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Introduction to Fuzzy Sets and Fuzzy Logic

Introduction to Fuzzy Sets and Fuzzy Logic

Department of Mathematics and Computer ScienceUniversity of Salerno

REASONPARK. Foligno, 17 - 19 September 2009

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Introduction to Fuzzy Sets and Fuzzy Logic

Contents of the Course

PART I

Introduction

Fuzzy sets

Operations with fuzzy sets

t-norms

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Introduction to Fuzzy Sets and Fuzzy Logic

Contents of the Course

PART II

What is a logic?

Propositional logic

Syntax

Semantics

Multiple truth values

Again t-norms

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Introduction to Fuzzy Sets and Fuzzy Logic

Contents of the Course

PART III

Many-valued logic

Standard completeness

Prominent many-valued logics

Functional interpretation

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Introduction to Fuzzy Sets and Fuzzy Logic

Contents of the Course

PART IV

Ulam game

Pavelka style fuzzy logic

(De)Fuzzyfication

Algebraic semantics

Algebraic Completeness

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Introduction to Fuzzy Sets and Fuzzy Logic

Part I

Fuzzy sets

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Introduction to Fuzzy Sets and Fuzzy Logic

Contents of part I

IntroductionWhat Fuzzy logic is?Fuzzy logic in broad senseFuzzy logic in the narrow sense

Fuzzy sets

Operations with fuzzy setsUnionIntersectionComplement

t-norms

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Introduction to Fuzzy Sets and Fuzzy Logic

Introduction

There are no whole truths; all truths are half- truths.It is trying to treat them as whole truths that plays thedevil.

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Introduction to Fuzzy Sets and Fuzzy Logic

Introduction

What Fuzzy logic is?

What Fuzzy logic is?

Fuzzy logic studies reasoning systems in which the notions oftruth and falsehood are considered in a graded fashion, incontrast with classical mathematics where only absolutly truestatements are considered.

From the Stanford Encyclopedia of Philosophy :

The study of fuzzy logic can be considered in twodifferent points of view: in a narrow and in a broadsense.

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Introduction to Fuzzy Sets and Fuzzy Logic

Introduction

Fuzzy logic in broad senseFuzzy logic in broad sense serves mainly as apparatus for fuzzycontrol, analysis of vagueness in natural language and severalother application domains.It is one of the techniques of soft-computing, i.e. computationalmethods tolerant to suboptimality and impreciseness(vagueness) and giving quick, simple and sufficiently goodsolutions.

Klir, G.J. and Yuan, B. Fuzzy sets and fuzzy logic: theory andapplications. Prentice-Hall (1994)

Nguyen, H.T. and Walker, E. A first course in fuzzy logic. CRC Press(2006)

Novak, V. and Novbak, V. Fuzzy sets and their applications. Hilger(1989)

Zimmermann, H.J. Fuzzy set theoryand its applications. KluwerAcademic Pub (2001)

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Introduction to Fuzzy Sets and Fuzzy Logic

Introduction

Fuzzy logic in the narrow sense

Fuzzy logic in the narrow senseFuzzy logic in the narrow sense is symbolic logic with acomparative notion of truth developed fully in the spirit ofclassical logic (syntax, semantics, axiomatization,truth-preserving deduction, completeness, etc.; bothpropositional and predicate logic).It is a branch of many-valued logic based on the paradigm ofinference under vagueness.

Cignoli, R. and DOttaviano, I.M.L. and Mundici, D. Algebraicfoundations of many-valued reasoning. Kluwer Academic Pub (2000)

Gottwald, S. A treatise on many-valued logics. Research Studies Press(2001)

Hajek, P. Metamathematics of fuzzy logic. Kluwer Academic Pub(2001)

Turunen, E. Mathematics behind fuzzy logic. Physica-VerlagHeidelberg (1999)

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Introduction to Fuzzy Sets and Fuzzy Logic

Fuzzy sets

Fuzzy sets and crisp sets

In classical mathematics one deals with collections of objectscalled (crisp) sets.Sometimes it is convenient to fix some universe U in whichevery set is assumed to be included. It is also useful to think ofa set A as a function from U which takes value 1 on objectswhich belong to A and 0 on all the rest.Such functions is called the characteristic function of A, A:

A(x) =def

{1 if x A0 if x / A

So there exists a bijective correspondence betweencharacteristic functions and sets.

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Introduction to Fuzzy Sets and Fuzzy Logic

Fuzzy sets

Crisp sets

Example

Let X be the set of all real numbers between 0 and 10 and letA = [5, 9] be the subset of X of real numbers between 5 and 9.This results in the following figure:

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Introduction to Fuzzy Sets and Fuzzy Logic

Fuzzy sets

Fuzzy sets

Fuzzy sets generalise this definition, allowing elements to belongto a given set with a certain degree.Instead of considering characteristic functions with value in{0, 1} we consider now functions valued in [0, 1].

