Introduction to Fuzzy Logic

Adnan YazıcıDept. of Computer Engineering,

Middle East Technical UniversityAnkara/Turkey

Introduction¾ “Mathematics that refers to reality is not certain and mathematics that is

certain does not refer to reality”Albert Einstein

¾ “While the mathematician constructs a theory in terms of ´perfect´objects, the experimental observes objects of which the properties demanded by theory are and can, in the very nature of measurement, be only approximately true”

Max Black

¾ “What makes society turn is science, and the language of science is math, and the structure of math is logic, and the bedrock of logic is Aristotle, and that is what goes out with fuzzy logic”

Bart Kosko

Introduction (cont.)

Introduction (cont.)

Introduction (cont.)� Data* refers to the results of experience, observation or experiment,

or a set of premises. This may consist of numbers, words, or images, particularly as measurements or observations of a set of variables.

� Information* as a concept bears a diversity of meanings, from everyday usage to technical settings.

� Knowledge* is defined (Oxford English Dictionary) variously as (i) expertise, and skills acquired by a person through experience or education; the theoretical or practical understanding of a subject, (ii) what is known in a particular field or in total; facts and information or (iii) awareness or familiarity gained by experience of a fact or situation.

*: Ref: Wikipedia

Introduction (cont.)� Knowledge is information at a higher level of abstraction.

Ex: Ali is 20 years old (fact)Ali is young (knowledge)

¾ Problems in many nontraditional applications need the following:� Decision� Management� Prediction

¾ Solutions are:� Faster access to more information and of increased aid in

analysis� Understanding – utilizing information available� Managing with information not avaliable

Introduction (cont.)� Uncertainty is produced when a lack of information exists. � The complexity also involves the degree of uncertainty.� It is possible to have a great deal of data (facts collected from

observations or measurements) and at the same time lack of information (meaningful interpretation and correlation of data that allows one to make decisions.)

Data Information

Database Intelligent information systems

ÆKnowledge & Intelligence

ÆKnowledge base & AI

Introduction (cont.)¾ Large amount of information with large amount of

uncertainty lead to complexity.

¾ Avareness of knowledge (what we know and what we do not know) and complexity goes together.

Example: Driving a car is complex, driving in a very icy and steep road is even more complex, since more knowledge is needed for driving in an iced road.

Introduction (cont.)

Fuzziness refers to vagueness and uncertainty, in particular to the vagueness related to human language and thinking.

• “… the set of heavy people.”• “… all people living very close to my home.”• “… all areas that are somewhat suitable for

growing corn.”• “… ”

Introduction (cont.)¾ Fuzzy logic provides a systematic basis for representation

of uncertainty, imprecision, vagueness, and/or incompletenes.

� Uncertain information: Information for which it is not possible to determine whether it is true or false. Ex: a person is “possibly 30 years old”

� Imprecise information: Information which is not available as precise as it should be. Ex: A person is “around 30 years old.”

� Vague information: Information that is inherently vague. Ex: A person is “young.”

Introduction (cont.)� Incomplete information: information for which data is

missing or data is partially available. Ex: A person’s age is “not known” or a person is “between 25 and 32 years old”

� Inconsistent information: Information containing two or more assertions that cannot be true at the same time. Ex: “Ali is 16” and “Ali is older than 17”

� Combination of the various types of such information may also exist.Ex: “possibly young”, “possibly around 30”, etc.

Introduction (cont.)

"As the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance become almost mutually exclusive characteristics."

Principle of Incompatibility (Zadeh, 1973)

Introduction (cont.)

UNCERTAINTY(Uncertainty-based information)

COMPLEXITY CREDIBILITY

USEFULNESS

Introduction (cont.)Example: When uncertainties like heavy traffic, unfamiliar

roads, unstable wheather conditions, etc. increase, the complexity of driving a car increases.

How do we go with the complexity?� We try to simplify the complexity by making a

satisfactory trade-off between information available to us and the amount of uncertainty we allow.

� We increase the amount of uncertainty by replacing some of the precise information with vague but more useful information.

