Classical and Fuzzy Logic Be Autonomous Latest

CLASSICAL LOGIC AND FUZZY LOGIC

Dr S.Natarajan Professor Department of Information Science and Engineering PESIT, Bangalore

Part I Logic

Classical Predicate Logic – tautologies, Contradictions, Equivalence, Exclusive Or Exclusive Nor, Logical Proofs, Deductive InferencesFuzzy Logic, Approximate Reasoning, Fuzzy Tautologies, Contradictions, Equivalence and Logical Proofs, Other forms of the Implication Operation

Part II Fuzzy Systems

Natural language processing, Lingustic Hedges, Rule Based Systems, Multiple conjunctve antecedents , Aggregation of Fuzzy Rules, Graphical techniques of inference

Classical Logic

What is

LOGIC- Small part of Human body to reason

LOGIC- means to compel us to infer correct answers

What is

NOT LOGIC- Not responsible for our creativity or ability to

remember

LOGIC helps in organizing words to form words- not

context dependent

Fuzzy Logic

FUZZY LOGIC is a method to formalize humancapacity to Imprecise learning called ApproximateReasoning

Such reasoning represents human ability to reason approximately and judge under uncertainty

In Fuzzy Logic --- all truths are partial or approximate Here, the reasoning has been termed as Interpolative reasoning

September 5, 2006 Applied Discrete MathematicsWeek 1: Logic and Sets

Negation (NOT)Negation (NOT)

Unary Operator, Symbol: Unary Operator, Symbol:

truetrue falsefalse

falsefalse truetrue

Conjunction (AND)Conjunction (AND)

Binary Operator, Symbol: Binary Operator, Symbol:

PP QQ PPQQ

truetrue truetrue truetrue

truetrue falsefalse falsefalse

falsefalse truetrue falsefalse

falsefalse falsefalse falsefalse

Disjunction (OR)Disjunction (OR)

PP QQ PPQQ

truetrue falsefalse truetrue

falsefalse truetrue truetrue

Exclusive Or (XOR)Exclusive Or (XOR)

PP QQ PPQQ

truetrue truetrue falsefalse

truetrue falsefalse truetrue

Implication (if - then)Implication (if - then)

PP QQ PPQQ

falsefalse falsefalse truetrue

Biconditional (if and only if)Biconditional (if and only if)

PP QQ PPQQ

falsefalse truetrue falsefalse

falsefalse falsefalse truetrue

Statements and OperatorsStatements and OperatorsStatements and operators can be combined in any Statements and operators can be combined in any

way to form new statements.way to form new statements.

PP QQ PP QQ ((P)P)((Q)Q)

truetrue truetrue falsefalse falsefalse falsefalse

truetrue falsefalse falsefalse truetrue truetrue

falsefalse truetrue truetrue falsefalse truetrue

falsefalse falsefalse truetrue truetrue truetrue

Statements and OperationsStatements and OperationsStatements and operators can be combined in any way Statements and operators can be combined in any way

to form new statements.to form new statements.

PP QQ PPQQ (P(PQ)Q) ((P)P)((Q)Q)

truetrue truetrue truetrue falsefalse falsefalse

truetrue falsefalse falsefalse truetrue truetrue

falsefalse truetrue falsefalse truetrue truetrue

falsefalse falsefalse falsefalse truetrue truetrue

Equivalent StatementsEquivalent Statements

PP QQ (P(PQ)Q) ((P)P)((Q)Q) (P(PQ)Q)((P)P)((Q)Q)

truetrue truetrue falsefalse falsefalse truetrue

truetrue falsefalse truetrue truetrue truetrue

falsefalse truetrue truetrue truetrue truetrue

falsefalse falsefalse truetrue truetrue truetrue

The statements The statements (P(PQ) and (Q) and (P)P)((Q) are Q) are logically equivalentlogically equivalent, ,

because because (P(PQ)Q)((P)P)((Q) is always true.Q) is always true.

Conditional (Implication)Conditional (Implication)

This one is probably the least intuitive. It’s only This one is probably the least intuitive. It’s only partly akin to the English usage of “if,then” or partly akin to the English usage of “if,then” or “implies”.“implies”.

DEF: DEF: p p q q is true if is true if q q is true, or if is true, or if pp is false. In is false. In the final case (the final case (pp is true while is true while qq is false) is false) p p q q is false.is false.

Semantics: “Semantics: “pp implies implies q q ” is true if one can ” is true if one can mathematically derive mathematically derive q q from from pp..

Truth Tables

P Q P P Q P Q P Q PQ

False False True False False True True

False True True False True True False

True False False False True False False

True True False True True True True

Tautologies and ContradictionsTautologies and Contradictions

A tautology is a statement that is always true.A tautology is a statement that is always true.

Examples: Examples: RR((R)R)(P(PQ)Q)((P)P)((Q)Q)

If SIf ST is a tautology, we write ST is a tautology, we write ST.T.If SIf ST is a tautology, we write ST is a tautology, we write ST. This symbol T. This symbol

is also used for logical equivalence.is also used for logical equivalence.

Tautologies and ContradictionsTautologies and Contradictions

A contradiction is a statement that is alwaysA contradiction is a statement that is always

false.false.

Examples: Examples:

RR((R)R)

(((P(PQ)Q)((P)P)((Q))Q))

The negation of any tautology is a contra-The negation of any tautology is a contra-

diction, and the negation of any contradiction is diction, and the negation of any contradiction is

a tautology.a tautology.

TAUTOLOGIES

Tautologies – Compound Propositions which are ALWAYS TRUE , irrespective of TRUTH VALUES of INDIVIDUAL SIMPLE PROPOSITIONS

APPLICATIONS- DEDUCTIVE REASONING, THEOREM PROVING , DEDUCTIVE INFERENCING ETC.,Example: A is a set of prime numbers given by (A1 = 1 , A2 = 2, A3 = 3, A4 = 5, A5 = 7, A6 = 11 …) on

the real line universe X, then the proposition Ai is not divisible by 6 is A TAUTOLOGY

Proof by Contradiction

• A method for proving A method for proving p p qq..

• Assume Assume pp, and prove that , and prove that pp ( (qq qq))

• ((qq qq) is a trivial contradiction, equal to ) is a trivial contradiction, equal to FF

• Thus Thus ppFF, which is only true if , which is only true if pp==FF

Contradiction Proof Example

• Definition:Definition: The real number The real number rr is is rational rational if there if there exist integers exist integers p p and and q q ≠≠ 0, 0, with no common factors with no common factors other than 1 (i.e., gcd(other than 1 (i.e., gcd(pp,,qq)=1), such that )=1), such that r=p/q.r=p/q. A A real number that is not rational is called real number that is not rational is called irrational.irrational.

• Theorem:Theorem: Prove that is irrational. Prove that is irrational.2

Classical Logic

• disjunction ( )∨• conjunction ( )∧• negation (−)• implication (→)• equivalence (↔)

Classical Logic & Fuzzy Logic

Classical predicate logic

T: uU [0,1]

U: universe of all propositions.

All elements u U are true for proposition P are called the truth set of P: T(P).

Those elements u U are false for P are called falsity set of P: F(P).

T(Y) = 1 T(Ø) = 0

Example 5.1. Let P be the proposition “The structural beam is an 18WF45” and let Q bethe proposition “The structural beam is made of steel.” Let X be the universe of structuralmembers comprising girders, beams, and columns; x is an element (beam); A is the set of all wide-flange (WF) beams; andB is the set of all steel beams. Hence, P : x is in A Q : x is in B

Classical Logic &Fuzzy Logic

Given a proposition P: xA, P: xA, we have the following logical connectives:

Disjunction PQ: x A or x B hence, T(PQ) = max(T(P),T(Q))Conjunction PQ: xA and xB

hence T(P Q)= min(T(P),T(Q))Negation If T(P) =1, then T(P) = 0 then T(P) =1Implication (P Q): xA or xB Hence , T(P Q)= T(P Q)(P ←→ Q) : T (P ←→ Q) = 1, for T (P) = T (Q) = 0, for T (P) ≠ T (Q)

Equivalence

1, for T(P) = T(Q)(P Q): T(PQ)=

0, for T(P) T(Q)

The logical connective implication, i.e.,P Q (P implies

Q) presented here is also known as the classical

implication.

