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| Fuzzy
Logic |
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- Fuzzy logic is a superset of conventional (Boolean) logic
that has been
extended to handle the concept of partial truth -- truth values
between
"completely true" and "completely false".
It was introduced by Dr. Lotfi
Zadeh of UC/Berkeley in the 1960's as a means to model the
uncertainty
of natural language.
- It is basically a multi-valued logic that allows intermediate
values to be defined between conventional boolean logic like
true/false, yes/no, black/white.
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| Fuzzy
set theory |
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| Just as there is a strong relationship between Boolean logic and
the
concept of a subset, there is a similar strong relationship between
fuzzy
logic and fuzzy subset theory.
In classical set theory, a subset U of a set S can be defined
as a
mapping from the elements of S to the elements of the set {0,
1},
U: S --> {0, 1}
This mapping may be represented as a set of ordered pairs, with
exactly
one ordered pair present for each element of S. The first element
of the
ordered pair is an element of the set S, and the second element
is an
element of the set {0, 1}. The value zero is used to represent
non-membership, and the value one is used to represent membership.
The
truth or falsity of the statement
x is in U
is determined by finding the ordered pair whose first element
is x. The
statement is true if the second element of the ordered pair is
1, and the
statement is false if it is 0.
Similarly, a fuzzy subset F of a set S can be defined as a set
of ordered
pairs, each with the first element from S, and the second element
from
the interval [0,1], with exactly one ordered pair present for
each
element of S. This defines a mapping between elements of the set
S and
values in the interval [0,1]. The value zero is used to represent
complete non-membership, the value one is used to represent complete
membership, and values in between are used to represent intermediate
DEGREES OF MEMBERSHIP. The set S is referred to as the UNIVERSE
OF
DISCOURSE for the fuzzy subset F. Frequently, the mapping is described
as a function, the MEMBERSHIP FUNCTION of F. The degree to which
the
statement
x is in F
is true is determined by finding the ordered pair whose first
element is
x. The DEGREE OF TRUTH of the statement is the second element
of the
ordered pair.
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| References |
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| [1] |
Shyue-Liang Wang, Jenn-Shing Tsai and Tzung-Pei
Hong, Mining Functional Dependencies from Fuzzy Relational
Databases, ACM SAC 2000, Como, Italy, March 2000. |
| [2] |
B.P. Buckles and F.E. Petry, “A fuzzy representation
of data for relational databases”, Fuzzy Sets and Systems,
7, 1982, 213-226. |
| [3] |
Chan Man Kuok, Ada Fu and Man Hon Wong, “Mining Fuzzy
Association Rules in Databases”, SIGMOD vol.27 I, March
1998. 41-46. |
| [4] |
M. Anvari, and G.F. Rose. “Fuzzy relational databases”,
in Bezdek, Ed., Analysis of Fuzzy Information. Vol II (CRC
Press, Boca Raton, FL, 1987). |
| [5] |
J.M. Medina, M.A. Vila, J.C. Cubero, and O. Pons, “
Towards the implementation of a generalized fuzzy relational
database model”, Fuzzy Sets and Systems, 75. 1995. 273-289.
|
| [6] |
Tzung-Pei Hong, Chan-Sheng Kuo, Sheng-Chai Chi and Shyue-Liang
Wang, “Mining Fuzzy Rules from Quantitative Data Based
on the AprioriTid Algorithm”, ACM SAC 2000 Como, Italy.
March 2000.490-493. |
| [7] |
K. V. S. V. N. Raju and A. K. Majumdar: "Fuzzy Functional
Dependencies and Lossless Join Decomposition of Fuzzy Relational
Database Systems", ACM Transactions on Database Systems,
Vol. 13, NO. 2, June 1988, 129-166. |
| [8] |
L.A. Zadeh and J. Kacprzyk: "Fuzzy Logic for the Management
of Uncertainty". John Wiley & Sons Inc., 1992. |
| [9] |
von Altrock, "Fuzzy Logic and NeuroFuzzy Applications
Explained", ISBN 0-1336-8465-2, Prentice Hall 1995 |
| [10] |
von Altrock, C. and Krause, B., "On-Line-Development
Tools for Fuzzy Knowledge-Base Systems of Higher Order",
2nd Int'l Conference on Fuzzy Logic and Neural Networks Proceedings,
IIZUKA, Japan (1992), ISBN 4-938717-01-8. |
| [11] |
R. Belohlavek. Lattices of fixed points of fuzzy Galois
connections. Mathematical Logic Quarterly, to appear. |
| [12] |
R. Belohlavek. Lattices generated by binary fuzzy relations.
Tatra Mountains Mathematical Publications, 16, 11–19,
1999 (special issue on fuzzy sets). |
| [13] |
R. Belohlavek. Fuzzy Galois connections. Mathematical Logic
Quarterly, 45, 497–504, 1999. |
| [14] |
P. Hajek. Metamathematics of Fuzzy Logic. Kluwer, 1998. |
| [15] |
G. J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic. Theory
and Applications. Prentice Hall, 1995. |
| [16] |
V. Novak, I. Perfilieva and J. Mockor. Mathematical Principles
of Fuzzy Logic. Kluwer, 1999. |
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