Fuzzy Logic
CC-BY
Fabian M. Suchanek
62
A large knowledge base
https://yago-knowledge.org
KB-LM
Natural language processing
https://suchanek.name/...
AMIE
Mining rules in knowledge bases
https://github.com/dig-team/amie
Professor at Télécom Paris, France.
I work on several topics broadly related to AI:
• Natural Language Processing
• Data Integration
• Knowledge Bases
• Automated Reasoning
Fabian Suchanek
Flagship projects:
Hiring a PhD
student now!
Statements with weights
Amal Clooney is a human rights lawyer.
true statement
3
Statements with weights
Amal Clooney is a human rights lawyer.
true statement
Amal Clooney is rich. [70%]
true only to a certain degree:
not as poor as us, but not as rich as Elon Musk
4
wealth
How could we measure this?
Statements with weights
Amal Clooney is a human rights lawyer.
true statement
Amal Clooney is rich. [70%]
true only to a certain degree:
not as poor as us, but not as rich as Elon Musk
5
wealth
How could we measure this? Maybe the percentage of money of the richest
person? Or the percentage of people who have less money?
Statements with weights
Amal Clooney is a human rights lawyer.
true statement
Amal Clooney is rich. [70%]
true only to a certain degree:
not as poor as us, but not as rich as Elon Musk
Amal Clooney retires next year. [20%]
true with a certain probability
6
How could we estimate this?
Statements with weights
Amal Clooney is a human rights lawyer.
true statement
Amal Clooney is rich. [70%]
true only to a certain degree:
not as poor as us, but not as rich as Elon Musk
Amal Clooney retires next year. [20%]
true with a certain probability
7
How could we estimate this? Maybe the percentage
of lawyers of her age who retire next year?
Statements with weights
Amal Clooney is a human rights lawyer.
true statement
Amal Clooney is rich. [70%]
true only to a certain degree:
not as poor as us, but not as rich as Elon Musk
Amal Clooney retires next year. [20%]
true with a certain probability
If X is a lawyer, then X is rich. [60%]
not always true
8
What is the degree of truth?
Statements with weights
Amal Clooney is a human rights lawyer.
true statement
Amal Clooney is rich. [70%]
true only to a certain degree:
not as poor as us, but not as rich as Elon Musk
Amal Clooney retires next year. [20%]
true with a certain probability
If X is a lawyer, then X is rich. [60%]
not always true
9
What is the degree of truth? Maybe the percentage of lawyers who are rich?
Statements with weights
Amal Clooney is a human rights lawyer.
true statement
Amal Clooney is rich. [70%]
true only to a certain degree:
not as poor as us, but not as rich as Elon Musk
If X is a lawyer, then X is rich. [60%]
not always true
Amal Clooney retires next year. [20%]
true with a certain probability
If X is a lawyer, then X is rich to degree >40%. [60%]
weights on both the formula
and the individual statements
10
Statements with weights
Amal Clooney is a human rights lawyer.
Propositional Logic or First‐Order Logic
Amal Clooney is rich. [70%]
Fuzzy Logic
If X is a lawyer, then X is rich. [60%]
Weighted Logics
Amal Clooney retires next year. [20%]
Probabilistic Logic
If X is a lawyer, then X is rich to degree >40%. [60%]
Probabilistic Soft Logic
->Markov-logic
11
->Propositional-logic
->Weighted-max-sat
->Probabilistic-soft-logic
Probability vs. degree of truth
Probability
12
I invite my friends to my place. I have the food but I cannot cook.
Amal can cook. [Probability 60%]
George can cook. [Probability 60%]
Elon can cook. [Probability 50%]
Priscilla can cook. [Probability 50%]
Will we have cooked food?
Probability vs. degree of truth
Probability
13
I invite my friends to my place. I have the food but I cannot cook.
Amal can cook. [Probability 60%]
George can cook. [Probability 60%]
Elon can cook. [Probability 50%]
Priscilla can cook. [Probability 50%]
Will we have cooked food?
Yes, with probability 96%.
(Assuming that the probabilities
are independent.)
Probability vs. degree of truth
Probability
14
I invite my friends to my place. I have the food but I cannot cook.
Amal can cook. [Probability 60%]
George can cook. [Probability 60%]
Elon can cook. [Probability 50%]
Priscilla can cook. [Probability 50%]
Will we have cooked food?