P is referred to as hypothesis or antecedent

Q is referred to as conclusion or consequent.

T(PQ)=(T(P)T(Q))Or PQ= (AB is true)T(PQ) = T(PQ is true) = max (T(P),T(Q))(AB)= (AB)= ABSo (AB)= ABOr AB false AB

Truth table for various compound propositions

P Q P PQ PQ PQ PQ

T(1) T(1) F(0) T(1) T(1) T(1) T(1)

T(1) F(0) F(0) T(1) F(0) F(0) F(0)

F(0) T(1) T(1) T(1) F(0) T(1) F(0)

F(0) F(0) T(1) F(0) F(0) T(1) T(1)

Classical LogicP : truth that x A∈Q : truth that x B where truth is measured in terms of the truth value, that ∈is,if x A, T (P)∈ = 1; otherwise, T (P) = 0if x B, T (Q) = 1; otherwise, T (Q) = 0∈or, using the characteristic function to represent truth (1) and falsity (0), the following notation results: χA(x) = 1 x A∈ = 0 x ∉ A

A notion of mutual exclusivity arises in this calculus. For the situation involving two propositions P and Q, where T (P) ∩ T (Q) = Ø, we have that the truth of P always implies the falsity of Q and vice versa; hence, P and Q are mutually exclusive propositions.

Logic(P ←→ Q) : T (P ←→ Q) = 1, for T (P) = T (Q) = 0, for T (P) ≠ T (Q)

LogicExample 5.2. (Similar to Gill, 1976). Consider the following four propositions:1. if 1 + 1 = 2, then 4 > 0; conclusion is T regardless of the Hypothesis2. if 1 + 1 = 3, then 4 > 0; conclusion is T regardless of the Hypothesis3. if 1 + 1 = 3, then 4 < 0; both propositions are false but this does not disprove the implication4. if 1 + 1 = 2, then 4 < 0. a true hypothesis cannot produce a false conclusionHence, the classical form of the implication is true for all propositions of P and Q except for those propositions that are in both the truth set of P and the false set of Q E EE E E E E E E E_͞E_͞_͞E_͞T (P → Q) = T (P) ∩ T (Q)

Logic __(P → Q) ≡ (A ∪ B is true) ≡ (either “not in A” or “in B”) _͞ _͞_͞_͞so that T (P → Q) = T (P Q) = max(T (P), T (Q))∨This expression is linguistically equivalent to the statement “P → Q is true” when either “not A” or “B” is true (logical or)Graphically, this implication and the analogous set operation are represented by the Venn diagram in Figure

Suppose the implication operation involves two different universes of discourse: P is a proposition described by set A, which is defined on universe X, and Q is a proposition described by set B, which is defined on universe Y. Then, the implication P → Q can be represented in set-theoretic terms by the relation R, where R is defined as

_͞R = (A × B) ∪ (A × Y) ≡ IF A, THEN B IF x A where ∈ x X and A X∈ ⊂THEN y B where ∈ y Y and B Y∈ ⊂The graphic in the figure below, represents the space of the Cartesian product X × Y showing typical sets A and B; superposed on this space is the set-theoretic equivalent ofthe implication. That is,

PQ: If x A, Then y B, or PQ AB

The shaded regions of the compound Venn diagram in the following figure represent the truth domain of the implication, If A, then B(PQ).

Two ways to interpret “If x is A then y is B”:

A entails By

A coupled with B

Fuzzy if-then rules (3.3) (cont.)

Two ways to interpret “If x is A then y is B”:

A coupled with B

Fuzzy if-then rules (cont.)

IF A, THEN B, or IF A , THEN CPREDICATE LOGIC (PQ)(PS)Where P: xA, AX

Q: yB, BYS: yC, CY

SET THEORETIC EQUIVALENT (A X B)(A X C) = R = relation ON X Y

Truth domain for the above compound proposition.

Classical Logic

Venn diagram for equivalence (darkened areas), that is, for T (A ↔ B).

Example:

Suppose we consider the universe of positive integers, X = {1 ≤ n ≤ 8}. Let P = “n is an even number” and let Q = “(3 ≤ n ≤ 7) ∧ (n = 6).” Then T (P) = {2, 4, 6, 8} and T (Q) = {3, 4, 5, 7}. The equivalence P ↔ Q has the truth set ___ ___T (P ←→ Q) = (T (P ) ∩ T (Q)) ∪ (T (P) ∩ T (Q)) = {4} ∪ {1} = {1, 4}

Classical LogicExample . Prove that P ↔ Q if P = “n is an integer power of 2 less than 7 and greater than zero” and Q = “n2− 6n + 8 = 0.” Since T (P) = {2, 4} and T (Q) = {2, 4}, it follows that P ↔ Q is an equivalence

Classical Logic

Exclusive OR grey areas

TautologiesModus ponens deduction, is a very common inference scheme used in forward- chaining rule-based expert systems It is an operation whose task is to find the truth value of a consequent in a production rule, given the truth value of the antecedent in the ruleModus ponens deduction concludes that given two propositions, P and P → Q, if both of which are true, then the truth of the simple proposition Q is automatically inferred.Modus tollens, an implication between two propositions is combined with a second proposition and both are used to imply a third proposition

Common Tautologies _ B ∪ B ←→ X. _ A ∪ X; A ∪ X ←→ X. (A ∧ (A → B)) → B (modus ponens) _ _ (B ∧ (A → B)) →A (modus tollens).

Exclusive orExclusive NorExclusive or P “” Q(AB) (AB)Exclusive Nor(P “” Q)(PQ)Logical proofsLogic involves the use of inference in everyday life.

In natural language if we are given some hypothesis it is often useful to make certain conclusions from them the so called process of inference (P1P2….Pn) Q is true.

Hypothesis : Engineers are mathematicians. Logical thinkers do not believe in magic. Mathematicians are logical thinkers.Conclusion : Engineers do not believe in magic.Let us decompose this information into individual propositionsP: a person is an engineerQ: a person is a mathematicianR: a person is a logical thinkerS: a person believes in magicThe statements can now be expressed as algebraic propositions as((PQ)(RS)(QR))(PS)It can be shown that the proposition is a tautology.ALTERNATIVE: proof by contradiction.

Deductive inferences

The modus ponens deduction ( for p q if p holds the q is inferred) is used as a tool for making inferences in rule based systems. This rule can be translated into a relation between sets A and B.

R = (AB)(AY) Y is the universe

Now suppose a new antecedent say A’ is known, since A implies B is defined on the cartesian space X Y, B can be found through the following set theoretic formulation

B= AR= A((AB)(AY))

Denotes the composition operation. Modus ponens deduction can also be used for compound rule.

Whether A is contained only in the complement of A or whether A’ and A overlap to some extent as described next:

IF AA, THEN y=B

IF AA THEN y =C

IF AA , AA, THEN y= BC

Fuzzy Logic

The restriction of classical propositional calculus to a two-valued logic has created many interesting paradoxes over the ages. For example, the barber of Seville is a classic paradox (also termed as Russell’s barber). In the small Spanish town of Seville, there is a rule that all and only those men who do not shave themselves are shaved by a barber. Who shaves the barber?

Another example comes from ancient Greece. Does the liar from Crete lie when he claims, “All Cretians are liars”? If he is telling the truth, then the statement is false. If the statement is false, he is not telling the truth.

Fuzzy Logic

Let S: the barber shaves himself

S’: he does not

S S’ and S’ S

T(S) = T(S’) = 1 – T(S)

T(S) = 1/2

But for binary logic T(S) = 1 or 0

Fuzzy propositions are assigned for fuzzy sets:

Some common tautologies follow:

BB X AX; A X X

AB (A(AB))B (modeus ponens)(B(AB))A (modus tollens)Proof:(A(AB)) B(A(AB)) B Implication((AA) (AB))B Distributivity((AB))B Excluded middle laws(AB)B Identity(AB)B Implication(AB)B Demorgans lawA(BB) AssociativityAX Excluded middle lawsX T(X) =1 Identity; QED

Classical Logic & Fuzzy Logic

Proof(B(AB))A(B(AB))A((BA)(BB)) A((BA))A(BA)A

(BA)AB(AA)BX = X T(X) =1 A B AB (A(AB) (A(AB)B

O 0 1 0 1

O 1 1 0 1

1 0 0 0 1

1 1 1 1 1

Truth table (modus ponens)

Exclusive orExclusive NorExclusive or P “” Q(AB) (AB)Exclusive Nor(P “” Q)(PQ)Logical proofsLogic involves the use of inference in everyday life.