A fuzzy subset F of a set X is a function F (x) assigning toevery element x of X the degree of membership of x to F :

x X 7 F (x) [0, 1].

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Introduction to Fuzzy Sets and Fuzzy Logic

Fuzzy sets

Fuzzy set

Example (Cont.d)

Let, as above, X be the set of real numbers between 1 and 10.A description of the fuzzy set of real numbers close to 7 couldbe given by the following figure:

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Introduction to Fuzzy Sets and Fuzzy Logic

Operations with fuzzy sets

Operations between sets

In classical set theory there are some basic operations definedover sets.Let X be a set and P(X) be the set of all subsets of X or,equivalently, the set of all functions between X and {0, 1}.The operation of union, intersection and complement aredefined in the following ways:

A B = {x | x A or x B} i.e. AB(x) = max{A(x), B(x)}A B = {x | x A and x B} i.e. AB(x) = min{A(x), B(x)}

A = {x | x / A} i.e. A(x) = 1 A(x)

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Introduction to Fuzzy Sets and Fuzzy Logic

Operations with fuzzy sets

Union

Operations between fuzzy sets: union

The lawAB(x) = max{A(x), B(x)}.

gives us an obvious way to generalise union to fuzzy sets.Let F and S be fuzzy subsets of X given by membershipfunctions F and S :

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Introduction to Fuzzy Sets and Fuzzy Logic

Operations with fuzzy sets

Union

Operations between fuzzy sets: union

We setFS(x) = max{F (x), S(x)}

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Introduction to Fuzzy Sets and Fuzzy Logic

Operations with fuzzy sets

Intersection

Operations between fuzzy sets: intersectionAnalogously for intersection:

AB(x) = min{A(x), B(x)}.

We setFS(x) = min{F (x), S(x)}

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Introduction to Fuzzy Sets and Fuzzy Logic

Operations with fuzzy sets

Complement

Operations between fuzzy sets: complementFinally the complement for characteristic functions is definedby,

A(x) = 1 A(x).

We setF (x) = 1 F (x).

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Introduction to Fuzzy Sets and Fuzzy Logic

t-norms

Operations between fuzzy sets 2

Lets go back for a while to operations between sets and focuson intersection.We defined operations between sets inspired by the operationson characteristic functions.Since characteristic functions take values over {0, 1} we had tochoose an extension to the full set [0, 1].

It should be noted, though, that also the product would do thejob, since on {0, 1} they coincide:

AB(x) = min{A(x), B(x)} = A(x) B(x).

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Introduction to Fuzzy Sets and Fuzzy Logic

t-norms

Operations between fuzzy sets 2

So our choice for the interpretation of the intersection betweenfuzzy sets was a little illegitimate.Further we have

AB(x) = min{A(x), B(x)} = max{0, A(x) + B(x) 1}

It turns out that there is an infinity of functions which havethe same values as the minimum on the set {0, 1}.This leads to isolate some basic property that the our functionsmust enjoy in order to be good candidate to interpret theintersection between fuzzy sets.

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Introduction to Fuzzy Sets and Fuzzy Logic

t-norms

t-normsIn order to single out these properties we look again back at thecrisp case: It is quite reasonable for instance to require thefuzzy intersection to be commutative, i.e.

F (x) S(x) = S(x) F (x),

or associative:

F (x) [S(x) T (x)] = [F (x) S(x)] T (x).

Finally it is natural to ask that if we take a set F bigger thanS than the intersection F T should be bigger or equal thanS T :

If for all x XF (x) S(x) then F (x)T (x) S(x)T (x)

.

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Introduction to Fuzzy Sets and Fuzzy Logic

t-norms

t-normsSumming up the few basic requirements that we make on afunction that candidates to interpret intersection are:

I To extend the {0, 1} case, i.e. for all x [0, 1].

1 x = x and 0 x = 0

I Commutativity, i.e., for all x, y, z [0, 1],

x y = y x

I Associativity, i.e., for all x, y, z [0, 1],

(x y) z = x (y z),

I To be non-decreasing, i.e., for all x1, x2, y [0, 1],

x1 x2 implies x1 y x2 y.

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Introduction to Fuzzy Sets and Fuzzy Logic

t-norms

t-normsObjects with such properties are already known in mathematicsand are called t-norms.

Example

(i) Lukasiewicz t-norm: x y = max(0, x+ y 1).(ii) Product t-norm: x y usual product between real num