Yes, with probability 96%.
(Assuming that the probabilities
are independent.)
Degree of truth
Amal is a great cook. [60%]
George is a great cook. [60%]
Elon is a great cook. [50%]
Priscilla is a great cook. [50%]
Will we have great cooked food?
Probability vs. degree of truth
Probability
15
I invite my friends to my place. I have the food but I cannot cook.
Amal can cook. [Probability 60%]
George can cook. [Probability 60%]
Elon can cook. [Probability 50%]
Priscilla can cook. [Probability 50%]
Will we have cooked food?
Yes, with probability 96%.
(Assuming that the probabilities
are independent.)
Degree of truth
Amal is a great cook. [60%]
George is a great cook. [60%]
Elon is a great cook. [50%]
Priscilla is a great cook. [50%]
Yes, of quality 60%.
(Assuming that the qualities
do not add up.)
Difference between probability and degree of truth: (1) semantic and (2) way of aggregation
Will we have great cooked food?
Degrees of truth
16
Amal is successful in law [90%].
Amal Clooney defended
Yulia Tymoshenko
(co-leader of the Orange
Revolution in Ukraine)
Nadia Murad (Iraqi-born
Yazidi human rights activist,
Nobel laureate)
Julian Assange
(Wikileaks founder)
Negated degrees of truth
17
Amal is successful in law [90%].
The degree of truth of the negation
of a statement is usually computed
as one minus the degree
of truth of that statement, ¬ｘ＝１－ｘ .
An alternative is the Yager‐negation
with
.
Amal is not successful in law [10%].
⇔
ｘ
¬ ｘ
ｘ
ｗ＝１
ｗ＞１
ｗ＜１
Combining degrees of truth
18
Amal is successful in law [90%].
George is successful in law [20%].
Washington (CNN) —
Police arrested actor George Clooney
and others Friday during a protest at
the Sudanese Embassy in Washington.
[CNN 2012-03-16]
19
Amal is successful in law [90%].
George is successful in law [20%].
Both are successful in law [
??%
].
Combining degrees of truth
20
Amal is successful in law [90%].
George is successful in law [20%].
Both are successful in law [
??%
].
Combining degrees of truth
55%
16%
-
Average:
(not sensitive to small values)
-
Harmonic mean:
)
(sensitive to small values, but one small value cannot annihilate the aggregation)
21
Amal is successful in law [90%].
George is successful in law [20%].
Both are successful in law [
??%
].
Combining degrees of truth
55%
16%
-
Average:
(not sensitive to small values)
-
Harmonic mean:
)
(sensitive to small values, but one small value cannot annihilate the aggregation)
These functions are not associative.
Therefore, they have to be applied to all conjuncts at once, not to pairs of conjuncts.
(Proof of non-commutativity by example)
Combining degrees of truth
22
Amal is successful in law [90%].
George is successful in law [20%].
Both are successful in law [
??%
].
A
T-Norm
is a function Ｔ： ［０， １］ × ［０， １］ → ［０， １］ that satsifies
- commutativity:
Ｔ（ａ， ｂ） ＝ Ｔ（ｂ， ａ）
- monotonicity:
Ｔ（ａ， ｃ） ≤ Ｔ（ｂ， ｃ） if ａ ≤ ｂ
- associativity:
Ｔ（ａ， Ｔ（ｂ， ｃ）） ＝ Ｔ（Ｔ（ａ， ｂ）， ｃ）
- neutral element:
Ｔ（ａ， １） ＝ ａ
It follows: Ｔ（ａ，０）＝０
One commonly uses a T-norm to compute the truth value of a a conjunction.
Common T-Norms
23
Amal is successful in law [90%].
George is successful in law [20%].
Both are successful in law [
??%
].
- Minimum t-norm:
Ｔ（ａ，ｂ）＝ｍｉｎ（ａ，ｂ）
- Product t-norm:
Ｔ（ａ，ｂ）＝ａ× ｂ
- Łukasiewicz t-norm:
Ｔ（ａ，ｂ）＝ｍａｘ（０，ａ＋ｂ－１）
- Drastic t-norm:
Ｔ（ａ，ｂ）＝（ａ＝１） ? ｂ ： （ｂ＝１） ? ａ：０
Common T-Norms
24
Amal is successful in law [90%].
