13 Fuzzy Logic (Us)

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Artificial Intelligence

Fuzzy logicFall 2008

professor: Luigi Ceccaroni


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Uncertainty

- Modeling with uncertainty requires morethan probability theory.

- There are problems where boundaries are

gradual.Examples: What is the boundary of Spain? What is

the area of Spain?

When does a tumor begin its formation? What is the habitat of rabbits? What is the depth of the sea 30 km east of

Barcelona?


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Representing vagueness: Fuzzysets and fuzzy logic

The linguistic term tall does not refer to asharp demarcation of objects into twoclasses.

There are degrees of tallness.

Fuzzy set theorytreats Tallas a fuzzypredicate and says that the truth value ofTall(Carla) is a number between 0 and 1,rather than being just true orfalse.


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Examples

In classical sets, either an element belongsor it does not. Examples: Set of integers a real number is an integer or not. You are either in an airplane or not. Your bank-account balance is 142 or not. The Bible is either the word of God, or it is not. Either Christ was divine, or he was not.

Fuzzy sets are sets that have gradations ofbelonging. Examples:

BIGNear

Tall


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Degrees of truth: example

The temperature is pleasant.

Variable: temperature Universe of values (U) ordomain: real

The distribution indicates thedegree of truth of thelinguistic term pleasant.

1

00 20 30 40 C


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Degrees of truth: example

The temperature is low.

The temperature is high.

0

1

5 10 15 20 25 30 35 C

low

0

1

5 10 15 20 25 30 35 C

high


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Possibility function

A : U [0, 1]

The possibility function, orcharacteristic

function, is often represented as atrapezoidalortriangularapproximation,which can be characterized by theabscises of the vertexes.

1

00 20 30 40

1

0 0 20 30 40

(0, 20, 30, 40)


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Degrees of fever: Fever is Low(37,37,37.6,38) Fever is Medium (37.6,38, 38.5,39)

Fever is High (38.5,39,43,43)

0

37

38

39

43

1.0

0.7

0.3

0.0

temperature

fever(temperature)

low medium high

Practical representation of fuzzysets


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[Temperature is Gelada]

[Temperature is Molt Freda]

[Temperature is Freda]

[Temperature is Fresca]

[Temperature isAgradable] [Temperature is Calorosa]

[Temperature is MoltCalorosa]

0 5 10 15 20 25 30 35

1

0

G MF F FS A C MC

Each variable has a domain and a set oflabels or linguistic terms. Each label corresponds to a characteristicfunction defined over the domain.



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-10 -7,5 0 -7,5 10

BM BP GI PP PM

-5 -1 0 1 5

PN GZ PP

s[VC isPoc Positiva]s[VC is Gaireb Zero]

s[VC isPoc Negativa]

Control variable

s[T Puja Molt]s[T Puja Poc]s[T Gaireb Igual]

s[T Baixa Poc]s[T Baixa Molt]

Variation of temperature



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Fuzzy logic: connectives

Fuzzy logical connectives are defined asfunctions in the [0,1] interval, whichgeneralize classic connectives:

Intersection of sets P Q T-norm (P,Q)Union of sets P Q T-conorm (P,Q)Complement of a set P Negation function (P)

Inclusion of sets P Q Implication function (P,Q)

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Fuzzy negation

Negation function N : [0,1] [0,1]

Properties

N(0) = 1 and N(1) = 0 boundary conditions N(p) N(q) if p q monotony N(N(p)) = p involution

Examples N(x) = 1-x Nw(x) = (1-xw)1/w w > 0 Yager family Nt(x) = (1-x) / (1+t*x) t > -1 Sugeno family

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Fuzzy conjunction

T-Norms T : [0,1] x [0,1] [0,1]

Properties T(p,q) = T(q,p)

commutability T(p,T(q,r)) = T(T(p,q),r) associativity T(p,q) (r,s) if p r q s monotony T(0,p) = T(p,0) = 0 absorbing element T(1,p) = T(p,1) = p neutral element

Examples T(x,y) = min (x,y) minimum T(x,y) = x*y algebraic product T(x,y) = max (0, x+y-1) bounded difference15


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Relations between T-norms andT-conorms

De Morgan laws:

N(T(x,y)) = S(N(x),N(y))

N(S(x,y)) = T(N(x),N(y))

T(x,y) = min (x,y) S(x,y) = max (x,y) T(x,y) = x*y S(x,y) = x+y - x*y T(x,y) = max (0, x+y-1) S(x,y) = min (x+y,1)

T(x,y) = x*y / (x+y-x*y) S(x,y) = (x+y -2x*y) / (1-x*y)

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Fuzzy connectives over thesame universe

If F [X is A] and G [X is B] withpossibility distributions A and B definedover the same universe U:

F G [X is A B] with AB(u) = T(A(u) ,B(u)) T-norm

F G [X is A B] with A B(u) = S(A(u) ,B(u)) T-conorm

F [X is A] with A(u) = N(A(u)) Negation function

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Fuzzy connectives over thesame universe


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Fuzzy connectives overdifferent universes

If F [X is A] and G [Y is B] withpossibility distributions A defined over U iB defined over V, UV:

F G [X is A] [Y is B] with AB(u,v) = T(A(u) ,B(v)) T-norm

F G [X is A] [Y is B] with A B(u,v) = S(A(u) ,B(v)) T-conorm

Functions are defined in two differentdimensions.

