| Title:
|
Program for generating fuzzy logical operations and its use in mathematical proofs (English) |
| Author:
|
Bartušek, Tomáš |
| Author:
|
Navara, Mirko |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
38 |
| Issue:
|
3 |
| Year:
|
2002 |
| Pages:
|
[235]-244 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $[0,1]$. (English) |
| Keyword:
|
$t$-norm |
| Keyword:
|
finitely valued conjunction |
| MSC:
|
03B52 |
| MSC:
|
03E72 |
| MSC:
|
28E10 |
| MSC:
|
68T37 |
| idZBL:
|
Zbl 1265.28041 |
| idMR:
|
MR1944306 |
| . |
| Date available:
|
2009-09-24T19:45:30Z |
| Last updated:
|
2015-03-25 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135460 |
| . |
| Reference:
|
[1] Bartušek T.: Fuzzy Operations on Finite Sets of Truth Values (in Czech).Diploma Thesis, Czech Technical University, Praha 2001 |
| Reference:
|
[2] Baets B. De, Mesiar R.: Triangular norms on product lattices.Fuzzy Sets and Systems 104 (1999), 61–75 Zbl 0935.03060, MR 1685810, 10.1016/S0165-0114(98)00259-0 |
| Reference:
|
[3] Baets B. De, Mesiar R.: Discrete triangular norms.In: Topological and Algebraic Structures (E. P. Klement and S. Rodabaugh, eds.), Universität Linz 1999, pp. 6–10 |
| Reference:
|
[4] Baets B. De, Mesiar R.: Discrete triangular norms.In: Topological and Algebraic Structures in Fuzzy Sets: Recent Developments in the Mathematics of Fuzzy Sets (S. Rodabaugh and E. P. Klement, eds.). Kluwer Academic Publishers, 2002, to appear Zbl 1037.03046, MR 2046749 |
| Reference:
|
[5] Drossos C., Navara M.: Matrix composition of t-norms.In: Enriched Lattice Structures for Many-Valued and Fuzzy Logics (S. Gottwald and E. P. Klement, eds.), Univ. Linz 1997, pp. 95–100 |
| Reference:
|
[6] Godo L., Sierra C.: A new approach to connective generation in the framework of expert systems using fuzzy logic.In: Proc. 11th Internat. Symposium on Multiple–Valued Logic, IEEE Computer Society Press, Palma de Mallorca 1988, pp. 157–162 |
| Reference:
|
[7] Klement E. P., Mesiar, R., Pap E.: Triangular Norms.Kluwer, Dordrecht – Boston – London 2000 Zbl 1087.20041, MR 1790096 |
| Reference:
|
[8] Mayor G., Torrens J.: On a class of operators for expert systems.Internat. J. Intell. Syst. 8 (1993), 771–778 Zbl 0785.68087, 10.1002/int.4550080703 |
| Reference:
|
[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, 10.1016/S0020-7373(75)80002-2 |
| Reference:
|
[10] Mesiar R., Navara M.: Diagonals of continuous triangular norms.Fuzzy Sets and Systems 104 (1999), 34–41 Zbl 0972.03052, MR 1685807, 10.1016/S0165-0114(98)00256-5 |
| Reference:
|
[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 Zbl 0988.93050 |
| Reference:
|
[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, MR 1685804, 10.1016/S0165-0114(98)00253-X |
| . |
