Traditional (crisp) logic - site.iugaza.edu.pssite.iugaza.edu.ps/mahir/files/2010/02/presentation1.pdf · What is fuzzy logic?What is fuzzy logic? Fuzzy logic is a superset of conventional

A rose is either RED

or not RED.

A rose is either RED

or not RED.or not RED.

Traditional (crisp) logicTraditional (crisp) logicTraditional (crisp) logic

What about this rose?What about this rose?What about this rose?

Traditional (crisp) logicTraditional (crisp) logicTraditional (crisp) logic

What color is this leopard?What color is this leopard?What color is this leopard?

Is this glass full or empty? Is this glass full or empty? Is this glass full or empty?

At what point short people become tall? At what point short At what point short people become tall? people become tall?

What is fuzzy logic?What is fuzzy logic?What is fuzzy logic?

Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth –the truth values between "completely true" and "completely false".

Fuzzy logic is a superset of conventional Fuzzy logic is a superset of conventional (Boolean) logic that has been extended (Boolean) logic that has been extended to handle the to handle the concept of partial truthconcept of partial truth ––the truth values between "the truth values between "completely completely truetrue" and "" and "completely falsecompletely false".".

A type of logic that recognizes more than simple true and false values. With fuzzy logic, propositions can be represented with degrees of truthfulness and falsehood. For example, the statement, today is sunny, might be 100% true if there are no clouds, 80% true if there are a few clouds, 50% true if it's hazy and 0% true if it rains all day.

A type of logic that recognizes more than A type of logic that recognizes more than simple true and false values. With fuzzy simple true and false values. With fuzzy logic, propositions can be represented logic, propositions can be represented with degrees of truthfulness and with degrees of truthfulness and falsehood. For example, the statement, falsehood. For example, the statement, today is sunny, might be 100% true if today is sunny, might be 100% true if there are no clouds, 80% true if there are there are no clouds, 80% true if there are a few clouds, 50% true if it's hazy and 0% a few clouds, 50% true if it's hazy and 0% true if it rains all day. true if it rains all day.


“ A form of knowledge representation suitable for notions that cannot be defined precisely, but which depend upon their context. It enables computerized devices to reason more like humans”

““ A form of knowledge representation A form of knowledge representation suitable for notions that cannot be suitable for notions that cannot be defined precisely, but which depend defined precisely, but which depend upon their context. It enables upon their context. It enables computerized devices to reason more computerized devices to reason more like humanslike humans””


Fuzzy Logic: MotivationsFuzzy Logic: Motivations• Alleviate difficulties in developing and

analyzing complex systems encountered by conventional mathematical tools.

• Observing that human reasoning can utilize concepts and knowledge that do not have well-defined, sharp boundaries.

•• Alleviate difficulties in developing and Alleviate difficulties in developing and analyzing complex systems encountered analyzing complex systems encountered by conventional mathematical tools. by conventional mathematical tools.

•• Observing that human reasoning can Observing that human reasoning can utilize concepts and knowledge that do utilize concepts and knowledge that do not have wellnot have well--defined, sharp boundaries.defined, sharp boundaries.

Fuzzy Logic: MotivationsFuzzy Logic: Motivations

Fuzzy Logic: MotivationsFuzzy Logic: Motivations

Fuzzy Logic: MotivationsFuzzy Logic: MotivationsFuzziness is beneficial for: - Complex systems that are difficult or

impossible to model - Systems controlled by human

experts or systems that use human observations as inputs

- Systems that naturally vague (behavioral and social sciences)

Fuzziness is beneficial for: Fuzziness is beneficial for: -- Complex systems that are difficult or Complex systems that are difficult or

impossible to model impossible to model -- Systems controlled by human Systems controlled by human

experts or systems that use human experts or systems that use human observations as inputs observations as inputs

-- Systems that naturally vague Systems that naturally vague (behavioral and social sciences)(behavioral and social sciences)

History of Fuzzy LogicHistory of Fuzzy Logic1964: Lotfi A. Zadeh, UC Berkeley, introduced the paper on fuzzy sets.

– Idea of grade of membership was born – Sharp criticism from academic community

• Name! • Theory’s emphasis on imprecision

– Waste of government funds!

