunctions differs substantially from that obtained for $$t$$-norms on $$[0,1]$$.

##### MSC:
 2.8e+11 Fuzzy measure theory 3e+72 Theory of fuzzy sets, etc.
##### Keywords:
$$t$$-norm; finitely-valued conjunction
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##### References:
 [1] Bartušek T.: Fuzzy Operations on Finite Sets of Truth Values (in Czech). Diploma Thesis, Czech Technical University, Praha 2001 [2] Baets B. De, Mesiar R.: Triangular norms on product lattices. Fuzzy Sets and Systems 104 (1999), 61-75 · Zbl 0935.03060 [3] Baets B. De, Mesiar R.: Discrete triangular norms. Topological and Algebraic Structures (E. P. Klement and S. Rodabaugh, Universität Linz 1999, pp. 6-10 [4] Baets B. De, Mesiar R.: Discrete triangular norms. Topological and Algebraic Structures in Fuzzy Sets: Recent Developments in the Mathematics of Fuzzy Sets (S. Rodabaugh and E. P. Klement. Kluwer Academic Publishers, 2002, to appear · Zbl 1037.03046 [5] Drossos C., Navara M.: Matrix composition of t-norms. Enriched Lattice Structures for Many-Valued and Fuzzy Logics (S. Gottwald and E. P. Klement, Univ. Linz 1997, pp. 95-100 [6] Godo L., Sierra C.: A new approach to connective generation in the framework of expert systems using fuzzy logic. Proc. 11th Internat. Symposium on Multiple-Valued Logic, IEEE Computer Society Press, Palma de Mallorca 1988, pp. 157-162 [7] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer, Dordrecht - Boston - London 2000 · Zbl 1087.20041 [8] Mayor G., Torrens J.: On a class of operators for expert systems. Internat. J. Intell. Syst. 8 (1993), 771-778 · Zbl 0785.68087 [9] Mamdani E. H., Assilian S.: An experiment in linguistic synthesis with a fuzzy logic controller. J. Man-Machine Stud. 7 (1975), 1-13 · Zbl 0301.68076 [10] Mesiar R., Navara M.: Diagonals of continuous triangular norms. Fuzzy Sets and Systems 104 (1999), 34-41 · Zbl 0972.03052 [11] Moser B., Navara M.: Conditionally firing rules extend the possibilities of fuzzy controllers. Proc. Internat. Conf. Computational Intelligence for Modelling, Control and Automation (M. Mohammadian, IOS Press, Amsterdam 1999, pp. 242-245 · Zbl 0988.93050 [12] Viceník P.: A note to a construction of t-norms based on pseudo-inverses of monotone functions. Fuzzy Sets and Systems 104 (1999), 15-18 · Zbl 0953.26009
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