George is successful in law [20%].
Both are successful in law [
??%
].
- Minimum t-norm:
Ｔ（ａ，ｂ）＝ｍｉｎ（ａ，ｂ）
- Product t-norm:
Ｔ（ａ，ｂ）＝ａ× ｂ
- Łukasiewicz t-norm:
Ｔ（ａ，ｂ）＝ｍａｘ（０，ａ＋ｂ－１）
- Drastic t-norm:
Ｔ（ａ，ｂ）＝（ａ＝１） ? ｂ ： （ｂ＝１） ? ａ：０
The minimum T-norm is the largest possible T-norm, the Drastic one the smallest.
T-norms fulfill the weak law of contradiction: Ｔ（ａ，１－ａ）≤ ０．５ .
Proof: min is the largest T-norm, and ｍｉｎ（ａ，１－ａ）≤ ０．５ .
20%
18%
10%
0%
T-conorms compute disjunctive truth values
25
Amal is successful in law [90%].
George is successful in law [20%].
The
T-conorm
of a T-norm Ｔ is the function
⊥（ａ，ｂ）＝１－Ｔ（１－ａ，１－ｂ） .
It follows:
- commutativity:
⊥（ａ， ｂ） ＝ ⊥（ｂ， ａ）
- monotonicity:
⊥（ａ， ｃ） ≤ ⊥（ｂ， ｃ） if ａ ≤ ｂ
- associativity:
⊥（ａ， ⊥（ｂ， ｃ）） ＝ ⊥（⊥（ａ， ｂ）， ｃ）
- identity element:
⊥（ａ，０）＝ａ
- ⊥（１，０）＝１， ⊥（０，０）＝０， ⊥（１，１）＝１
One commonly uses a T-conorm to compute the truth value of a disjunction.
At least one of them is successful in law [
??%
].
(Proof by definition of ⊥ )
Common T-conorms
26
Amal is successful in law [90%].
George is successful in law [20%].
- Maximum t-conorm:
⊥（ａ， ｂ）＝ｍａｘ（ａ， ｂ）
- Probabilistic sum:
⊥（ａ， ｂ）＝１－（１－ａ）×（１－ｂ）
- Bounded sum:
⊥（ａ， ｂ）＝ｍｉｎ（ａ＋ｂ， １）
- Drastic t-conorm:
⊥（ａ， ｂ）＝ａ＝０?ｂ ： ｂ＝０?ａ ： １
At least one of them is successful in law [
??%
].
Common T-conorms
27
Amal is successful in law [90%].
George is successful in law [20%].
- Maximum t-conorm:
⊥（ａ， ｂ）＝ｍａｘ（ａ， ｂ）
- Probabilistic sum:
⊥（ａ， ｂ）＝１－（１－ａ）×（１－ｂ）
- Bounded sum:
⊥（ａ， ｂ）＝ｍｉｎ（ａ＋ｂ， １）
- Drastic t-conorm:
⊥（ａ， ｂ）＝ａ＝０?ｂ ： ｂ＝０?ａ ： １
The largest t-conorm is the drastic t-conorm,
the smallest is the maximum t-conorm.
At least one of them is successful in law [
??%
].
90%
98%
100%
100%
(with Ｔ＝ｍｉｎ , proof by case split ａ＜ｂ )
Implications
28
George is successful in law [20%].
Amal is successful in law [90%].
The truth value of an implication ｘ ⇒ ｙ can be
given by the
S-implication
, i.e., the
truth value of the logical transformation ¬ ｘ ∨ ｙ :
- Maximum t-conorm:
ｍａｘ（１－ｘ，ｙ）
- Probabilistic sum:
１－ｘ×（１－ｙ）
- Bounded sum:
ｍｉｎ（１－ｘ＋ｙ，１）
If George is successful, then Amal is successful [
??%
].
Implications
29
George is successful in law [20%].
Amal is successful in law [90%].
If George is successful, then Amal is successful [
??%
].
90%
98%
100%
An S-implication can assign a truth value smaller than one to ｘ⇒ ｙ even if ｘ＜ｙ !