In natural language if we are given some hypothesis it is often useful to make certain conclusions from them the so called process of inference (P1P2….Pn) Q is true.

Hypothesis : Engineers are mathematicians. Logical thinkers do not believe in magic. Mathematicians are logical thinkers.Conclusion : Engineers do not believe in magic.Let us decompose this information into individual propositionsP: a person is an engineerQ: a person is a mathematicianR: a person is a logical thinkerS: a person believes in magicThe statements can now be expressed as algebraic propositions as((PQ)(RS)(QR))(PS)It can be shown that the proposition is a tautology.ALTERNATIVE: proof by contradiction.

Deductive inferences

The modus ponens deduction is used as a tool for making inferences in rule based systems. This rule can be translated into a relation between sets A and B.

R = (AB)(AY)

Now suppose a new antecedent say A’ is known, since A implies B is defined on the cartesian space X Y, B can be found through the following set theoretic formulation B= AR= A((AB)(AY))

Denotes the composition operation. Modus ponens deduction can also be used for compound rule.

Whether A is contained only in the complement of A or whether A’ and A overlap to some extent as described next:

IF AA, THEN y=B

IF AA THEN y =C

IF AA , AA, THEN y= BC

Fuzzy Logic

1 PTPT

QTPTQPT

BorAxQP

QTPTQPT

BandAxQP

~~~~~~

,max QTPTQPTQPT

Negation

Disjunction

Conjunction

Implication [Zadeh, 1973]

Fuzzy Logic

When the logical conditional implication is of the compound form,

IF x is , THEN y is , ELSE y is

Then fuzzy relation is:

whose membership function can be expressed as:

~~~~~CABAR

yxyxyx CABAR

1,max,

Fuzzy Logic

Rule-based format to represent fuzzy information.

Rule 1: IF x is , THEN y is , where and represent fuzzy propositions (sets)

Now suppose we introduce a new antecedent, say, and we consider the following rule

Rule 2: IF x is , THEN y is

RAB ''~~

Fuzzy Logic

Suppose we use A in fuzzy composition, can we get

The answer is: NO

Example:

For the problem in pg 127, let

A’ = AB’ = A’ R = A R = {0.4/1 + 0.4/2 + 1/3 + 0.8/4 + 0.4/5 + 0.4/6} ≠ B

RBB ~~

Logic connectives

Disjunction Conjunction Negation –Implication Equivalence

If xA, T(P) =1 otherwise T(P) = 0OrxA(x)={ 1 if x A, otherwise it is 0 }

If T(p)T()=0 implies P true, false, or true P false. P and are mutually exclusive propositions.

General format:– If x is A then y is B (where A & B are linguistic

values defined by fuzzy sets on universes of discourse X & Y).

• “x is A” is called the antecedent or premise• “y is B” is called the consequence or

conclusion– Examples:

• If pressure is high, then volume is small.• If the road is slippery, then driving is dangerous.• If a tomato is red, then it is ripe.• If the speed is high, then apply the brake a little.

Fuzzy if-then rules

– Meaning of fuzzy if-then-rules (A B)

• It is a relation between two variables x & y; therefore it is a binary fuzzy relation R defined on X * Y

• There are two ways to interpret A B:–A coupled with B–A entails B

if A is coupled with B then:

Fuzzy if-then rules (cont.)

If A entails B then:

R = A B = A B ( material implication)

R = A B = A (A B) (propositional calculus)

R = A B = ( A B) B (extended propositional calculus)

Fuzzy if-then rules (3.3) (cont.)

Spring 2003 CMSC 203 - Discrete Structures 68

Rules of InferenceRules of Inference

Rules of inferenceRules of inference provide the justification of provide the justification of the steps used in a proof.the steps used in a proof.

One important rule is called One important rule is called modus ponensmodus ponens or the or the law of detachmentlaw of detachment. It is based on the . It is based on the tautology tautology (p (p (p (p q)) q)) q. We write it in the following q. We write it in the following way:way:

ppp p q q________ qq

The two The two hypotheseshypotheses p and p p and p q q are are written in a column, and the written in a column, and the conclusionconclusionbelow a bar, where below a bar, where means means “therefore”.“therefore”.

Spring 2003 CMSC 203 - Discrete Structures 69

Rules of InferenceRules of Inference

The general form of a rule of inference is:The general form of a rule of inference is:

pp22 .. .. .. ppnn________ qq

The rule states that if pThe rule states that if p11 andand p p22 andand … … andand p pnn are all true, then q is true as are all true, then q is true as well.well.

Each rule is an established tautology Each rule is an established tautology ofof pp11 p p22 … … p pnn q q

These rules of inference can be used These rules of inference can be used in any mathematical argument and do in any mathematical argument and do not not require any proof.require any proof.

CS 173 Proofs - Modus Ponens

I am Mila.If I am Mila, then I am a great swimmer.

I am a great swimmer!

Tautology:

(p (p q)) q

Inference Rule:

Modus Ponens

CS 173 Proofs - Modus Tollens

I am not a great skater.If I am Erik, then I am a great skater.

I am not Erik!

Tautology:

(q (p q)) p

Inference Rule:

Modus Tollens

Fuzzy Logic

The restriction of classical propositional calculus to a two-valued logic has created many interesting paradoxes over the ages. For example, the barber of Seville is a classic paradox (also termed as Russell’s barber). In the small Spanish town of Seville, there is a rule that all and only those men who do not shave themselves are shaved by a barber. Who shaves the barber?

Another example comes from ancient Greece. Does the liar from Crete lie when he claims, “All Cretians are liars”? If he is telling the truth, then the statement is false. If the statement is false, he is not telling the truth.

Fuzzy Logic

Let S: the barber shaves himself

S’: he does not

S S’ and S’ S

T(S) = T(S’) = 1 – T(S)

T(S) = 1/2

But for binary logic T(S) = 1 or 0

Fuzzy propositions are assigned for fuzzy sets:

Fuzzy Logic

1 PTPT

QTPTQPT

BorAxQP

QTPTQPT

BandAxQP

~~~~~~

,max QTPTQPTQPT

Negation

Disjunction

Conjunction

Implication [Zadeh, 1973]

Fuzzy Logic

ABAR~~~~

1,max,~~~~

Example:

= medium uniqueness =

= medium market size =

Then…

Fuzzy Logic

When the logical conditional implication is of the compound form,

IF x is , THEN y is , ELSE y is

Then fuzzy relation is:

whose membership function can be expressed as:

~~~~~CABAR

yxyxyx CABAR

1,max,

Fuzzy Logic

Rule-based format to represent fuzzy information.

Rule 1: IF x is , THEN y is , where and represent fuzzy propositions (sets)

Now suppose we introduce a new antecedent, say, and we consider the following rule

Rule 2: IF x is , THEN y is

RAB ''~~

Fuzzy Logic

Suppose we use A in fuzzy composition, can we get

The answer is: NO

Example:

For the problem in pg 127, let

A’ = AB’ = A’ R = A R = {0.4/1 + 0.4/2 + 1/3 + 0.8/4 + 0.4/5 + 0.4/6} ≠ B

RBB ~~

Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs

The following form of the implication operator show different techniques for obtaining the membership function values of fuzzy relation defined on the Cartesian product space X × Y:

The following common methods are among those proposed in the literature for the composition operation , where is the input, or antecedent defined on the universe X, is the output, or consequent defined on the universe Y, and is a fuzzy relation characterizing the relationship between specific inputs (x) and specific outputs (y):

