Geodesic
Program for generating fuzzy logical operations and its use in mathematical proofs
Kybernetika, Tome 38 (2002) no. 3, p. [235]

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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 $[0,1]$.
Classification : 03B52, 03E72, 28E10, 68T37
Keywords: $t$-norm; finitely valued conjunction 
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     author = {Bartu\v{s}ek, Tom\'a\v{s} and Navara, Mirko},
     title = {Program for generating fuzzy logical operations and its use in mathematical proofs},
     journal = {Kybernetika},
     pages = {[235]},
     publisher = {mathdoc},
     volume = {38},
     number = {3},
     year = {2002},
     mrnumber = {1944306},
     zbl = {1265.28041},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2002__38_3_a1/}
}
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Bartušek, Tomáš; Navara, Mirko. Program for generating fuzzy logical operations and its use in mathematical proofs. Kybernetika, Tome 38 (2002) no. 3, p. [235]. http://geodesic.mathdoc.fr/item/KYB_2002__38_3_a1/