The truth value of an implication ｘ ⇒ ｙ can be
given by the
S-implication
, i.e., the
truth value of the logical transformation ¬ ｘ ∨ ｙ :
- Maximum t-conorm:
ｍａｘ（１－ｘ，ｙ）
- Probabilistic sum:
１－ｘ×（１－ｙ）
- Bounded sum:
ｍｉｎ（１－ｘ＋ｙ，１）
Implications
30
The truth value of an implication ｘ ⇒ ｙ is usually
given by the
R-implication/residuum
.
If ｘ≤ｙ , then the value is one, otherwise:
- Minimum t-norm:
ｙ
- Product t-norm:
ｙ／ｘ
- Łukasiewicz t-norm:
１－ｘ＋ｙ
An implication is
satisfied
if ｘ≤ｙ .
100%
100%
100%
The residuum coincides with the disjunction on crisp truth values, but does not satisfy Modus Tollens.
If George is successful, then Amal is successful [
??%
].
George is successful in law [20%].
Amal is successful in law [90%].
Implications
31
20%
22%
30%
The residuum coincides with the disjunction on crisp truth values, but does not satisfy Modus Tollens.
Amal is successful in law [90%].
If Amal is successful, then George is successful [
??%
].
George is successful in law [20%].
The truth value of an implication ｘ ⇒ ｙ is usually
given by the
R-implication/residuum
.
If ｘ≤ｙ , then the value is one, otherwise:
- Minimum t-norm:
ｙ
- Product t-norm:
ｙ／ｘ
- Łukasiewicz t-norm:
１－ｘ＋ｙ
An implication is
satisfied
if ｘ≤ｙ .
Interpretation
32
Amal is successful in law [90%].
An
interpretation
is a function that assigns a degree of truth to a statement.
Given a T-norm Ｔ , a given interpretation Ｉ can be extended to formulas as follows:
- Ｉ（¬ ａ）＝
１－ａ
- Ｉ（ａ ∧ ｂ）＝
Ｔ（ａ，ｂ）
- Ｉ（ａ ∨ ｂ）＝
１－Ｔ（１－ａ，１－ｂ）
- Ｉ（ａ ⇒ ｂ）＝
We can now compute the truth value of a formula from the truth value of its components:
I(Amal is successful in law)=90%
George is successful in law [20%].
I((Amal is successful in law ⇒ George is not successful in law) ∧ Amal is successful in law)
Interpretation
33
Amal is successful in law [90%].
An
interpretation
is a function that assigns a degree of truth to a statement.
Given a T-norm Ｔ , a given interpretation Ｉ can be extended to formulas as follows:
- Ｉ（¬ ａ）＝
１－ａ
- Ｉ（ａ ∧ ｂ）＝
Ｔ（ａ，ｂ）
- Ｉ（ａ ∨ ｂ）＝
１－Ｔ（１－ａ，１－ｂ）
- Ｉ（ａ ⇒ ｂ）＝
We can now compute the truth value of a formula from the truth value of its components:
I(Amal is successful in law)=90%
George is successful in law [20%].
I((Amal is successful in law ⇒ George is not successful in law) ∧ Amal is successful in law)
＝Ｉ（．２２ ∧ ．９）＝０．１９８ (with product T-norm )
Two main tasks on logical formulas
34
Amal is successful in law [90%].
George is successful in law [20%].
1)
Interpretation: Given statements with given truth values, compute the truth value of a formula
I((Amal is successful in law ⇒ George is not successful in law) ∧ Amal is successful in law)
＝Ｉ（．２２ ∧ ．９）＝０．１９８ (with product T-norm )
Two main tasks on logical formulas
35
Amal is successful in law [90%].
George is successful in law [20%].
1)
Interpretation: Given statements with given truth values, compute the truth value of a formula
2)
Deduction/Inference: Given statements with given truth values and formulas, compute the
truth value
of another statement.
I((Amal is successful in law ⇒ George is not successful in law) ∧ Amal is successful in law)
＝Ｉ（．２２ ∧ ．９）＝０．１９８ (with product T-norm )
Amal is successful in law [90%].
Amal is successful in law ⇒ George successful in law.
George is successful in law [
???
].
Two main tasks on logical formulas
36
Amal is successful in law [90%].
George is successful in law [20%].
1)
Interpretation: Given statements with given truth values, compute the truth value of a formula
2)
Deduction/Inference: Given statements with given truth values and formulas, compute the
truth value
of another statement.