Refer fig on next slide…

~~~RAB

The extension of truth operations for tautologies, contradictions, equivalence, and logical proofs is no different for fuzzy sets; the results, however, can differ considerably from those in classical logic. If the truth values for the simple propositions of a fuzzy logic compound proposition are strictly true (1) or false (0), the results follow identically those in classical logic. However, the use of partially true (or partially false) simple propositions in compound propositional statements results in new ideas termed quasi tautologies, quasi contradictions, and quasi equivalence. Moreover, the idea of a logical proof is altered because now a proof can be shown only to a “matter of degree”. Some examples of these will be useful.

Truth table (approximate modus ponens)

A B AB (A(AB)) (A(AB))B

.3 .2 .7 .3 .7

.3 .8 .8 .3 .8 Quasi tautology

.7 .2 .3 .3 .7

.7 .8 .8 .7 .8

Truth table (approximate modus ponens)

A B AB (A(AB)) (A(AB))B

.4 .1 .6 .4 .6

.4 .9 .9 .4 .9 Quasi tautology

.6 .1 .4 .4 .6

.6 .9 .9 .6 .9

The following form of the implication operator show different techniques for obtaining the membership function values of fuzzy relation defined on the Cartesian product space X × Y:

Part I Logic

Classical Predicate Logic – tautologies, Contradictions, Equivalence, Exclusive Or Exclusive Nor, Logical Proofs, Deductive InferencesFuzzy Logic, Approximate Reasoning, Fuzzy Tautologies, Contradictions, Equivalence and Logical Proofs, Other forms of the Implication Operation

Part II Fuzzy Systems

Natural language processing, Lingustic Hedges, Rule Based Systems, Multiple conjunctve antecedents , Aggregation of Fuzzy Rules, Graphical techniques of inference

NATURAL LANGUAGE

● Is perhaps the most powerful form of conveying information.

● Despite its vagueness and ambiguity it is the vehicle for human communication, and it seems appropriate that a mathematical theory that deals with fuzziness and ambiguity is also the same tool used to express and interpret the linguistic character of our language. Natural language is used in the expression of knowledge form known as RULE BASED SYSTEMS

NATURAL LANGUAGE● Cognitive scientists tell us that human base their

thinking primarily on conceptual patterns and mental images rather than on any numerical quantities.

● In fact the expert system paradigm known as “Frames” is based on the notion of a cognitive picture in one's mind.

● Furthermore, humans communicate with their own natural language by referring to previous mental images with rather vague but simple terms.

● Despite the vagueness and ambiguity in natural language, humans communicating in a common language have very little trouble in basic understanding.

● Since a vast amount of information involved in human communication involves natural language terms that, by their very nature, are often vague, imprecise, ambiguous, and fuzzy, we will propose the use of fuzzy sets as the mathematical foundation of our Natural Language (NL).

● NL consists of - atomic terms :The fundamental terms. Ex: slow, medium, young, beautiful etc. - composite : a collection of of atomic terms or set of terms. Ex: very slow horse, heavy-weight female, fairly beautiful painting, etc

Natural Language

The time interval x was the period exhibiting a 100 percent maximum ofpossible values as measured along some arbitrary social scale, [and]the interval x was the period of time exhibiting 100 percent minimum ofthese values as measured along the same scale.

Crisp version of this passage is identical to that posed by the law ofexcluded middle of probability theory.

The decomposition of compound rules into canonical (equivalent) forms and the treatment of rules forms as logical propositions. The

characterization of the confidence in a particular rule is addressed by using the truth qualifications. The expression of rules as a collection of logical implications manipulated by inference schemes

Knowledge & Mappings

• Knowledge is a collection of “facts” from some domain.

• What we need is a representation of facts that can be manipulated by a program.– Some symbolic representation is necessary.– Need to be able to map facts to symbols.– Need to be able to map symbols to facts?

A.I. Problems & K.R.

• Game playing - need rules of the game, strategy, heuristic function(s).

• Expert Systems - list of rules, methods to extract new rules.

• Learning - the space of all things learnable (domain representation), concept representation.

• Natural Language - symbols, groupings, semantic mappings, ...

Representation Properties

Representational Adequacy - Is it possible to represent everything of interest ?

Inferential Adequacy - Can new information be inferred?

Inferential Efficiency - How easy is it to infer new knowledge?

Acquisitional Efficiency - How hard is it to gather information (knowledge)?

Search and State Representation

• Each state could be represented as a collection of facts.

• Keeping many such states in memory may be impossible.

• Most facts will not change when we move from one state to another.

The Frame Problem

• Determining how to best represent facts that change from state to state along with those facts that do not change is the Frame Problem.

• Sometimes the hard part is determining which facts change and which do not.

Fuzzy Rule-based systems

Using fuzzy sets as a calculus to interpret natural language. It is vague, imprecise, ambiguous and fuzzy.

Fundamental terms atoms

Collection of atomic terms composite or set of terms

An atomic term (a linguistic variable) can be interpreted using fuzzy sets.

An atomic term in the universe of natural language, X.

Define a fuzzy set in the universe of interpretations or meanings, Y as a specific meaning of .

Suppose we define a specific atomic term in the universe of NL, X, as element and we define fuzzy set A in the universe of interpretations , or meanings, Y, as a specific meaning for the term . then NL can be expressed as a mapping, M from a set of atomic terms in X to a corresponding set of interpretations defined on Y.

● Each atomic term in X corresponds to a fuzzy set A in Y, which is the “interpretation” of

X Y ~~AM

Mapping of a linguistic atom to a cognitive interpretation ~A

Composite

LINGUISTIC HEDGES

● In linguistics, fundamental atomic terms are often modified with adjectives (nouns) or adverbs (verbs) . like very, low, slight, almost, more-or-less, etc

● Using fuzzy sets as the calculus of interpretation, these linguistic hedges have the effect of modifying the membership function for a basic atomic term

Define = a(y)/y, theny

● “very” = 2 =ʃ [μα(y)]2/y

● “Very, very” = 4

● “plus” =1.25

● “slightly” =sqrt () = ʃ [ μα(y))]0.5/y

● “minus” = 0.75

● The first three equations are called “concentrations”

● Another operation on linguistic fuzzy sets is known as intensification.

● Intensification can be expressed by numerous algorithms, one of which, proposed by Zadeh, is

● “intensify” a = = ● Combination of concentration and dilation

● Parentheses may be used to change the precedence order and ambiguities may be resolved by the use of association-to-the-right. For example, “plus very minus” as plus(very(minus))

22(y) for 0<= (y) <= 0.5

1-2[1-(y)]2 for 0.5 <= (y) <= 1

Concentration of A Dilation of A

Intensification of A

Concentration – reduces the degree of membership of the elements

which are “partly” in the set

Hedge “very” with membership of .9 reduced by 10 percent to a value

.81 – whereas, membership value of .1 is reduced by an order of

magnitude .01.

Decrease – Manifestation of the properties of the properties of the membership value itself for 0 ≤µ ≤ 1 then µ >= µ2

Dilation- Stretch or dilate fuzzy set by increasing the membership

of elements that are ‘partly’ in the set

For the hedge ‘slightly’ membership value .81 is increased by 11% to get 0.9 and the membership value of 0.01 is increased by an order of magnitude to 0.1

Precedence Operation

First Hedge, not

Second and

Third or

Precedence for linguistic hedges and logical operations

Linguistic Hedges

15.0121

Intensify

Slightly

veryVery

It increases contrast.

Precedence of the Operations

Example:

Suppose we have a universe of integers, Y = {1,2,3,4,5}. We define the following linguistic terms as a mapping onto Y:

“small” =

“large” =