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Fuzzy connectives over differentuniverses


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Fuzzy connectives over differentuniverses: conjunction


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Fuzzy connectives over differentuniverses: conjunction


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Fuzzy connectives over differentuniverses: disjunction


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Fuzzy connectives over differentuniverses: disjunction


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Fuzzy inference with crisp data:connectives

Interpretation of connectives:

Negation: N(x) = 1-x Conjunction: fuzzy intersection = T-norms (T) Disjunction: fuzzy union = T-conorms (S)

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1. Evaluation of the antecedent of the rules

2. Evaluation of the conclusions of the

rules3. Combination of the conclusions of the

rules

4. Defuzzification

Fuzzy inference with crispdata: phases


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The membership degree of each atom of the antecedent iscalculated and combined according to the interpretationof the connectives.

Phase 1: Evaluation of rulesantecedents

-10 -7,5 0 -7,5 10

BM BP GI PP PM

0 5 10 15 20 25 30

1 G MF F FS A CMC

26 8,5

min

0,2

0,4 0,40,2

Ex.: R1: If T = MC and T = PM then ...

Input: T = 26C, T = 8.5C


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The fuzzy set of the conclusion is truncated with thevalue coming from antecedent calculations.

Phase 2: Evaluation of rulesconclusions

0,2

-5 -1 0 1 5

PN

R1. If ... then VC = PN

Control Variable

0,2

-5 -1 0 1 5

Control Variable

Control variable

s [Poc Positiva]

s [Gaireb Zero]s [Poc Negativa]


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Phase 3: Combination of rulesconclusions

Conclusions are

always

disjunctively combined.

0,2

-5 -1 0 1 5

Control Variable

-5 -1 0 1 5

min

0,4

0,8

0,4

-5 -1 0 1 5

R1

R2

R3...

R40,6

-5 -1 0 1 5


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Defuzzification

xD

(x) x

xD (x)

Phase 4: defuzzification

-5 -1 0 1 5 -5 1.5 0 1 5

mass center


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Inferncia difusa amb dadesdifuses

Tenim una base de regles on els toms sonetiquetes difuses [X = A]

Si Temperatura = Agradable llavors Gir = Una mica a ladreta

Volem fer inferncia: amb encadenament cap endavant; a partir detiquetes difuses per les variables.


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Inferncia difusa amb dadesdifuses: Fases

Avaluaci de la relaci expressada en cadaregla (grau o funci dimplicaci)

I(x,y)

3. Avaluaci de la funci de pertinena delantecedent

A(x)

Composici inferencial (Modus ponens) de laimplicaci i lantecedent

CI(x,y) = f(I(x,y), A(x))


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Inferncia difusa amb dadesdifuses: Implicaci

Funcions dimplicaci, I : [0,1] x [0,1] [0,1]

PropietatsI(p,q) s creixent respecte a la primera variable i

decreixent respecte a la segona variableI(0,p) = 1I(1,p) = pI(p,I(q,r)) = I(q,I(p,r)) (intercanvi dantecedents)R-implicacions / S-implicacions

S-implicacionsa b = a bI(x,y) = S(N(x),y)Exemples de S-implicacionsI(x,y) = mx (1-x,y)I(x,y) = 1-x+x*yI(x,y) = mn (1-x+y,1)


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Les inferncies en lgica difusa utilitzen el modus ponens com a

regla bsica:

Si [X s A] llavors [Y s B ]

[X s A]

[Y s B ]

La combinaci de funcions de possibilitat es realitza mitjanantla funci generadora de modus ponens.

Inferncia difusa amb dadesdifuses: Modus ponens


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Emprant algunes de les definicions anteriors dimplicaci i el modusponens, s possible definir el procs dinferncia difusa:

p (p q) q

Donades les certeses dun antecedent i de la implicaci, la determinacide la certesa del conseqent es realitza en base a una funci generadoradel modus ponens que denominem mod().

La funci mod() t una srie de propietats, algunes de les quals sn lessegents:a) mod(u, imp(u, v)) v

b) mod(1, 1) = 1

c) mod(0, u) = v

d) u v mod(u, w) mod(v, w)

a) La funci mod() t com a

cota superior la certesa delconseqent.

b) Aquest s el lmit boole del

modus ponens

c) Dun antecedent

completamentfals es pot concloure qualsevolcosa.

d) La funci mod() ha de ser

montona creixent amb lacertesa

de lantecedent.

Modus ponens


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Funciones generadoras del modus ponens que resultan de lasdefiniciones de la funcin implicacin presentadasanteriormente:

Modus ponens


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La lgica difusa gaudeix de lavantatge de poder fer inferncies

sense tenir exactament lantecedent duna regla:

Si [X s A] llavors [Y s B]

[X s A]

[Y s B ]

Sha de calcular com es modifica la conclusi, ja que no tenim elmateix antecedent de la regla. Aix es fa mitjanant la

transformaci de la funci de possibilitat de A en la de A

Generalitzaci del modus ponens

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13 Fuzzy Logic (Us)