1964: Lotfi A. Zadeh, UC Berkeley, introduced the paper on fuzzy sets.

– Idea of grade of membership was born – Sharp criticism from academic community

• Name! • Theory’s emphasis on imprecision

– Waste of government funds!

1965-1975: Zadeh continued to broaden the foundation of fuzzy set theory

– Fuzzy multistage decision-making – Fuzzy similarity relations – Fuzzy restrictions – Linguistic hedges

1970s: research groups were formed in Japan

19651965--1975:1975: ZadehZadeh continued to broaden the continued to broaden the foundation of fuzzy set theory foundation of fuzzy set theory

–– Fuzzy multistage decisionFuzzy multistage decision--making making –– Fuzzy similarity relations Fuzzy similarity relations –– Fuzzy restrictions Fuzzy restrictions –– Linguistic hedges Linguistic hedges

1970s:1970s: research groups were formed in Japanresearch groups were formed in Japan

History of Fuzzy LogicHistory of Fuzzy Logic

1974: Mamdani, United Kingdom, developed the first fuzzy logic controller (steam engine control)

1982: First commercial control system using fuzzy logic (cement kiln, Holmblad and Ostergaard)

1974: Mamdani, United Kingdom, developed the first fuzzy logic controller (steam engine control)

1982: First commercial control system using fuzzy logic (cement kiln, Holmblad and Ostergaard)


1976-1987: Industrial application of fuzzy logic in Japan and Europe

1987- Present: Fuzzy Boom

2003: First class on fuzzy logic is held at Clarkson University

1976-1987: Industrial application of fuzzy logic in Japan and Europe

1987- Present: Fuzzy Boom

2003: First class on fuzzy logic is held at Clarkson University


Aerospace – Altitude control of spacecraft, satellite altitude

control, flow and mixture regulation in aircraft deicing vehicles.

Automotive – Trainable fuzzy systems for idle speed control,

shift scheduling method for automatic transmission, intelligent highway systems, traffic control, improving efficiency of automatic transmissions

–

Aerospace Aerospace –– Altitude control of spacecraft, satellite altitude Altitude control of spacecraft, satellite altitude

control, flow and mixture regulation in aircraft control, flow and mixture regulation in aircraft deicing vehicles. deicing vehicles.

Automotive Automotive –– Trainable fuzzy systems for idle speed control, Trainable fuzzy systems for idle speed control,

shift scheduling method for automatic shift scheduling method for automatic transmission, intelligent highway systems, transmission, intelligent highway systems, traffic control, improving efficiency of traffic control, improving efficiency of automatic transmissionsautomatic transmissions

––

Fuzzy Logic ApplicationsFuzzy Logic Applications

Business – Decision-making support systems, personnel

evaluation in a large company – Data mining systems

Chemical Industry – Control of pH, drying, chemical distillation

processes, polymer extrusion production, a coke oven gas cooling plant

•

Business – Decision-making support systems, personnel

evaluation in a large company – Data mining systems

Chemical Industry – Control of pH, drying, chemical distillation

processes, polymer extrusion production, a coke oven gas cooling plant

•


Defense – Underwater target recognition, automatic

target recognition of thermal infrared images, naval decision support aids, control of a hypervelocity interceptor, fuzzy set modeling of NATO decision making.

Electronics – Control of automatic exposure in video

cameras, humidity in a clean room, air conditioning systems, washing machine timing, microwave ovens, vacuum cleaners.

Defense – Underwater target recognition, automatic

target recognition of thermal infrared images, naval decision support aids, control of a hypervelocity interceptor, fuzzy set modeling of NATO decision making.

Electronics – Control of automatic exposure in video

cameras, humidity in a clean room, air conditioning systems, washing machine timing, microwave ovens, vacuum cleaners.