“very small” = “small”2 = {1/1 + 0.64/2 + 0.36/3 + 0.16/4 +0.04/5}“Not very small” = 1- “very small” = {0/1 + 0.36/2 + 0.64/3+ 0.84/4+ 0.96/5}Thus we construct a phrase, or a composite term:α = “not very small and not very very large” which involves the following set-theoretic operations:

Example (contd)

“Intensely small” =

6.0121

8.0121

Rule-based systems

● In the field of AI there are various ways to represent knowledge.IF premise (antecedent), THEN conclusion

(consequent)● Commonly referred to as the IF-THEN rule-based

form● The rule-based system is distinguished from expert

systems in the sense that the rules comprising a rule-based system might derive from sources other that human experts and, in this context, are distinguished from expert systems.

Rule-based Systems

IF-THEN rule based form

Canonical Rule Forms

1. Assignment statementsx = large, x y

2. Conditional statementsIf A then B,If A then B, else C

3. Unconditional statementsstopgo to 5

unconditional can beIf any conditions, then stopIf condition Ci, then restrict Ri

Canonical Rule Forms

● Assignment statements x = largebanana’s color = yellow

x approx= s● Conditional statements

IF the tomato is red THEN the tomato is ripe IF x is very hot THEN stop● Unconditional statements

go to 9 stop divide by x

turn the pressure higher

● The rule base under consideration could be described using a collection of conditional restrictive statements. These statements may also be modeled as fuzzy conditional statements, such as

IF condition C1 THEN restriction R1.● The unconditional restrictions might be in the form R1: The output is B1

AND R2: The output is B2

AND etc. Where B1, B2, …. Are fuzzy consequents.

● The canonical form for a fuzzy rule-based system

Rule 1: IF cond C1, THEN restriction R1

Rule 2: IF cond C2, THEN restriction R2

.Rule n: IF cond Cn, THEN restriction Rn.

● Example if the temperature is hot, then the pressure is

rather high. if the temperature is cold, then the pressure is

very low.● The vague term “rather high” in the first statement

places a fuzzy restriction on the pressure , based on a fuzzy “hot” temperature condition in the antecedent.

Decomposition of Compound Rule

Any compound rule structure can be decomposed and reduced to a number of simple canonical rules.

The most commonly used techniques

Multiple Conjunctive Antecedents

If x is and , then y is

Define

~A LAA

~LS AAAA

xxx LS AAA

The rule can be rewritten.

IF THENSA

Decomposition of Compound Rules

● A linguistic statement expressed by a human might involve compound rule structures

as an example, consider a rule-based for a simple home temperature control problem, which might contain the following rules.

IF it is raining hard THEN close the window.IF the room temp is very hot,THEN IF the heat is on THEN turn the heat lower AND IF it is not raining hard THEN open the window. etc..

Multiple Disjunctive Antecedents

If x is or or … or

then y is

LS AAA

,,max 1

IF THENSA

● Multiple conjunctive antecedents IF x is A1 and A2 … and AL THEN y is Bs

As = A1 ∩ A2 ∩ … ∩ AL

As(x) = min [A1(x), … , Al(x)]

IF As THEN Bs

● Multiple disjunctive antecedentsIF x is A1 or A2 … or AL THEN y is Bs

As = A1 A2 … A L

As(x) = max [A1(x), … , Al(x)]

IF x is As THEN y is Bs

Condition Statements

1. IF THEN ( ELSE ) decomposed into:

IF THEN or IF NOT THEN

2. IF (THEN ) unless decomposed into:IF THEN or IF NOT THEN NOT

3. IF THEN ( ELSE IF THEN ( )) decomposed into:

IF THEN or IF NOT and THEN NOT

4. Nested IF-THEN rules IF THEN (IF , THEN ( )) becomes IF and THEN

Each canonical form is an implication, and we can reduce the rules to a series of relations.

Condition Statements

“likely” “very likely” “highly likely” “true” “fairly true” “very true” “false” “fairly false” “very false”

anything

Let be a fuzzy truth value “very true” “true” “fairly true” “fairly false” “false”

A truth qualification proposition can be expressed as:“x is is ”

orx is is =

Aggregation of fuzzy rule

The process of obtaining the overall consequent (conclusion) from the individual consequent contributed by each rule in the rule-base is known as aggregation of rules.

Conjunctive System of Rules:

,,min 1

Disjunctive System of Rules:

,,max 1

“highly unlikely” = “minus very very unlikely” = “(very very unlikely)0.75” = 1/0 + 1/.1 + 1/.2 + .5/.3 + .3/.4}

Ex: if a fuzzy variable x has a membership value equal to .85 in the fuzzy set A i.e., (x)=.85 as shown in the figure then its membership values for the following truth qualification statements are determined from figure

: x is A is true : x is A is false: x is A isfairly true: x is A is very false

0 .85 1 a

A(x) = .85

A(x) = .15

A(x) = .96

A(x) = .04

Aggregation of Fuzzy Rules

● Conjunctive system of rules. y = y1 and y2 and … and yr

Or y = y1 ∩y2 ∩ … ∩ yr

Defined by y(y) = min (y1(y),…yr(y)) for y belongs to Y

● Disjunctive system of rules y = y1 or y2 or … or yr

Or y = y1 U y2 U … U yr

Defined by y(y) = max (y1(y), … y(r-1)(y), yr(y)) for y belongs to Y

Slide Slide 187187

Fuzzy Rule BaseFuzzy Rule Base Fuzzy rules can be formulated:Fuzzy rules can be formulated:

• from human expert’s knowledge or experiencefrom human expert’s knowledge or experience• by statistical analysis of numerical data by statistical analysis of numerical data

obtained from experimentationobtained from experimentation• through neuro-fuzzy optimisation (learning) through neuro-fuzzy optimisation (learning)

process – ANFIS (Adaptive Neuro-Fuzzy process – ANFIS (Adaptive Neuro-Fuzzy Inference System), FuNe (Neuro-Fuzzy learning Inference System), FuNe (Neuro-Fuzzy learning network with rule generation)network with rule generation)

Artificial Intelligence – CS364Artificial Intelligence – CS364Fuzzy LogicFuzzy Logic

Graphical Techniques of InferenceGraphical Techniques of Inference• The most commonly used fuzzy inference technique is the

so-called Mamdani method.

• In 1975, Professor Ebrahim Mamdani of London University built one of the first fuzzy systems to control a steam engine and boiler combination. He applied a set of fuzzy rules supplied by experienced human operators.

Slide Slide 190190

Fuzzy Inference Systems (FIS)Fuzzy Inference Systems (FIS) Inspiration: lexical imprecision in natural language Inspiration: lexical imprecision in natural language

reasoningreasoning

““price of crude oil which has price of crude oil which has edged higheredged higher in in recent recent weeksweeks after being after being remarkably stableremarkably stable through through much much of the yearof the year, may , may fluctuatefluctuate as much as a dollar a as much as a dollar a barrel in the barrel in the months aheadmonths ahead, but , but abrupt changesabrupt changes are are not likelynot likely, many analysts , many analysts believebelieve.”.”

Almost all our everyday reasoning is approximate Almost all our everyday reasoning is approximate in nature.in nature.

Slide Slide 191191

FIS: InspirationFIS: Inspiration Exploit the tolerance for imprecision.Exploit the tolerance for imprecision. High precision entails high cost.High precision entails high cost.

• park the carpark the car• park the car 10cm from the curbpark the car 10cm from the curb

High precision entails low tractabilityHigh precision entails low tractability• reduce the precision of information to make a reduce the precision of information to make a

complex problem more tractablecomplex problem more tractable

Slide Slide 192192

FIS: ApplicationsFIS: Applications Replacement of human operator by a FIS:Replacement of human operator by a FIS:

• Sendai subway (Hitachi), Elevator control (Hitachi, Sendai subway (Hitachi), Elevator control (Hitachi, Toshiba)Toshiba)

• Nuclear reactor control (Hitachi)Nuclear reactor control (Hitachi)• Automobile transmission (Nissan, Subaru, Honda)Automobile transmission (Nissan, Subaru, Honda)• Video image stabilisation (Canon, Minolta)Video image stabilisation (Canon, Minolta)

Replacement of human expert by a FIS:Replacement of human expert by a FIS:• medical diagnosis medical diagnosis • SecuritiesSecurities• Fault diagnosisFault diagnosis• Credit worthinessCredit worthiness

Slide Slide 193193

FISFIS

Defuzzifier

Slide Slide 194194

Fuzzy OperationsFuzzy Operations Union operation (OR)Union operation (OR)

Intersection operation (AND)Intersection operation (AND)

Complement operation (NOT)Complement operation (NOT)

BABA x ,max)(

BABA x ,min)(

AA x 1)(

Slide Slide 195195