I((Amal is successful in law ⇒ George is not successful in law) ∧ Amal is successful in law)
＝Ｉ（．２２ ∧ ．９）＝０．１９８ (with product T-norm )
Amal is successful in law [90%].
Amal is successful in law ⇒ George successful in law.
George is successful in law [90%].
There is no universally accepted calculus (proof system) for fuzzy formulas...
...except...
37
A
simple positive fuzzy inference problem
is a set of rules of the form ｐｒｅｍｉｓｅｓ ⇒ ｃｏｎｃｌｕｓｉｏｎ
where
are variables, and
is a conjunctive aggregation function.
A rule is
satisfied
if
.
All variables have initial values in [0,1]. A
solution
is an assignment of each variable to a value
so that (1) the value is greater or equal to the initial one, (2) all rules are satisfied, and
(3) no output variable can be assigned a smaller value.
1. Amal successful in law [90%] ⇒ Amal successful
2. couple successful ⇒ George successful
3. couple successful ⇒ Amal successful
4. George great actor [80%] ⇒ George successful
5. Amal successful ∧ George successful ⇒ couple successful
Simple Positive Fuzzy Inference Problems
[Peng, Bonald, Suchanek: “FLORA: Unsupervised Knowledge Graph Alignment by Fuzzy Logic”, ISWC 2025]
Simple Positive Fuzzy Inference Problems
38
Theorem:
The solution of a simple fuzzy inference problem can be computed by iteratively
satisfying all rules in any order until they are all satisfied.
(If the aggregation functions are monotonic and continuous.)
1. Amal successful in law [90%] ⇒ Amal successful
2. couple successful ⇒ George successful
3. couple successful ⇒ Amal successful
4. George great actor [80%] ⇒ George successful
5. Amal successful ∧ George successful ⇒ couple successful
[Peng, Bonald, Suchanek: “FLORA: Unsupervised Knowledge Graph Alignment by Fuzzy Logic”, ISWC 2025]
Simple Fuzzy Inference Problems
39
If we allow negated premises or conclusions, a solution that satisfies all rules may not exist,
and iteration may oscillate.
Amal successful in law [90%] ⇒ Amal successful
Amal successful ⇒ George successful
George successful ⇒ Amal
not
successful
A recursive problem
Fuzzy sets
40
A
fuzzy set
over a set Ｕ (the
universe of discourse
) is a function μ：Ｕ→ ［０，１］ .
We say ｘ
belongs
to the set μ to degree μ（ｘ） .
-
support
:
｛ｘ：μ（ｘ）＞０｝
-
core
:
｛ｘ：μ（ｘ）＝１｝
-
α ‐cut
:
｛ｘ：μ（ｘ）≥α｝
beautiful people
Fuzzy sets
41
A
fuzzy set
over a set Ｕ (the
universe of discourse
) is a function μ：Ｕ→ ［０，１］ .
We say ｘ
belongs
to the set μ to degree μ（ｘ） .
-
support
:
｛ｘ：μ（ｘ）＞０｝
-
core
:
｛ｘ：μ（ｘ）＝１｝
-
α ‐cut
:
｛ｘ：μ（ｘ）≥α｝
beautiful people
α＝０．４
Fuzzy set intersections and unions
42
Intersections and unions of fuzzy sets can be computed by point‐wise application of
t-norms and t-conorms, respectively.
rich people
beautiful people
Fuzzy set intersections and unions
43
Intersections and unions of fuzzy sets can be computed by point‐wise application of
t-norms and t-conorms, respectively.
beautiful and rich people
rich people
beautiful people
Capping
44
An
α -capping
of a fuzzy set μ is the fuzzy set
.
rich people
α＝０．７
Capping
45
An
α -capping
of a fuzzy set μ is the fuzzy set
.
α＝０．７
rich people
Equality relations
46
An
equality relation
for a T-norm Ｔ is a function
that satisfies
- commutativity:
Ｅ（ｘ，ｙ）＝Ｅ（ｙ，ｘ）
- reflexivity:
Ｅ（ｘ，ｘ）＝１
- transitivity:
Ｔ（Ｅ（ｘ，ｙ）， Ｅ（ｙ，ｚ）） ≤ Ｅ（ｘ，ｚ）
0.01
???