Financial – Banknote transfer control, fund management,

stock market predictions. Industrial

– Cement kiln controls (dating back to 1982), heat exchanger control, activated sludge wastewater treatment process control, water purification plant control, quantitative pattern analysis for industrial quality assurance, control of constraint satisfaction problems in structural design, control of water purification plants

Financial – Banknote transfer control, fund management,

stock market predictions. Industrial

– Cement kiln controls (dating back to 1982), heat exchanger control, activated sludge wastewater treatment process control, water purification plant control, quantitative pattern analysis for industrial quality assurance, control of constraint satisfaction problems in structural design, control of water purification plants


Marine – Autopilot for ships, optimal route selection,

control of autonomous underwater vehicles, ship steering.

Medical – Medical diagnostic support system, control of

arterial pressure during anesthesia, multivariable control of anesthesia, modeling of neuropathological findings in Alzheimer's patients, radiology diagnoses, fuzzy inference diagnosis of diabetes and prostate cancer.

Marine – Autopilot for ships, optimal route selection,

control of autonomous underwater vehicles, ship steering.

Medical – Medical diagnostic support system, control of

arterial pressure during anesthesia, multivariable control of anesthesia, modeling of neuropathological findings in Alzheimer's patients, radiology diagnoses, fuzzy inference diagnosis of diabetes and prostate cancer.


Mining and Metal Processing – Sinter plant control, decision making in metal

forming.

Robotics – Fuzzy control for flexible-link manipulators,

robot arm control.

Securities – Decision systems for securities trading.

Mining and Metal Processing – Sinter plant control, decision making in metal

forming.

Robotics – Fuzzy control for flexible-link manipulators,

robot arm control.

Securities – Decision systems for securities trading.


Signal Processing and Telecommunications – Adaptive filter for nonlinear channel

equalization control of broadband noise

Transportation – Automatic underground train operation, train

schedule control, railway acceleration, braking, and stopping

Signal Processing and Telecommunications – Adaptive filter for nonlinear channel

equalization control of broadband noise

Transportation – Automatic underground train operation, train

schedule control, railway acceleration, braking, and stopping


Fuzzy setsFuzzy setsA fuzzy set is a set with a smooth boundary.

An element of a fuzzy set can belong to that set partially to a degree and the set does not have crisp boundaries.

A fuzzy set is defined by a functions that maps objects in a domain of concern into their membership value in a set.

Such a function is called the membership function.

A fuzzy set is a set with a smooth boundary.

An element of a fuzzy set can belong to that set partially to a degree and the set does not have crisp boundaries.

A fuzzy set is defined by a functions that maps objects in a domain of concern into their membership value in a set.

Such a function is called the membership function.

Fuzzy setsFuzzy sets

Definition: let X be a non-empty set and be called the universe of discourse. A fuzzy set A⊂X is characterized by the membership function

where µA(x) is a grade (degree) of membership of x in set A.

Definition: let X be a non-empty set and be called the universe of discourse. A fuzzy set A⊂X is characterized by the membership function

where µA(x) is a grade (degree) of membership of x in set A.

]1,0[: →XAµ ]1,0[: →XAµ

Fuzzy setsFuzzy sets

Note : since {0,1}∈[0,1] all crisp sets are fuzzy sets!

Note : since {0,1}∈[0,1] all crisp sets are fuzzy sets!

]1,0[: →XAµ ]1,0[: →XAµ

Fuzzy setsFuzzy setsDefinition of fuzzy sets:

Fuzzy set A can be represented as a set of ordered pairs

Definition of fuzzy sets:

Fuzzy set A can be represented as a set of ordered pairs

( ){ }XxxxA A ∈= )(, µ( ){ }XxxxA A ∈= )(, µ

Fuzzy setsFuzzy setsDiscrete example: µA = 0.1/x1 + 0.4/x2 +0.8/x3 + 1.0/x4 +

0.8/x5 + 0.4/x6 + 0.1/x7

Discrete example: µA = 0.1/x1 + 0.4/x2 +0.8/x3 + 1.0/x4 +

0.8/x5 + 0.4/x6 + 0.1/x7

Fuzzy setsFuzzy setsContinuous example: Continuous example:

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+∈−

−

−∈−

+

=

otherwise ,0

],[ ,1

],[ ,1

)( hccxh

cx

chcxh

cx

xAµ

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+∈−

−

−∈−

+

=

otherwise ,0

],[ ,1

],[ ,1

)( hccxh

cx

chcxh

cx

xAµ

Support : support of a fuzzy set A is a crisp set that contains all elements of A with non-zero membership grade:

Properties of fuzzy setsProperties of fuzzy sets

{ }0)()(supp >∈= xXxA Aµ{ }0)()(supp >∈= xXxA Aµ

Core: comprises those elements x of the universe such that µA (x) = 1.