Introduction (cont.)

Introduction (cont.)

Precise but IrrelevantParking a car:1. set the steering wheel to 0.00 degrees, go back,2. turn the steering wheel 12.8 degrees

counterclockwise,3. stop, you are 20.0 cm from the car on your left,4. pull 78 cm forward . . .

Introduction (cont.)

Imprecise (Ambiguous) but RelevantParking a car:1. go straight back,2. turn your steering wheel slightly left,3. now you are a bit too close to the car

next to you,4. pull a little forward . . .

Introduction (cont.)More Examples: � Travel directions: try to do it in mm terms (or turn the

wheel % 23 left, etc.), which is very precise and complex but not very useful. So replace mm information with city blocks, which is not as precise but more meaningful (and/or useful) information.

� Describing wheather of a day: try to do it in % cloud cover, which is very precise and complex but not very useful. So replace % cloud information with vague terms (very cloudy, sunny etc.), which is not as precise but more meaningful (or useful) information.

Introduction (cont.)

Level of Detail

Introduction (cont.)

1. Rigorous versus approximate modeling.2. Detailed models are not necessarily accurate.3. Precision costs time and money

(drop some, but when and how much?).

Fuzzy techniques worthwhile also for data analysis (data mining).

Fuzzy Logic¾ Fuzzy logic has been used for two different

senses:� In a narrow sense: refers to logical system

generalizing crisp logic for reasoning uncertainty.� In a broad sense: refers to all of the theories and

technologies that employ fuzzy sets, which are classes with imprecise boundaries.

�The broad sense of fuzzy logic includes the narrow sense of fuzzy logic as a branch.

Fuzzy Logic (cont.)� A possible application areas for the use of Fuzzy Logic

include:� fuzzy control, � fuzzy pattern recongnition, � fuzzy arithmetic, � fuzzy probability theory, � fuzzy decision analysis, � fuzzy databases, � fuzzy expert systems, � fuzzy computer SW and HW, etc.

Fuzzy Logic (cont.)�Control: “If the temperature is very high and the presure

is decreasing rapidly, then reduce the heat significantly.”�Database: “Retrieve the names of all candidates that are

fairly young, have a strong background in algorithms, and a modest administrative experience.”

�Medicine: Hepatitis is characterized by the followingstatement, ‘Total proteins are usually normal, albumin is decreased, α-globulins are slightly decreased, β-globulins are slightly decreased, γ-globulins are increased’

�More..

Fuzzy Logic (cont.)

With Fuzzy Logic, one can accomplish two things:

� Ease of describing human knowledge involving vague concepts

� Enhanced ability to develop a cost-effective solution to real-world

In another word, fuzzy logic not only provides a cost effective way to model complex systems involving numeric variables but also offers a quantitative description of the system that is easy to comprehend.

Inference Mechanism

Washing Machine Example

Fuzzy Logic (cont.)Fuzzy Logic was motivated by two objectives:

� First objective, it aims to alleviate difficulties in developing and analyzing complex systems encountered by conventional mathematical tools. This motivation requires fuzzy logic to workin quantitative and numeric domains.

� Second, it is motivated by observing that human reasoning can utilize concepts and knowledge that do not have well defined, sharp boundaries (i.e., vague concepts). This motivation enablesfuzzy logic to have a descriptive and qualitative form. This is related to AI.

Fuzzy Logic (cont.)

Components of Fuzzy Logic

� Fuzzy Predicates: tall, small, kind, expensive,...� Predicates modifiers (hedges): very, quite, more or less,

extremely,..� Fuzzy truth values: true, very true, fairly false,...� Fuzzy quantifiers: most, few, almost, usually, ..� Fuzzy probabilities: likely, very likely, highly likely,...

Fuzzy Logic (cont)

� The behaviour of a fuzzy system is completely deterministic.

� Fuzzy logic differs from multivalued logic by introducing concepts such as linguistic variables and hedges to capture human linguistic reasoning.

Probability theory vs fuzzy set theory� Probability measures the likelihood of a future

event, based on something known now. Probability is the theory of random events.