1
0.8
Equality relations
47
0.8
≥ 0.01 with Ｔ＝ｍｉｎ
An
equality relation
for a T-norm Ｔ is a function
that satisfies
- commutativity:
Ｅ（ｘ，ｙ）＝Ｅ（ｙ，ｘ）
- reflexivity:
Ｅ（ｘ，ｘ）＝１
- transitivity:
Ｔ（Ｅ（ｘ，ｙ）， Ｅ（ｙ，ｚ）） ≤ Ｅ（ｘ，ｚ）
0.01
Fuzzification
48
Ａ
An
equality relation
for a T-norm Ｔ is a function
that satisfies
- commutativity:
Ｅ（ｘ，ｙ）＝Ｅ（ｙ，ｘ）
- reflexivity:
Ｅ（ｘ，ｘ）＝１
- transitivity:
Ｔ（Ｅ（ｘ，ｙ）， Ｅ（ｙ，ｚ）） ≤ Ｅ（ｘ，ｚ）
The
fuzzification
of a crisp set Ａ by
an equality relation Ｅ is the fuzzy set
μ（ｘ）＝ｓｕｐ ｛ Ｅ（ｘ，ｙ） ： ｙ ∈ Ａ ｝
0.01
Fuzzification
49
The
fuzzification
of a crisp set Ａ by
an equality relation Ｅ is the fuzzy set
μ（ｘ）＝ｓｕｐ ｛ Ｅ（ｘ，ｙ） ： ｙ ∈ Ａ ｝
Ａ
0.01
1
An
equality relation
for a T-norm Ｔ is a function
that satisfies
- commutativity:
Ｅ（ｘ，ｙ）＝Ｅ（ｙ，ｘ）
- reflexivity:
Ｅ（ｘ，ｘ）＝１
- transitivity:
Ｔ（Ｅ（ｘ，ｙ）， Ｅ（ｙ，ｚ）） ≤ Ｅ（ｘ，ｚ）
0.01
Defuzzification
50
A
defuzzification
is a function that maps a fuzzy set to an element of its universe.
We focus on real-valued universes.
1900
1
life
quality
1914
When was “the best year”?
1935
1960
2020
Defuzzification
51
A
defuzzification
is a function that maps a fuzzy set to an element of its universe.
We focus on real-valued universes.
1900
1
life
quality
1914
When was “the best year”?
1935
1960
2020
First of Maxima
is the defuzzification ｄ（μ）＝ｍｉｎ ｛ｃ ｜ ∀ ｘ： μ（ｃ） ≥ μ（ｘ）｝
Defuzzification
52
A
defuzzification
is a function that maps a fuzzy set to an element of its universe.
We focus on real-valued universes.
1900
1
life
quality
1914
When was “the best year”?
First of Maxima
is the defuzzification ｄ（μ）＝ｍｉｎ ｛ｃ ｜ ∀ ｘ： μ（ｃ） ≥ μ（ｘ）｝
1935
1960
2020
But what about the good years 1960-2020?
Defuzzification
53
A
defuzzification
is a function that maps a fuzzy set to an element of its universe.
We focus on real-valued universes.
1900
1
life
quality
1914
When was “the best year”?
Mean of Maxima
is the defuzzification ｄ（μ）＝ａｖｇ ｛ｃ ｜ ∀ ｘ： μ（ｃ） ≥ μ（ｘ）｝
1935
1960
2020
Defuzzification
54
A
defuzzification
is a function that maps a fuzzy set to an element of its universe.
We focus on real-valued universes.
1900
1
life
quality
1914
When was “the best year”?
Mean of Maxima
is the defuzzification ｄ（μ）＝ａｖｇ ｛ｃ ｜ ∀ ｘ： μ（ｃ） ≥ μ（ｘ）｝
1935
1960
2020
Problem: this may land on a value that is itself not a maximum
Defuzzification
55
A
defuzzification
is a function that maps a fuzzy set to an element of its universe.
We focus on real-valued universes.
1900
1
life
quality
1914
When was “the best year”?
Median
is the defuzzification ｄ（μ）＝ｘ , s.t.
1935
1960
2020
ｘ
Defuzzification
56
A
defuzzification
is a function that maps a fuzzy set to an element of its universe.
