# 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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topBartušek, 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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