� Fuzziness is the uncertainty resulting from imprecision of meaning of a concept expressed by a linguistic term in NL, such as “tall” or “warm” etc.

� Probability is not capable of capturing uncertainty resulting from vagueness of linguistic terms.

� Fuzziness is not the uncertainty of expectation.

Probability theory vs fuzzy set theory (cont.)

n Probability gives us an indication about the likelihood that an event will occur. Whether it is going to happen or not, it is not known.

n Fuzziness is an indication to what degree something belongs to a class. We know that it exists. What we do not know, however, is its extent, i.e., to which degree members of a given universe belong to the class.

Probability theory vs fuzzy set theory (cont)

�Whereas probability theory makes statements about a collection of objects from which one is selected; therefore, modeling global uncertainty.

� Fuzzy set theory makes statements about one concrete object; therefore, modeling local vagueness,

� Fuzzy logic and probability complement each other. Ex: “highly probable” is a concept that involves both randomness and fuziness.

Fuzzy Logic Concepts¾ Even though the broad sense of fuzzy logic covers a wide range

of theories and techniques, its core technique is based on four basic concepts:

1. Fuzzy sets: sets with smooth boundaries;2. Linguistic variables: variables whose values are both qualitatively

and quantitatively described by a fuzzy set;3. Possibility distribution: constraints on the value of a linguistic

variable imposed by assigning it a fuzzy set; and4. Fuzzy if-then rules: a knowledge representation scheme for

describing a functional mapping (fuzzy mapping rules) or a logical formula that generalizes an implication in two-valued logic (fuzzy implication rules).

9 The first three concepts are fundamental for all subareas in fuzzy logic, but the fourth one is also important.

Fuzzy Set� Mathematically speaking, a fuzzy set is

characterized by mapping from its universe of discourse into the interval, [0,1].

� Each fuzzy (sub-) set is defined in terms of a relevant universal set U by a membership function, denoted as μA(u).

� The universe is not fuzzy!� Formally, membership functions are the functions of

the formμA: U Æ [0,1] is called the membership function of A.

� The set A={(u, μA(u)) | u∈U} is called a fuzzy set in U.

Membership function

Membership functionn The membership function must be a real valued

function whose values are between 0 and 1.n The membership values should be 1 at the center of

the set, i.e., for those members that definitely belong to the set.

n The membership function should fall off in an appropriate way from the center through the boundary.

n The points with membership value 0.5 (crossover point) should be at the boundary of the crisp set, i.e., if we would apply a crisp classification, the classboundary should be represented by the crossover points.

Membership function

Choice of membership functionn The membership function depends on the

application.

n Example: moderate elevation may be defined differently in the Netherlands than in Tibet.

Fuzzy Sets (cont.)

0 50 100 150 100 250

Triangle Trapezoid Gaussian1

0.5

0

Types of membership functions

The most commonly used membership functions in practice are triangles, trapezoids, bell curves, Gaussian, and sigmoid functions.

� Triangular membership function is specified by three parameters {a,b,c}as follows:

� Trapezoidal membership function is specified by four parameters {a,b,c,d} as follows:

� A Gaussian membership function is specified by two parameters {m,δ) as follows:

Gaussian (x:m,δ) = exp (-(x-m)2/δ2)

where m and δ denote the center and width of the function, respectively. We control the shape of the function by adjusting the parameter δ. A small δ will generate a “thin” membership function, while a big δ will lead to a “flat” membership function.

Introduction (cont.)

Introduction (cont.)

Introduction (cont.)

How to Get Membership Functions?1. prior knowledge: expert(s), designer, user, etc. Interview

those who are familiar with the underlying concepts and later adjust it based on a tuning strategy.

2. statistical information (conditional probabilities)3. comparison of data to prototypical objects. Learn it based

on feedback from the system performance.

4. fuzzy clustering (automatic search for prototypes), construct it automatically from data.

Fuzzy Sets (cont.)

The guidelines for membership function design� Use parameterizable functions that can be defined by a

small number of parameters. Parameterizablemembership functions reduce the system design time and facilitate the automated tuning of the system.

