3. Representing Vague Knowledge – Fuzzy Logic
Objectives of this class:
•
Extension of binary logic•
Vague knowledge, its representation•
Fuzzy inferenceFuzzy Sets
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Founded in 1965 by L. Zadeh.●
Very popular eighties/nineties of the last century.●
Membership values for (ordinary) sets: [0, 1].●
Membership values for fuzzy sets: [0,0 .. 1,0].●
Examples for fuzzy sets
Room temperature (cold, warm, hot)
Length of men (tall, short, dwarf)
Length of holiday (short, long)
(Your) Comprehension of the classes (well, less, zero)Membership Function µ
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M: a fuzzy set; x Є M.●
µ: M → [0,0 .. 1,0] (membership function).●
x → µ(x) (degree of membership).●
Operations
(µ1 U µ2) (x) := max {(µ1(x), µ2(x) }
(µ1 П µ2) (x) := min {(µ1(x), µ2(x) }
¬µ (x) := 1 - µ(x) (Complement)
(µ1 subset µ2) ::= For all x Є M: (µ1(x) <= µ2(x)) (Inclusion)Fuzzy Logic
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Also founded in 1965 by L. Zadeh.●
Variable “degrees of truth”.●
Values of Truth – binary (dual) logic: [true, false]●
Values of Truth – Fuzzy logic: [0,0 .. 1,0]●
Examples for fuzzy logic statements
Room temperature is “high”.
The bearing damage is “moderate”
Christmas holiday is “short”.
The learning efficiency of this class is “very high”.
Klaus’s account is unbalanced.Membership Function / Truth Values – Visualization I
State hot
cold
Room temperature
µ
cold heiß
chilly
perfect
warm hot
baking
Room temperature State
µ
100%
70%
0%
warm hot baking
0
Room temperature in Grade C
Membership Function / Truth Values – Visualization II
µ
50% chilly perfect
20 60%
cold
cold chilly perfect
50%
60%
20
Fuzzy Inference
