Program for generating fuzzy logical operations and its use in mathematical proofs

Kybernetika (2002)

• Volume: 38, Issue: 3, page [235]-244
• ISSN: 0023-5954

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Abstract

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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 $\left[0,1\right]$. 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 $\left[0,1\right]$.

How to cite

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Bartušek, Tomáš, and Navara, Mirko. "Program for generating fuzzy logical operations and its use in mathematical proofs." Kybernetika 38.3 (2002): [235]-244. <http://eudml.org/doc/33579>.

@article{Bartušek2002,
abstract = {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 $[0,1]$.},
author = {Bartušek, Tomáš, Navara, Mirko},
journal = {Kybernetika},
keywords = {$t$-norm; finitely valued conjunction; -norm; finitely-valued conjunction},
language = {eng},
number = {3},
pages = {[235]-244},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Program for generating fuzzy logical operations and its use in mathematical proofs},
url = {http://eudml.org/doc/33579},
volume = {38},
year = {2002},
}

TY - JOUR
AU - Bartušek, Tomáš
AU - Navara, Mirko
TI - Program for generating fuzzy logical operations and its use in mathematical proofs
JO - Kybernetika
PY - 2002
PB - Institute of Information Theory and Automation AS CR
VL - 38
IS - 3
SP - [235]
EP - 244
AB - 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 $[0,1]$.
LA - eng
KW - $t$-norm; finitely valued conjunction; -norm; finitely-valued conjunction
UR - http://eudml.org/doc/33579
ER -

References

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9. Mamdani E. H., Assilian S., 10.1016/S0020-7373(75)80002-2, J. Man-Machine Stud. 7 (1975), 1–13 (1975) Zbl0301.68076DOI10.1016/S0020-7373(75)80002-2
10. Mesiar R., Navara M., 10.1016/S0165-0114(98)00256-5, Fuzzy Sets and Systems 104 (1999), 34–41 (1999) Zbl0972.03052MR1685807DOI10.1016/S0165-0114(98)00256-5
11. Moser B., Navara M., Conditionally firing rules extend the possibilities of fuzzy controllers, In: Proc. Internat. Conf. Computational Intelligence for Modelling, Control and Automation (M. Mohammadian, ed.), IOS Press, Amsterdam 1999, pp. 242–245 (1999) Zbl0988.93050
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