� The parameterizable membership functions most commonly used in practice are the triangular and trapezoidal membership functions, because of their simplicity.

� If you want to learn the membership function using neural network learning techniques, choose a differentiable (or even continuous differentiable) membership function (e.g., Gaussian).

Fuzzy Sets (cont.)

Designing antecedent membership functions¾ The membership functions of an input variable’s

fuzzy sets should usually be designed in a way that the following conditions are satisfied:� Unless there is a good reason, use symmetric membership

functions. This guideline has an additional benefit from the viewpoint of stability analysis.

� Each membership function overlaps only with the closest neighboring membership functions;

Ai ∩ Aj = ∅ ∀ j ≠ i, j+1, i-1, where Ai are fuzzy sets.� For any possible input data, its membership values in all

relevant fuzzy sets should sum to one (1) (or nearly so), ∑i μAi (x) ≅ 1.

Fuzzy Sets (cont.)

Fuzzy Sets (cont.)

� αA = {u | μA(u) ≥ α} is called α-cut.α1A ⊆ α2A and α1+A ⊆ α2+A, when α2≤ α1, which implies that the set of all distinct α-cuts (as well as strong α-cuts) is always a nested family of crisp sets.

� α+A = {u | μA(u) > α} is called strong α-cut.

Fuzzy Sets (cont.)

� 0+A = {u | μA(u) > 0} is called support of A.� 1A = {u | μA(u) = 1} is called core of A.� When the core of A is not empty, A is

called normal; otherwise, it is called subnormal.

� The largest value of A is called the heightof A, denoted as hA.

� The set of distinct values of μA(u),∀u∈ U is called the level set of A and denoted as ΛA.

Fuzzy Sets (cont.)

Core

α2A

α1A

α1

α2

1

μA(u)

u

hA

(A is normal)

Fuzzy Sets (cont.)• The significance of α-cut representation of fuzzy sets is that it

connects fuzzy sets with crisp sets.• While each crisp set is a collection of objects that are

conceived as a whole, each fuzzy set is a collection of nested crisp sets that are also conceived as a whole.

• Fuzzy sets are thus wholes of a higher category.

Example: A = 0.2/x1 +0.4/x2+0.6/x3+0.8/x4+1/x5• Its level set is ΛA = {0.2,0.4,0.6, 0.8,1}, so it is associated with

only 5-distinct α-cuts, which are defined as follows:0.2A = 1/x1+1/x2+1/x3+1/x4+1/x50.4A = 0/x1+1/x2+1/x3+1/x4+1/x50.6A = 0/x1+0/x2+1/x3+1/x4+1/x50.8A = 0/x1+0/x2+0/x3+1/x4+1/x51A = 0/x1+0/x2+0/x3+0/x4+1/x5

Fuzzy Sets (cont.)

� Theorem: (Decomposition theorem of fuzzy sets): For any A ∈ F(X),

A = ∪α∈[0,1] αA � We now convert each of the α-cuts to a special fuzzy set

αA defined for each u∈A by the formula αA = α.αμA(u). We obtain the following results:0.2A = 0.2/x1+0.2/x2+0.2/x3+0.2/x4+0.2/x5

0.4A = 0/x1+0.4/x2+0.4/x3+0.4/x4+0.4/x5

0.6A = 0/x1+0/x2+0.6/x3+0.6/x4+0.6/x5

0.8A = 0/x1+0/x2+0/x3+0.8/x4+0.8/x5

1A = 0/x1+0/x2+0/x3+0/x4+1/x5

� The union of these five special fuzzy set is exactly the original fuzzy set A, that is, A = 0.2A ∪ 0.4 A ∪ 0.6 A ∪ 0.8 A ∪ 1A

A = 0.2/x1 +0.4/x2+0.6/x3+0.8/x4+1/x5

Fuzzy Sets (cont.)

� Any property of fuzzy sets that is derieved from classical set theory is called a cutworthy property.

