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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:
28E10 Fuzzy measure theory
03E72 Theory of fuzzy sets, etc.
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