Introduction to Fuzzy Sets and Fuzzy .Introduction to Fuzzy Sets and Fuzzy Logic ... Introduction

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    Introduction to Fuzzy Sets and Fuzzy Logic

    Introduction to Fuzzy Sets and Fuzzy Logic

    Luca Spada

    Department of Mathematics and Computer ScienceUniversity of Salerno

    www.logica.dmi.unisa.it/lucaspada

    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

    A theorem about continuous t-norm

    www.logica.dmi.unisa.it/lucaspada

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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

    A theorem about continuous t-norm

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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.

    - Alfred North Whitehead

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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 sense

    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