Examples:A = B iff μA(u) = μB(u), ∀u ∈U, similarly,A = B iff αA = αB, ∀α ∈[0,1] A ⊆ B iff αA ⊆ αB, ∀α ∈[0,1]

� The convexity of fuzzy sets: A fuzzy set defined on the set of real numbers (or more generally, on any n-dim Euclidean space) is said to be convex iff all of its α-cuts are convex in the classical sense. For a fuzzy set to be convex the graph must have just one peak.

convex non convex

Basic operations on Fuzzy Sets� Set union: A ∪ B ⇔ { u, μA∪B(u) | (u∈A ∨

u∈B) ∧ μ(A∪B) (u) = Max (μA(u), μB(u))}

Basic operations on Fuzzy Sets

Basic operations on Fuzzy Sets

� Set intersection: A∩B⇔ {u, μA∩B (u)| (u∈A ∧u∈B) ∧ μ(A∩B) (u) = Min (μA(u), μB(u))}

Basic operations on Fuzzy Sets

Basic operations on Fuzzy Sets� Set Complement:� ¬Α= {u,μ¬A (u) | (μ¬A (u) = (1- μA(u))}

� Set equality: A=B⇔ {u,μA (u)|(u∈A ∧ u∈B) ∧ μA(u) = μB(u)}

� Set containment: A⊆B⇔ {u|∀u(u∈A→u∈B) ∧ μA(u) ≤ μB(u)}

� Concentration: CON(A)={u,μCON(A)(u)(u∈A ∧ μCON(A)(u)= (μA(u))2}

� Dilation: DIL(A) = {u, μDIL(A)(u) | (u∈A ∧μ DIL(A)(u) = (μA(u))1/2}

Basic operations on Fuzzy Sets

Fuzzy Sets (cont.)

Fuzzy Sets (cont.)

Fuzzy Sets (cont.)n Linguistic variable: A linguistic

variable is a variable that assumes linguistic values (linguistic terms).

Examplen Linguistic Variable: heightn Linguistic Values: short, average-

height, tall

Fuzzy Sets (cont.)

n Hedgesn hedge h functions as modifier of a

meaning of a term x, thus resulting in acomposite term hx, e.g., “very steep.”

n Examples for hedges are “very”, “sort of”, “slightly”, etc.

n They are implemented with operators on fuzzy sets.

Fuzzy Sets (cont.)

μMore-or-Less High u)=0.9

μHigh (u)= 0.80

μVery High (u)=0.3

u

Very High

High

More-or-Lessμ

μvery A(u) ≤ μA(u) ≤ μMore-or-Less A(u)u=28°C

Fuzzy Sets (cont.)

1

0.8

0.45

0.40.3

0.2

00.8 (for u) 1

Very True

True

False

μtv(a)

Fairly False Fairly TrueVery False

Absolutley TrueAbsolutley False

0

Fuzzy Sets (cont.)� Failing of the Law of Excluded Middle and the

Law of Contradiction� Fuzzy set theory generalizes the black-and-white

situations of set membership to allow various “gray areas”.

� It violates two fundamental laws of set theory� The law of excluded middle, A ∪⎯A = U� The law of contradiction, A ∩⎯A = Φ

� However, any element partially belonging to a fuzzy set is also partial member of its complement.For example, μBald (Ali) = 0.2 and μ∼Bald (Ali) = 0.8.

� So, the law of excluded middle and the law of contradiction are not axioms of fuzzy set theory.

Extension Principle� In order to develop computation with fuzzy sets, we

need to take crisp functions and fuzzify them. A principle for fuzzyfying crisp functions is called the Extension Principle.

f: XÆY, where X and Y are crisp sets. � We say that the function is fuzzified when it is

extended to act on fuzzy sets defined on X and Y. Formally, the fuzzified function, f’, has the form

f’: F(X) Æ F(Y), where F(X) and F(Y) denote the fuzzy power set (the set of all fuzzy subsets) of X and Y, respectively.

� To qualify as a fuzzified version of f, function f’ must conform to f within the extended domain F(X) and F(Y). This is guaranteed when a principle is employed that is called the extension principle.