We focus on real-valued universes.
1900
1
life
quality
1914
When was “the best year”?
Median
is the defuzzification ｄ（μ）＝ｘ , s.t.
1935
1960
2020
Problem: this may also land on a value that is itself not a maximum
ｘ
=> The
first of maxima
is generally a safe choice
Mamdani-Style Fuzzy Inference System
57
A
Mamdani-Style Fuzzy Inference System
is a set of rules
where
are input variables,
are fuzzy
sets, ｃ is the output variable, and Ｃ is a fuzzy set.
Given items for the input variables,
such a system is applied as follows:
1)
Fuzzify the inputs, i.e., compute
to what degree
belongs to
.
2)
Compute the “firing strength” α of the rule,
i.e., the T-norm of the degrees of belonging.
3)
Cap the set Ｃ at α .
4)
Defuzzify the capped Ｃ to a single value ｃ ,
output ｃ .
Mamdani-Style Fuzzy Inference System
58
ｔｅｍｐ ｉｓ ＨＯＴ ∧ ｒｏｏｍ ｉｓ ＣＲＯＷＤＥＤ
⇒ ｆａｎｓｐｅｅｄ ｉｓ ＦＡＳＴ
A
Mamdani-Style Fuzzy Inference System
is a set of rules
where
are input variables,
are fuzzy
sets, ｃ is the output variable, and Ｃ is a fuzzy set.
Given items for the input variables,
such a system is applied as follows:
1)
Fuzzify the inputs, i.e., compute
to what degree
belongs to
.
2)
Compute the “firing strength” α of the rule,
i.e., the T-norm of the degrees of belonging.
3)
Cap the set Ｃ at α .
4)
Defuzzify the capped Ｃ to a single value ｃ ,
output ｃ .
Mamdani-Style Fuzzy Inference System
59
ｔｅｍｐ ｉｓ ＨＯＴ ∧ ｒｏｏｍ ｉｓ ＣＲＯＷＤＥＤ
⇒ ｆａｎｓｐｅｅｄ ｉｓ ＦＡＳＴ
ｒｏｏｍ＝１０ ｉｓ ＣＲＯＷＤＥＤ： ０．６
α＝ｍｉｎ（０．９， ０．６）＝０．６
A
Mamdani-Style Fuzzy Inference System
is a set of rules
where
are input variables,
are fuzzy
sets, ｃ is the output variable, and Ｃ is a fuzzy set.
Given items for the input variables,
such a system is applied as follows:
1)
Fuzzify the inputs, i.e., compute
to what degree
belongs to
.
2)
Compute the “firing strength” α of the rule,
i.e., the T-norm of the degrees of belonging.
3)
Cap the set Ｃ at α .
4)
Defuzzify the capped Ｃ to a single value ｃ ,
output ｃ .
Mamdani-Style Fuzzy Inference System
60
ｔｅｍｐ ｉｓ ＨＯＴ ∧ ｒｏｏｍ ｉｓ ＣＲＯＷＤＥＤ
⇒ ｆａｎｓｐｅｅｄ ｉｓ ＦＡＳＴ
ＦＡＳＴ
0 rotations/s 100
1
0
A
Mamdani-Style Fuzzy Inference System
is a set of rules
where
are input variables,
are fuzzy
sets, ｃ is the output variable, and Ｃ is a fuzzy set.
Given items for the input variables,
such a system is applied as follows:
1)
Fuzzify the inputs, i.e., compute
to what degree
belongs to
.
2)
Compute the “firing strength” α of the rule,
i.e., the T-norm of the degrees of belonging.
3)
Cap the set Ｃ at α .
4)
Defuzzify the capped Ｃ to a single value ｃ ,
output ｃ .
ｒｏｏｍ＝１０ ｉｓ ＣＲＯＷＤＥＤ： ０．６
α＝ｍｉｎ（０．９， ０．６）＝０．６
Mamdani-Style Fuzzy Inference System
61
ｔｅｍｐ ｉｓ ＨＯＴ ∧ ｒｏｏｍ ｉｓ ＣＲＯＷＤＥＤ
⇒ ｆａｎｓｐｅｅｄ ｉｓ ＦＡＳＴ
ＦＡＳＴ
0 rotations/s 100
1
0
α