Core: comprises those elements x of the universe such that µA (x) = 1.


{ }1)()(core =∈= xXxA Aµ{ }1)()(core =∈= xXxA Aµ

Boundary : boundaries comprise those elements x of the universe such that 0< µA (x) <1

Boundary : boundaries comprise those elements x of the universe such that 0< µA (x) <1


{ }1)(0)(bnd <<∈= xXxA Aµ{ }1)(0)(bnd <<∈= xXxA Aµ

Height : the height of a fuzzy set A is defined as

Set A is called normal if hgt(A)=1and subnormal if hgt(A)<1

Height : the height of a fuzzy set A is defined as

Set A is called normal if hgt(A)=1and subnormal if hgt(A)<1


XxA xA

∈= )(max)(hgt µ

XxA xA

∈= )(max)(hgt µ

Question?

Is the fuzzy set defined as

Normal or subnormal?

Question?

Is the fuzzy set defined as

Normal or subnormal?


2

/11)( xA ex −−=µ

2

/11)( xA ex −−=µ

52.50-2.5-5

1

0.75

0.5

0.25

x

y

x

y

1 - 1/exp(x^2)

Convex Fuzzy set: a fuzzy set A is convex iff

Convex Fuzzy set: a fuzzy set A is convex iff


( ))(),(min))1((]1,0[ and ,

yxyxXyx

AAA µµλλµλ

≥−+∈∀∈∀

( ))(),(min))1((]1,0[ and ,

yxyxXyx

AAA µµλλµλ

≥−+∈∀∈∀

Fuzzy number: a fuzzy set A is a fuzzy number if the fuzzy set is – Convex – Normal – The core consists of one value only – MF is piecewise continuous

Example: fuzzy 3

Fuzzy number: a fuzzy set A is a fuzzy number if the fuzzy set is – Convex – Normal – The core consists of one value only – MF is piecewise continuous

Example: fuzzy 3



x0

1

µA(x)

100

1

Set “positive number”Set “positive number” Set “positive number not exceeding 10”Set “positive number not exceeding 10”

Set “number near 0”Set “number near 0” Set “number near 10”Set “number near 10”

0

1

10

1

1. Empty set

2. Basic set (universe)

3. Identity

1. Empty set

2. Basic set (universe)

3. Identity

Operations on fuzzy setsOperations on fuzzy sets

0≡∅µ 0≡∅µ

1≡Xµ 1≡Xµ

XxxxBA BA ∈∀=⇔= )()( µµ XxxxBA BA ∈∀=⇔= )()( µµ

4. Subset4. Subset


Xx xxBA BA ∈∀≤⇔⊂ )()( µµ Xx xxBA BA ∈∀≤⇔⊂ )()( µµ

BA1.0

A

B

5. Union 5. Union


( ) xxxXx BABA

BABA

)(),(max)(: ], ,max[ ] ,U[

µµµµµµµ

=∈∀=

∪ ( ) xxxXx BABA

BABA

)(),(max)(: ], ,max[ ] ,U[

µµµµµµµ

=∈∀=

∪

6. Intersection 6. Intersection


( ) xxxXxI

BABA

BABA

)(),(min)(:],min[],[

µµµµµµµ

=∈∀=

∩ ( ) xxxXxI

BABA

BABA

)(),(min)(:],min[],[

µµµµµµµ

=∈∀=

∩

7. Complement Standard complement function: 7. Complement Standard complement function:


xxXxaaC

AA )(1)(:1)(

µµ −=∈∀−=

xxXxaaC

AA )(1)(:1)(

µµ −=∈∀−=

C(a) = 1 - a

a1

C(a)

1

7. Complement

Note: the laws of excluded middle

and the law of contradiction

Are not valid for fuzzy sets!