Extension Principle

�According to this principle,B(y) = f’(A) is determined for any given fuzzy set A∈F(X) via the following formula:

B(y) = maxx|y=f(x)A(x)= maxx|y=f(x)μA(x)/f(x), ∀y ∈Y.

�When the maximum does not exist, it is replaced with the supremum.

� Fuzzy set B captures the meaning of the specified linguistic expression.

Fuzzy Sets (cont.)

Example: Employees’ ages and their salaries

Example Query: What is a young employee’s salary?

Answer: We use the extension principle here. Let us have a function f: XÆ Y, where X = {20,25,30,35,40,45,50,55,60,65} and

Y= {2.5, 3, 3.5, 4.0, 4.5, 5.0}.

Age in years 20 25 30 35 40 45 50 55 60 65

Salary $ in K 2.5 2.5 3.0 3.5 3.5 4.0 4.0 4.5 4.5 5.0

Fuzzy Sets (cont.)

� First step: Formulate the meaning of the concept ‘young’ as a fuzzy set A of general form A = μA(x) / x for all x ∈X. Assume that Ayoung=1/20+1/25+0.8/30+0.6/35+0.4/40+0.2/45+0/50+0/55+0/60+0/65

� Second step: Use the fuzzy set A and information in the table to determine an appropriate fuzzy set B that captures the meaning of the linguistic expression young employee’s salary.

� This fuzzy set is dependent on A via function f which for each x in X assigns a particular y = f(x) in Y. This dependency is expressed by the general form

B(y) = max x|y=f(x) A(x) = max x|y=f(x) μA(x) / f(x)

μA(x) / f(x) = 1/f(20) + 1/f(25) + 0.8/f(30) + 0.6/f(35) + 0.4/f(40) + 0.2/f(45) + 0/f(50) + 0/f(55) + 0/f(60) + 0/f(65)

=1/2.5+1/2.5+0.8/3+0.6/3.5+0.4/3.5+0.2/4+0/4+0/4.5+ 0/4.5+0/5

� Third step: B(y) = max x|y=f(x) A(x) = 1/2.5 +0.8/3+0.6/3.5+0.2/4+0/4.5+0/5, which denotes the salary of young employes in the company.

Fuzzy Sets (cont.)

� The inverse function, f’-1, is from F(Y) to F(X).f’-1 : F(Y) Æ F(X).

� According to the extension principle, for any B∈F(Y),[f’-1(B)] (x) = B(f(x))/x = B(y), ∀x ∈X,

where y= f(x).

Fuzzy Sets (cont.)

� Example Query: Now let us answer the query: Who are (ages of) employees with low salary?

Answer: � First, assume that

Blow = 1/2.5 + 0.75/3 + 0.5/3.5 + 0.25/4 + 0/4.5 + 0/5[f’-1(B)] (x) = B(f(x))/x =

B(f(20))/20+B(f(25))/25+B(f(30))/30+B(f(35))/35+ B(f(40))/40+B(f(45))/45+B(f(50))/50+B(f(55))/55+ B(f(60))/60+ B(f(65))/65

=1/20+1/25+0.75/30+0.5/35+0.5/40+0.25/45+0.25/50+0/55+0/60+0/65

� This fuzzy set is defined on X and represents the age of employees with low salaries.

Age in years 20 25 30 35 40 45 50 55 60 65

Salary $ in K 2.5 2.5 3.0 3.5 3.5 4.0 4.0 4.5 4.5 5.0

Fuzzy Sets (cont.)

A(x) B(x)

Age Salary

0 0.25 0.5 0.75 11 0.8 0.6 0.4 0.2 0 2.5

3.0

3.5

4.0

4.5

5.0

20

25

30

35

40

45

50

55

60

65

Illustration of the extension principle

¾ A linguistic variable enables its value to be described both qualitatively by a linguistic term (i.e., a symbol serving as the name of a fuzzy set) and quantitatively by a corresponding membership function, (which express the meaning of the fuzzy set).