Mamdani Fuzzy InferenceMamdani Fuzzy Inference Single rule with single antecedentSingle rule with single antecedent

Rule:Rule: if x is A then y is B if x is A then y is B

Fact:Fact: x is A’ x is A’

Inference:Inference: y is B’ y is B’ Graphical Representation:Graphical Representation:

A’ B

x is A’

Xy is B’

Slide Slide 196196

FIS: Mamdani ProcedureFIS: Mamdani Procedure iithth rule: rule: if xif x11 is A is A1i1i and … and x and … and xnn is A is Ani ni then y is Bthen y is Bii

1. Determine the degree of membership of each input to different fuzzy terms Aji:

2. Determine the strength of each rule antecedent

3. Determine the contribution of each rule

4. Rule aggregation

5. Defuzzification

j runs on each fuzzy term

i runs on each rule

Mamdani Fuzzy InferenceMamdani Fuzzy Inference• The Mamdani-style fuzzy inference process is performed

in four steps:

1. Fuzzification of the input variables

2. Rule evaluation (inference)

3. Aggregation of the rule outputs (composition)

4. Defuzzification.

Mamdani Fuzzy InferenceMamdani Fuzzy InferenceWe examine a simple two-input one-output problem that includes three rules:

Rule: 1 Rule: 1IF x is A3 IF project_funding is adequateOR y is B1 OR project_staffing is smallTHEN z is C1 THEN risk is low

Rule: 2 Rule: 2IF x is A2 IF project_funding is marginalAND y is B2 AND project_staffing is largeTHEN z is C2 THEN risk is normal

Rule: 3 Rule: 3IF x is A1 IF project_funding is inadequateTHEN z is C3 THEN risk is high

Step 1: FuzzificationStep 1: Fuzzification• The first step is to take the crisp inputs, x1 and y1 (project funding and

project staffing), and determine the degree to which these inputs belong to each of the appropriate fuzzy sets.

Crisp Inputy1

Crisp Input

0.20.5

A1 A2 A3

(x = A1) = 0.5

(x = A2) = 0.2

(y = B1) = 0.1

(y = B2) = 0.7

Project Funding Project Staffing

Step 2: Rule EvaluationStep 2: Rule Evaluation• The second step is to take the fuzzified inputs,

(x=A1) = 0.5, (x=A2) = 0.2, (y=B1) = 0.1 and (y=B2) = 0.7,

and apply them to the antecedents of the fuzzy rules.

• If a given fuzzy rule has multiple antecedents, the fuzzy operator (AND or OR) is used to obtain a single number that represents the result of the antecedent evaluation.

• This number (the truth value) is then applied to the consequent membership function.

Step 2: Rule EvaluationStep 2: Rule Evaluation

RECALL:

To evaluate the disjunction of the rule antecedents, we use the OR fuzzy operation. Typically, fuzzy expert systems make use of the classical fuzzy operation union:

AB(x) = max [A(x), B(x)]

Similarly, in order to evaluate the conjunction of the rule antecedents, we apply the AND fuzzy operation intersection:

AB(x) = min [A(x), B(x)]

Rule 3:

C30.5 0.5

OR(max)

AND(min)

OR THENRule 1:

AND THENRule 2:

IF x is A3 (0.0) y is B1 (0.1) z is C1 (0.1)

IF x is A2 (0.2) y is B2 (0.7) z is C2 (0.2)

IF x is A1 (0.5) z is C3 (0.5)

Step 2: Rule EvaluationStep 2: Rule Evaluation• Now the result of the antecedent evaluation can be applied

to the membership function of the consequent.

• There are two main methods for doing so:– Clipping

– Scaling

Step 2: Rule EvaluationStep 2: Rule Evaluation• The most common method of correlating the rule

consequent with the truth value of the rule antecedent is to cut the consequent membership function at the level of the antecedent truth. This method is called clipping (lambda-cut).

• Since the top of the membership function is sliced, the clipped fuzzy set loses some information.

• However, clipping is still often preferred because it involves less complex and faster mathematics, and generates an aggregated output surface that is easier to defuzzify.

Step 2: Rule EvaluationStep 2: Rule Evaluation• While clipping is a frequently used method, scaling offers

a better approach for preserving the original shape of the fuzzy set.

• The original membership function of the rule consequent is adjusted by multiplying all its membership degrees by the truth value of the rule antecedent.

• This method, which generally loses less information, can be very useful in fuzzy expert systems.

Degree ofMembership1.0

Degree ofMembership

clipping scaling

Step 3: Aggregation of the rule outputsStep 3: Aggregation of the rule outputs• Aggregation is the process of unification of the outputs of

all rules.

• We take the membership functions of all rule consequents previously clipped or scaled and combine them into a single fuzzy set.

• The input of the aggregation process is the list of clipped or scaled consequent membership functions, and the output is one fuzzy set for each output variable.

Step 3: Aggregation of the rule outputsStep 3: Aggregation of the rule outputs

Cz is 1 (0.1)

Cz is 2 (0.2)

Cz is 3 (0.5)

Step 4: DefuzzificationStep 4: Defuzzification• The last step in the fuzzy inference process is

defuzzification.

• Fuzziness helps us to evaluate the rules, but the final output of a fuzzy system has to be a crisp number.

• The input for the defuzzification process is the aggregate output fuzzy set and the output is a single number.

Step 4: DefuzzificationStep 4: Defuzzification• There are several defuzzification methods, but probably

the most popular one is the centroid technique. It finds the point where a vertical line would slice the aggregate set into two equal masses. Mathematically this centre of gravity (COG) can be expressed as:

Step 4: DefuzzificationStep 4: Defuzzification• Centroid defuzzification method finds a point representing

the centre of gravity of the fuzzy set, A, on the interval, ab.

• A reasonable estimate can be obtained by calculating it over a sample of points.

160 170 180 190 200

Step 4: DefuzzificationStep 4: Defuzzification

4.675.05.05.05.02.02.02.02.01.01.01.0

5.0)100908070(2.0)60504030(1.0)20100(

0 20 30 40 5010 70 80 90 10060

Degree ofMembership

Sugeno Fuzzy InferenceSugeno Fuzzy Inference• Mamdani-style inference, as we have just seen, requires us

to find the centroid of a two-dimensional shape by integrating across a continuously varying function. In general, this process is not computationally efficient.

• Michio Sugeno suggested to use a single spike, a singleton, as the membership function of the rule consequent.

• A singleton, or more precisely a fuzzy singleton, is a fuzzy set with a membership function that is unity at a single particular point on the universe of discourse and zero everywhere else.

Sugeno Fuzzy InferenceSugeno Fuzzy Inference• Sugeno-style fuzzy inference is very similar to the

Mamdani method. Sugeno changed only a rule consequent. Instead of a fuzzy set, he used a mathematical function of the input variable. The format of the Sugeno-style fuzzy rule is

IF x is AAND y is BTHEN z is f(x, y)

where x, y and z are linguistic variables; A and B are fuzzy sets on universe of discourses X and Y, respectively; and f(x, y) is a mathematical function.

Sugeno Fuzzy InferenceSugeno Fuzzy Inference• The most commonly used zero-order Sugeno fuzzy

model applies fuzzy rules in the following form:

IF x is AAND y is BTHEN z is k

where k is a constant.

• In this case, the output of each fuzzy rule is constant. All consequent membership functions are represented by singleton spikes.

Sugeno Rule EvaluationSugeno Rule Evaluation

IF x is A1 (0.5) z is k3 (0.5)Rule 3:

0.5 0.5

OR(max)

AND(min)

OR y is B1 (0.1) THEN z is k1 (0.1)Rule 1:

IF x is A2 (0.2) AND y is B2 (0.7) THEN z is k2 (0.2)Rule 2:

IF x is A3 (0.0)

IF x is A1 (0.5) z is k3 (0.5)Rule 3:

0.5 0.5

OR(max)

AND(min)

IF x is A3 (0.0)

Sugeno Aggregation of the Rule OutputsSugeno Aggregation of the Rule Outputs

z is k1 (0.1) z is k2 (0.2) z is k3 (0.5) 0

0.1Z 0

Zk1 k2 k3 0

0.1Zk1 k2 k3

0.20.5

Graphical Technique of Inference

Case 1 : CRISP SETS max-min

Graphical Technique of InferenceCase 2: CRISP SETS: Using max-product (or correlation product) implication technique, aggregated output for r rules would be:

jinputiinputy kkk AAkB

max2~1~~

max rk ,,2,1

Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 220

Mamdani-style inference, as we have just seen, Mamdani-style inference, as we have just seen, requires us to find the centroid of a two-dimensional requires us to find the centroid of a two-dimensional shape by integrating across a continuously varying shape by integrating across a continuously varying function. In general, this process is not function. In general, this process is not computationally efficient.computationally efficient.

