Fuzzy Implications
Various fuzzy implications Each of these is a fuzzy logic generalization of the classical implication operation.
fimplication_minimal(x, y) fimplication_maximal(x, y) fimplication_kleene(x, y) fimplication_lukasiewicz(x, y) fimplication_reichenbach(x, y) fimplication_fodor(x, y) fimplication_goguen(x, y) fimplication_goedel(x, y) fimplication_rescher(x, y) fimplication_weber(x, y) fimplication_yager(x, y)
x |
numeric vector with elements in [0,1] |
y |
numeric vector of the same length as |
A function I: [0,1]\times [0,1]\to [0,1] is a fuzzy implication if for all x,y,x',y'\in [0,1] it holds: (a) if x≤ x', then I(x, y)≥ I(x', y); (b) if y≤ y', then I(x, y)≤ I(x, y'); (c) I(1, 1)=1; (d) I(0, 0)=1; (e) I(1, 0)=0.
The minimal fuzzy implication is given by I_0(x, y)=1 iff x=0 or y=1, and 0 otherwise.
The maximal fuzzy implication is given by I_1(x, y)=0 iff x=1 and y=0, and 1 otherwise.
The Kleene-Dienes fuzzy implication is given by I_{KD}(x, y)=max(1-x, y).
The Lukasiewicz fuzzy implication is given by I_{L}(x, y)=min(1-x+y, 1).
The Reichenbach fuzzy implication is given by I_{RB}(x, y)=1-x+xy.
The Fodor fuzzy implication is given by I_F(x, y)=1 iff x≤ y, and max(1-x, y) otherwise.
The Goguen fuzzy implication is given by I_{GG}(x, y)=1 iff x≤ y, and y/x otherwise.
The Goedel fuzzy implication is given by I_{GD}(x, y)=1 iff x≤ y, and y otherwise.
The Rescher fuzzy implication is given by I_{RS}(x, y)=1 iff x≤ y, and 0 otherwise.
The Weber fuzzy implication is given by I_{W}(x, y)=1 iff x<1, and y otherwise.
The Yager fuzzy implication is given by I_{Y}(x, y)=1 iff x=0 and y=0, and y^x otherwise.
Numeric vector of the same length as x and y.
The ith element of the resulting vector gives the result
of calculating I(x[i], y[i]).
Klir G.J, Yuan B., Fuzzy sets and fuzzy logic. Theory and applications, Prentice Hall PTR, New Jersey, 1995.
Gagolewski M., Data Fusion: Theory, Methods, and Applications, Institute of Computer Science, Polish Academy of Sciences, 2015, 290 pp. isbn:978-83-63159-20-7
Other fuzzy_logic: fnegation_yager,
tconorm_minimum,
tnorm_minimum
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