¾ For example, if TradingQuantity is Heavy, the fuzzy set Heavy describes the quantity of the stock market trading in one day. The variable TradingQuantitydemonstrates the linguistic variable.

Linguistic Variables

¾ A linguistic variable is like a composition of a symbolic variable (whose value is a symbol, e.g., Shape is Cylinder)) and a numeric variable (whose value is a number, e.g., Height = 1.80 m)).

¾ Using the notion of the linguistic variable to combine these two kinds of variables into a uniform framework is, in fact, one of the main reasons that fuzzy logic has been successful in offering intelligent approaches in engineering and many other areas that deal with continuous problem domains.

Linguistic Variables (cont.)

¾ A possibility distribution, Π, maps a given domain of definition into the interval [0,1].

¾ We can view a possibility distribution as a mechanism for interpreting factual statements involving fuzzy sets.

¾ Example: the statement, “Temperature is High”, where High is defined as μHigh : T Æ [0,1], translates into a possibility distribution, Π(T) = μHigh (T).

¾ For more complex statement, “Temperature is High but not too high” translates into a possibility distribution in terms of conjunction of the terms High and Not VeryHigh:

Π(T)=min(μHigh(T),μNotVeryHigh(T))=min[μHigh(T),(1-μHigh(T))2].

Possibility Distributions

¾ Fuzzy logic offers an appealing alternative, such as assigningthe fuzzy set Young to the age of the suspect. Thus, we obtain a distribution about the possibility degree of the suspect’s age (e.g., the possibility that the suspect is 19 is 0.8, 20 is 0.9 while the possibility of 21 - 28 is 1.0, etc),

ΠAge(suspect) (x) = μYoung (x), where Π denotes a possibility distribution of the suspect’s age, and x is a variable representing a person’s age.

¾ Pos(A|ΠX) denotes the possibility of the condition “X is A”given the possibility distribution ΠX

¾ Nec(A|ΠX) denotes the necessity of the condition “X is A”given the possibility distribution ΠX.

Possibility Distributions (cont.)

¾ The possibility and necessity are two related measures:1a.Total necessity implies total possibility,

Nec(A|ΠX) =1ÆPos(A|ΠX) = 1(If a variable is necessarily A, then it is possibly A. The reverse is not true.)1b. No possibility implies no necessity,

Pos(A|ΠX) = 0 Æ Nec(A|ΠX) = 02a. A variable is not possible to be NOT A iff it is necessarily A

1- Pos(¬A|ΠX) = 1 ↔ Nec(A|ΠX) = 1,2b. Pos(¬A|ΠX) = 1 ↔ 1 - Nec(A|ΠX) = 1,

¾ we can review 2b as follows:2b’. 1 - Pos(¬A|ΠX) = 0 ↔ Nec(A|ΠX) = 0.

(A variable is possible to be NOT A iff it is not necessarily A)

Possibility Distributions

¾ These observations can provide insights on the general relationships between the two measures. The relationships 1a and 1b can be generalized to

Nec (A|ΠX) ≤ Pos(A|ΠX) ¾ The relationships 2a and 2b’ can be generalized to

1- Pos(¬A|ΠX) = Nec(A|ΠX). Thus, one can automatically derive necessity measure using a possibility measure.

¾ In general, when we assign a fuzzy set A to a variable X, the assignment results in a possibility distribution of X, which is defined by A’s membership function: ΠX (x) = μA (x).

Possibility Distributions (cont.)

¾ The possibility measure for a variable X to satisfy the condition “X is A” given a possibility distribution ΠX is defined to be

Pos(A|ΠX) = sup xi∈U [(μA ⊗ ΠX ),where ⊗ denotes a fuzzy intersection (i.e., a fuzzy conjunction) operator.

¾ A common choice of the fuzzy intersection operator for calculating the possibility measure is the min operator. Thus,

Pos(A|ΠX) = supxi∈U [min (μA (xi), ΠX (xi))].¾ The corresponding formula for the necessity measure:

Nec(A|ΠX) = infxi∈U [max (μA (xi), 1-ΠX (xi))].