Michio SugenoMichio Sugeno suggested to use a single spike, a suggested to use a single spike, a singletonsingleton, as the membership function of the rule , as the membership function of the rule consequent. A consequent. A singletonsingleton, or more precisely a , or more precisely a fuzzy fuzzy singletonsingleton, is a fuzzy set with a membership function , is a fuzzy set with a membership function that is unity at a single particular point on the that is unity at a single particular point on the universe of discourse and zero everywhere else.universe of discourse and zero everywhere else.

Sugeno fuzzy inferenceSugeno fuzzy inference

Sugeno-style fuzzy inference is very similar to the Sugeno-style fuzzy inference is very similar to the Mamdani method. Sugeno changed only a rule Mamdani method. Sugeno changed only a rule consequent. Instead of a fuzzy set, he used a consequent. Instead of a fuzzy set, he used a mathematical function of the input variable. The mathematical function of the input variable. The format of the format of the Sugeno-style fuzzy ruleSugeno-style fuzzy rule is is

IFIF xx is is AAANDAND yy is is BBTHEN THEN zz is is f f ((x, yx, y))

where where xx, , yy and and zz are linguistic variables; are linguistic variables; AA and and BB are are fuzzy sets on universe of discourses fuzzy sets on universe of discourses XX and and YY, , respectively; and respectively; and f f ((x, yx, y) is a mathematical function.) is a mathematical function.

The most commonly used The most commonly used zero-order Sugeno fuzzy zero-order Sugeno fuzzy modelmodel applies fuzzy rules in the following form: applies fuzzy rules in the following form:

IFIF xx is is AAANDAND yy is is BBTHEN THEN zz is is kk

where where kk is a constant. is a constant.

In this case, the output of each fuzzy rule is constant. In this case, the output of each fuzzy rule is constant. All consequent membership functions are represented All consequent membership functions are represented by singleton spikes.by singleton spikes.

IF x is A1 (0.5) z is k3 (0.5)Rule 3:

0.5 0.5

OR(max)

AND(min)

IF x is A3 (0.0)

Sugeno-style rule evaluationSugeno-style rule evaluation

Sugeno-style Sugeno-style aggregation of the rule outputsaggregation of the rule outputs

z is k1 (0.1) z is k2 (0.2) z is k3 (0.5) 0

0.1Z 0

Zk1 k2 k3 0

0.1Zk1 k2 k3

0.20.5

Weighted average (WA):Weighted average (WA):

655.02.01.0

805.0502.0201.0

)3()2()1(

3)3(2)2(1)1(

kkkkkkWA

Crisp Outputz1

Sugeno-style Sugeno-style defuzzificationdefuzzification

How to make a decision on which method How to make a decision on which method to apply to apply Mamdani or Sugeno? Mamdani or Sugeno? Mamdani method is widely accepted for capturing Mamdani method is widely accepted for capturing

expert knowledge. It allows us to describe the expert knowledge. It allows us to describe the expertise in more intuitive, more human-like expertise in more intuitive, more human-like manner. However, Mamdani-type fuzzy inference manner. However, Mamdani-type fuzzy inference entails a substantial computational burden. entails a substantial computational burden.

On the other hand, Sugeno method is On the other hand, Sugeno method is computationally effective and works well with computationally effective and works well with optimisation and adaptive techniques, which makes optimisation and adaptive techniques, which makes it very attractive in control problems, particularly it very attractive in control problems, particularly for dynamic nonlinear systems.for dynamic nonlinear systems.

Building a fuzzy expert system: case studyBuilding a fuzzy expert system: case study

A service centre keeps spare parts and repairs failed A service centre keeps spare parts and repairs failed ones. ones.

A customer brings a failed item and receives a spare A customer brings a failed item and receives a spare of the same type. of the same type.

Failed parts are repaired, placed on the shelf, and Failed parts are repaired, placed on the shelf, and thus become spares. thus become spares.

The objective here is to advise a manager of the The objective here is to advise a manager of the service centre on certain decision policies to keep service centre on certain decision policies to keep the customers satisfied.the customers satisfied.

Process of developing a fuzzy expert systemProcess of developing a fuzzy expert system

1. Specify the problem and define linguistic variables.1. Specify the problem and define linguistic variables.

2. Determine fuzzy sets.2. Determine fuzzy sets.

3. Elicit and construct fuzzy rules.3. Elicit and construct fuzzy rules.

4. Encode the fuzzy sets, fuzzy rules and procedures 4. Encode the fuzzy sets, fuzzy rules and procedures

to perform fuzzy inference into the expert system.to perform fuzzy inference into the expert system.

5. Evaluate and tune the system.5. Evaluate and tune the system.

Step Step 11: Specify the problem and define: Specify the problem and define linguistic variableslinguistic variables

There are four main linguistic variables: average There are four main linguistic variables: average waiting time (mean delay) waiting time (mean delay) mm, repair utilisation , repair utilisation factor of the service centre factor of the service centre , number of servers , number of servers ss, , and initial number of spare parts and initial number of spare parts nn..

Linguistic variables and their rangesLinguistic variables and their ranges

Step Step 22: Determine fuzzy sets: Determine fuzzy sets

Fuzzy sets can have a variety of shapes. However, Fuzzy sets can have a variety of shapes. However, a triangle or a trapezoid can often provide an a triangle or a trapezoid can often provide an adequate representation of the expert knowledge, adequate representation of the expert knowledge, and at the same time, significantly simplifies the and at the same time, significantly simplifies the process of computation.process of computation.

Fuzzy sets of Fuzzy sets of Mean Delay mMean Delay m

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Mean Delay (normalised)

Degree of Membership

Fuzzy sets of Fuzzy sets of Number of Servers sNumber of Servers s

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Number of Servers (normalised)

Fuzzy sets of Fuzzy sets of Repair Utilisation Factor Repair Utilisation Factor

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Repair Utilisation Factor

Fuzzy sets of Fuzzy sets of Number of Spares nNumber of Spares n

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S RSVS M RL L VL

Number of Spares (normalised)

Step Step 33: Elicit and construct fuzzy rules: Elicit and construct fuzzy rules

To accomplish this task, we might ask the expert to To accomplish this task, we might ask the expert to describe how the problem can be solved using the describe how the problem can be solved using the fuzzy linguistic variables defined previously.fuzzy linguistic variables defined previously.

Required knowledge also can be collected from Required knowledge also can be collected from other sources such as books, computer databases, other sources such as books, computer databases, flow diagrams and observed human behaviour. flow diagrams and observed human behaviour.

The square FAM representationThe square FAM representation

VS S M

The rule tableThe rule table

Rule Base 1Rule Base 1

Cube FAM of Rule Base 2Cube FAM of Rule Base 2

VS VS VSVS VS VS

VS VS VS

VL L M

VS VS VSVS VS VS

VS VS VSM

VS VS VSVS VS VS

S S VSL

LVS S M

VS VS VS

LVS S M

VS VS VSM

S S VSL

S VS VS

MVS S M

VS S M

RS S VSM

M RS SL

M M SM

RL M RSL

Step Step 44: Encode the fuzzy sets, fuzzy rules: Encode the fuzzy sets, fuzzy rules and procedures to perform fuzzyand procedures to perform fuzzy inference into the expert systeminference into the expert systemTo accomplish this task, we may choose one of To accomplish this task, we may choose one of two options: to build our system using a two options: to build our system using a programming language such as C/C++ or Pascal, programming language such as C/C++ or Pascal, or to apply a fuzzy logic development tool such as or to apply a fuzzy logic development tool such as MATLAB Fuzzy Logic Toolbox or Fuzzy MATLAB Fuzzy Logic Toolbox or Fuzzy Knowledge Builder.Knowledge Builder.

Step Step 55: Evaluate and tune the system: Evaluate and tune the system

