Conditional propositions and
inference
G. J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and
Applications, Prentice-Hall, chapter 8
Type of fuzzy propositions
1. Unconditional and unqualified propositions
2. Unconditional and qualified propositions
3. Conditional and unqualified propositions
4. Conditional and qualified propositions
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Classical existential quantifier
„
Existential quantification of a predicate P(x) is expressed
by the form
(∃x) P ( x),
"There exists an individual x (in the universal set X) s.t. x
is P". We have the following equality:
(∃x) P( x) = V P ( x)
x∈ X
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Classical universal quantifier
„
Universal quantification of a predicate P(x) is
expressed by the form.
(∀x) P ( x),
“For every individual x (in the universal set) x is
P". Clearly, the following equality holds:
(∀x) P ( x) = ∧ P ( x)
x∈ X
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Fuzzy quantifiers
„
„
Fuzzy quantifiers are fuzzy numbers that take part in fuzzy propositions
There are two types:
‰ Type 1: (absolute quantifiers)
‰
Type 2: (relative quantifiers)
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Fuzzy quantifiers
„
There are two basic forms of propositions that contain fuzzy
quantifiers of type 1.
‰
The first basic form:
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3
8.4 Fuzzy quantifiers
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Fuzzy quantifiers
8
4
Fuzzy quantifiers
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Fuzzy quantifiers
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5
Fuzzy quantifiers
„
Fuzzy quantifiers of type 2.
‰
‰
These are quantifiers such as “almost all,” “about half,” “most,”
and so on.
They are represented by fuzzy numbers on the unit interval [0,1].
Conditional proposition or implication
„
Canonical form
p : If X is A, then Y is B,
where X, Y are variables whose values are in sets X, Y,
respectively, and A, B are linguistic terms (fuzzy sets) on
X, Y, respectively.
„
These propositions may also be viewed as
X , Y is R,
where R is a fuzzy relation on XxY
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Recall: Linguistic variables
„
Values of a linguistic variable are linguistic terms,
represented as fuzzy sets, and characterize concepts
Linguistic terms
Membership
values
1
Young
Middle-aged
Old
0.5
0
25
40
55
Age
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Implication, example
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Implication
In general
where ψ is some implication operator and
Implication
The membership function for N rules is given by
Where ϕ is an aggregation operator, such as a t-conorm
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Implications, different types of
Boolean implication
Lukasiewicz implication
Zadeh implication
Mamdani implication
Larsen implication
Boolean implication
For the case of Ν rules,
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Lukasiewicz implication
Bounded sum
For the case of Ν rules,
Zadeh implication
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Inference from conditional fuzzy propositions
„
„
Inference rules in classical logic are based on tautologies.
In this section, we describe generalizations of three classical
inference rules:
(a ∧ (a ⇒ b)) ⇒ b (modus ponens),
(b ∧ (a ⇒ b)) ⇒ a (modus tollens),
((a ⇒ b) ∧ (b ⇒ c)) ⇒ (a ⇒ c) (hypothetical syllogism).
„
These generalizations are based on the compositional rule of
inference.
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Inference: modus ponens
‰
Classical
A, A → B
∴B
‰
Fuzzy (compositional rule of inference)
A1 , A2 → B
∴ A1ο ( A2 → B)
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Compositional rule of inference
Compositional rule of inference
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Compositional rule of inference
Inference
two fuzzy implication inference rules
1.
Generalized Modus Ponens (or GMP)
use in all fuzzy controllers.
For the special case
Α’=Α and Β’=Β then GMP reduces to Modus Ponens.
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Inference
two fuzzy implication inference rules
2. Generalized Modus Tollens (or GMT)
application in expert systems
For the special case
then GMT reduces to Modus Tollens
Compositional Rule of Inference
the membership function of the resultant compositional rule of inference
Mamdani implication
Larsen implication
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Inference for Mamdani implication
For N rules
Inference for Larsen implication
For N rules
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Compositional Rule of Inference
the max-min operators
max-product operators:
Compositional Rule of Inference, example
Slow
Fast
determine the outcome if A = ‘slightly Slow’ for which there no rule exists
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Compositional Rule of Inference
The first step is to compute the Cartesian product and using the min operator
Compositional Rule of Inference
The second step using the fuzzy compositional inference rule:
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Compositional Rule of Inference
The final operation
Compositional Rule of Inference
using the max-product rule of compositional inference:
Using the the Mamdani compositional rule
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Compositional Rule of Inference, check point
0.7 1.0 0.4 0
⎧~
⎪ A1 = a + a + a + a
⎪
1
2
3
4
⎨
0 0.2 0.6 1
~
⎪A
=
+
+
+
⎪⎩ 2 a1 a 2
a3 a4
Given If inputs：
⎧ ~ 1.0 0.6 0
⎪ B1 = b + b + b
⎪
1
2
3
⎨
0.1 0.4 1
~
⎪B
=
+
+
⎪⎩ 2 b1
b2 b3
then outputs：
if input
0.3 0.6 1.0 0.2
~
A3 =
+
+
+
a1 a 2 a 3 a 4
Then output ?
~
B3
Example: Air Conditioning (direct) controller
Temperature
Speed
Controller
Advantages of fuzzy controllers
„
„
„
Minimal mathematical formulation
Intuitive design
Competitive performance
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Example: Air Conditioning Controller
0
100
90
80
If Hot
then
Blast
st
Bla
Fa
st
If Warm
then
Fast
70
60
M ed
50
40
If Just Right
then
Medium
ium
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Rule base:
IF Cold then Stop
If Cool then Slow
If OK then Medium
If Warm then Fast
IF Hot then Blast
if Cold
then Stop
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op
St
10
„
IF Cool
then
Slow
Sl
ow
0
Ho
t
W
ar
m
Jus
Rig t
ht
ld
Co
Co
ol
1
0
45
50
55
60
65
70
75
80
85
90
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