Possibility Distributions (cont.)

Example: Let the universe of discourse of a person’s age be {10,15,20,25,30,35,40,45,50}, and

Let the age possibility distribution of a suspect (denoted J) be:ΠAge (J) = 0.2/15 + 0.5/20 + 1/25 + 0.8 /30

Suppose that the membership function for the linguistic term Young is defined as a discrete fuzzy set as follows:

μYoung(Age) = 1/10 + 1/15 + 1/20 + 0.8 / 25 + 0.4 /30 + 0.2 /35Using the equation;

Pos(μA|ΠX)=Pos(μYoung(Age) | ΠAge (J))=supxi∈U[(μYoung(Age)⊗ΠAge (J)]=Pos(μYoung(Age) | ΠAge (J)) = max {min (μYoung, ΠAge (J))}

= max {0.2∧1, 0.5∧1, 1∧0.8, 0.8∧0.4}= max {0.2, 0.5,0.8, 0.4}

Pos(μYoung |ΠAge (J)) = 0.8

Possibility Distributions

¾ To calculate the necessity measure, we first calculate the complement of the possibility distribution of a suspect J’s age:1-ΠAge(J)= 1/10+0.8/15+0.5/20+0/25+0.2/30+1/35+1/40+1/45+1/50μYoung(Age) = 1/10 + 1/15 + 1/20 + 0.8 / 25 + 0.4 /30 + 0.2 /35

¾ The necessity measure is obtained by Nec(A|ΠX)= infxi∈U [ μA(xi) ⊕ 1-ΠX (xi)]=Nec(μYoung(Age) |ΠAge (J)) = infxi∈U[max(μYoung(Age) , 1- ΠAge (J)]Nec(μYoung(Age) |ΠAge(J))=min{1∨1,1∨0.8,1∨0.5,0.8∨0,0.4∨0.2,

0.2∨1,0∨1,0∨1,0∨1}= min {1, 1, 1, 0.8, 0.4, 1, 1,1, 1} = 0.4.

Therefore, the possibility that suspect J being young is 0.8, while the necessity that he/she is young is 0.4.

Possibility Distributions (cont.)

¾ Possibility and probability measures are two different kinds of uncertainty.

¾ Possibility measures the degree of ease for a variable to take a value, while probability measures the likelihood for a variable to take a value.

¾ The notion of possibility distribution generalizes the notion of interval value to smooth the boundary between the possible values and impossible values so that possibility becomes a matter of degree.

¾ Possibility theory handles imprecision, and probability theory handles likelihood of occurence.

¾ This fundamental difference leads to different mathematical properties of their distributions.

¾ Being a measure of occurence, the probability distribution of a variable must add up to exactly 1.

¾ The possibility distribution, however, is not subject to this restriction since a variable can have multiple possible values.¾ Ex: ΠHeight of a suspect(J) = somewhat tall

Possibility versus Probability

¾ There are two different kinds of fuzzy rules: Fuzzy mapping rules and Fuzzy implication rules.

¾ A fuzzy mapping rule describes an association; therefore, its fuzzy relation is constructed from the Cartesian product of its antecedent fuzzy condition and its consequent fuzzy condition.

¾ A fuzzy implication rule, however, describes a generalized logical implication; therefore, its fuzzy relation needs to be constructed from the semantics of a generalization to implication in multi-valued logic.

Fuzzy If-Then Rules

¾ The difference between the semantics of fuzzy mapping rules and fuzzy implication rules can be seen from the difference in th denotes the necessity of the condition “X is A” given the possibility distribution ΠX eir inference behavior. Even though these two types of rules behave the same when their antecedents are satisfied, they behave differently when their antecedents are not satisfied.

Example:Implication rule, Mapping rule

¾ (logic representation) (procedural representation)Rule: x ∈ [1,3] Æ y ∈ [7,8], Rule: IF x∈[1,3] THEN y∈[7,8]Input: x=5 Variable value: x = 5Infer: y is unkown (y ∈ [0,10]) Execution result: no action

Fuzzy If-Then Rules