7. Complement

Note: the laws of excluded middle

and the law of contradiction

Are not valid for fuzzy sets!


∅=∩ AA ∅=∩ AA

XAA =∪ XAA =∪

Properties of fuzzy operations: Properties of fuzzy operations:


Properties of fuzzy operations: Properties of fuzzy operations:


Power of fuzzy sets: Power of fuzzy sets:

mmAA

AA

Xxxx

Xxxx

m ∈∀=

∈∀=

,)]([)(

,)]([)( 22

µµ

µµmm

AA

AA

Xxxx

Xxxx

m ∈∀=

∈∀=

,)]([)(

,)]([)( 22

µµ

µµ


Do not sleep in class!!! For sleep-deprived: SL119:Linear SleepAdvanced class in mattress manipulation. Topics include unconsciousness and hibernation.

Do not sleep in class!!! For sleep-deprived: SL119:Linear SleepAdvanced class in mattress manipulation. Topics include unconsciousness and hibernation.

Fuzzy adviceFuzzy advice

5. Union

Axioms for union function U : [0,1] × [0,1] → [0,1] µA∪B(x) = U[µA(x), µB(x)]

• U(0,0) = 0, U(0,1) = 1, U(1,0) = 1, U(1,1) = 1

• U(a,b) = U(b,a) (Commutativity)

• If a ≤ a’ and b ≤ b’, U(a, b) ≤ U(a’, b’) (monotonicity).

• U(U(a, b), c) = U(a, U(b, c)) (Associativity)

• Function U is continuous.

• U(a, a) = a (idempotency)

5. Union

Axioms for union function U : [0,1] × [0,1] → [0,1] µA∪B(x) = U[µA(x), µB(x)]

• U(0,0) = 0, U(0,1) = 1, U(1,0) = 1, U(1,1) = 1

• U(a,b) = U(b,a) (Commutativity)

• If a ≤ a’ and b ≤ b’, U(a, b) ≤ U(a’, b’) (monotonicity).

• U(U(a, b), c) = U(a, U(b, c)) (Associativity)

• Function U is continuous.

• U(a, a) = a (idempotency)


6. Intersection Axioms for intersection function

I:[0,1] × [0,1] → [0,1] µA∩B(x) = I[µA(x), µB(x)]

• I(1, 1) = 1, I(1, 0) = 0, I(0, 1) = 0, I(0, 0) = 0

• I(a, b) = I(b, a), Commutativity.

• If a ≤ a’ and b ≤ b’, I(a, b) ≤ I(a’, b’),monotonicity.

• I(I(a, b), c) = I(a, I(b, c)), Associativity.

• I is a continuous function.

• I(a, a) = a, idempotency.

6. Intersection Axioms for intersection function

I:[0,1] × [0,1] → [0,1] µA∩B(x) = I[µA(x), µB(x)]

• I(1, 1) = 1, I(1, 0) = 0, I(0, 1) = 0, I(0, 0) = 0

• I(a, b) = I(b, a), Commutativity.

• If a ≤ a’ and b ≤ b’, I(a, b) ≤ I(a’, b’),monotonicity.

• I(I(a, b), c) = I(a, I(b, c)), Associativity.

• I is a continuous function.

• I(a, a) = a, idempotency.


7. Complement

Axioms for Complement function C: [0,1] → [0,1]

• Boundary conditions C(0) = 1, C(1) = 0

• C is monotonic non-increasinga,b ∈ [0,1] if a < b, then C(a) ≥ C(b)

• C is a continuous function.

• C is involutive C(C(a)) = a for all a ∈ [0,1]

7. Complement

Axioms for Complement function C: [0,1] → [0,1]

• Boundary conditions C(0) = 1, C(1) = 0

• C is monotonic non-increasinga,b ∈ [0,1] if a < b, then C(a) ≥ C(b)

• C is a continuous function.

• C is involutive C(C(a)) = a for all a ∈ [0,1]


Traditional (crisp) logic - site.iugaza.edu.pssite.iugaza.edu.ps/mahir/files/2010/02/presentation1.pdf · What is fuzzy logic?What is fuzzy logic? Fuzzy logic is a superset of conventional

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