The last, and the most laborious, task is to evaluate The last, and the most laborious, task is to evaluate and tune the system. We want to see whether our and tune the system. We want to see whether our fuzzy system meets the requirements specified at fuzzy system meets the requirements specified at the beginning. the beginning.

Several test situations depend on the mean delay, Several test situations depend on the mean delay, number of servers and repair utilisation factor. number of servers and repair utilisation factor.

The Fuzzy Logic Toolbox can generate surface to The Fuzzy Logic Toolbox can generate surface to help us analyse the system’s performance.help us analyse the system’s performance.

Three-dimensional plots for Rule Base 1Three-dimensional plots for Rule Base 1

0.40.6

number_of_serversmean_delay

0.40.6

utilisation_factormean_delay

0.40.6

However, even now, the expert might not be However, even now, the expert might not be satisfied with the system performance. satisfied with the system performance.

To improve the system performance, we may use To improve the system performance, we may use additional sets additional sets Rather SmallRather Small and and Rather LargeRather Large on the universe of discourse on the universe of discourse Number of ServersNumber of Servers, , and then extend the rule base.and then extend the rule base.

Modified fuzzy sets of Modified fuzzy sets of Number of Servers sNumber of Servers s

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Number of Servers (normalised)

RS M RL LS

Cube FAM of Rule Base 3Cube FAM of Rule Base 3

VS VS VS

S S VS

VL L M

VL RL RS

RL M RS

L M RS

LVS S M

VS VS VS

S S VS

LVS S M

S VS VS

RS S VS

M RS S

MVS S M

VS S M

0.40.6

Tuning fuzzy systems Tuning fuzzy systems

1.1. Review model input and output variables, and if Review model input and output variables, and if required redefine their ranges.required redefine their ranges.

2.2. Review the fuzzy sets, and if required define Review the fuzzy sets, and if required define additional sets on the universe of discourse.additional sets on the universe of discourse. The use of wide fuzzy sets may cause the fuzzyThe use of wide fuzzy sets may cause the fuzzy system to perform roughly.system to perform roughly.

3.3. Provide sufficient overlap between neighbouring Provide sufficient overlap between neighbouring sets. It is suggested that triangle-to-triangle and sets. It is suggested that triangle-to-triangle and trapezoid-to-triangle fuzzy sets should overlap trapezoid-to-triangle fuzzy sets should overlap between 25% to 50% of their bases.between 25% to 50% of their bases.

4.4. Review the existing rules, and if required add new Review the existing rules, and if required add new rules to the rule base.rules to the rule base.

5.5. Examine the rule base for opportunities to write Examine the rule base for opportunities to write hedge rules to capture the pathological behaviour hedge rules to capture the pathological behaviour of the system.of the system.

6.6. Adjust the rule execution weights. Most fuzzy Adjust the rule execution weights. Most fuzzy logic tools allow control of the importance of rules logic tools allow control of the importance of rules by changing a weight multiplier.by changing a weight multiplier.

7.7. Revise shapes of the fuzzy sets. In most cases, Revise shapes of the fuzzy sets. In most cases, fuzzy systems are highly tolerant of a shape fuzzy systems are highly tolerant of a shape approximation.approximation.

Example:Rule 1: if x1 is and x2 is , then y is

Rule 2: if x1 is or x2 is , then y is input(i) = 0.35 input(j) = 55

If x1 is and x2 is then y is , k = 1,2,..., r

Graphical methods that emulate the inference process and make manual computations involving a few simple rules.

Case 1: inputs x1, and x2 are crisp.

Memberships1 x1 = input(i)

(x1) = (x1 – input(i)) = 0 otherwise

1 x2 = input(i)(x2) = (x2 – input(i)) = 0 otherwise

For r disjunctive rules:

A11 refers to the first fuzzy antecedent of the first rule.

A12 refers to the second fuzzy antecedent of the first rule.

,minmax2~1~~

Summary

• Fuzzy Modelling – subjectivity blessing rather than a curse

Vagueness present in the definition of the terms is consistent with the information contained in the conditional rules developed by

the Engineer when observing some complex process• Set of linguistic variables and their meanings is compatible and

consistent with set of conditional rules used, the outcome of the

qualitative process is translated into objective and quantifiable results• Fuzzy mathematical tools and the calculus of fuzzy IF-THEN

rule provide a most useful paradigm for the automation and

implementation of an extensive body of human knowledge which are

not embodied in the quantitative modelling process

October 2005October 2005 260

• Fuzzification: definition of fuzzy sets, and determination of the degree of membership of crisp inputs in appropriate fuzzy sets.

• Inference: evaluation of fuzzy rules to produce an output for each rule.

• Composition: aggregation or combination of the outputs of all rules.

• Defuzzification: computation of crisp output

Operation of a fuzzy expert system:

Example: Air Conditioner

Example: Cart Pole Problem

Case Study: Building a Fuzzy Expert System

Summary

Fuzzy Expert SystemsFuzzy Expert Systems

Example: Air ConditionerExample: Air Conditioner

1a. Specify the problem

Air-conditioning involves the delivery of air, which can be warmed or cooled and have its humidity raised or lowered.

An air-conditioner is an apparatus for controlling, especially lowering, the temperature and humidity of an enclosed space. An air-conditioner typically has a fan which blows/cools/circulates fresh air and has a cooler. The cooler is controlled by a thermostat. Generally, the amount of air being compressed is proportional to the ambient temperature.

1b. Define linguistic variables• Ambient Temperature

• Air-conditioner Fan Speed

2. Determine Fuzzy Sets: Temperature

Temp Temp ((00C).C).

COLDCOLD COOLCOOL PLEASANTPLEASANT WARMWARM HOTHOT

00 Y*Y* NN NN NN NN

55 YY YY NN NN NN

1010 NN YY NN NN NN

12.512.5 NN Y*Y* NN NN NN

1515 NN YY NN NN NN

17.517.5 NN NN Y*Y* NN NN

2020 NN NN NN YY NN

22.522.5 NN NN NN Y*Y* NN

2525 NN NN NN YY NN

27.527.5 NN NN NN NN YY

3030 NN NN NN NN Y*Y*

Temp Temp ((00C).C).

COLDCOLD COOLCOOL PLEASANTPLEASANT WARMWARM HOTHOT

0< (T)<1

2. Determine Fuzzy Sets: Temperature

Temperature Fuzzy Sets

0 5 10 15 20 25 30

Temperature Degrees C

Pleasent

0 5 10 15 20 25 30

Pleasent

2. Determine Fuzzy Sets: Fan Speed

Rev/secRev/sec

(RPM)(RPM)

MINIMALMINIMAL SLOWSLOW MEDIUMMEDIUM FASTFAST BLASTBLAST

00 Y*Y* NN NN NN NN

1010 YY NN NN NN NN

2020 YY YY NN NN NN

3030 NN Y*Y* NN NN NN

4040 NN YY NN NN NN

5050 NN NN Y*Y* NN NN

6060 NN NN NN YY NN

7070 NN NN NN Y*Y* NN

8080 NN NN NN YY YY

9090 NN NN NN NN YY

100100 NN NN NN NN Y*Y*

2. Determine Fuzzy Sets: Fan Speed

Speed Fuzzy Sets

0 10 20 30 40 50 60 70 80 90 100

MINIMAL

MEDIUM

3. Elicit and construct fuzzy rules

RULE 1: IFRULE 1: IF temptemp is is coldcold THEN THEN speedspeed is is minimalminimalRULE 2: IF RULE 2: IF temptemp is is coolcool THEN THEN speedspeed is is slowslowRULE 3: IF RULE 3: IF temptemp is is pleasantpleasant THEN THEN speedspeed is is mediummediumRULE 4: IF RULE 4: IF temptemp is is warm warm THEN THEN speedspeed is is fastfastRULE 5: IF RULE 5: IF temptemp is is hothot THEN THEN speedspeed is is blastblast

3. Encode into an Expert System

4. Evaluate and tune the system

Consider a temperature of 16oC, use the system to compute the optimal fan speed.

Operation of a Fuzzy Expert System

• Fuzzification• Inference• Composition • Defuzzification

• Fuzzification

Affected fuzzy sets: COOL and PLEASANT

COOL(T) = – T / 5 + 3.5

= – 16 / 5 + 3.5

PLSNT(T) = T /2.5 - 6

= 16 /2.5 - 6

Temp=16 COLD COOL PLEASANT WARM HOT

0 0.3 0.4 0 0

• Inference

RULE 1: IFRULE 1: IF temptemp is is coldcold THEN THEN speedspeed is is minimalminimalRULE 2: IF RULE 2: IF temptemp is is coolcool THEN THEN speedspeed is is slowslowRULE 3: IF RULE 3: IF temptemp is is pleasantpleasant THEN THEN speedspeed is is mediummediumRULE 4: IF RULE 4: IF temptemp is is warm warm THEN THEN speedspeed is is fastfastRULE 5: IF RULE 5: IF temptemp is is hothot THEN THEN speedspeed is is blastblast

RULE 2: IF temp is cool (0.3) THEN speed is slow (0.3)

RULE 3: IF temp is pleasant (0.4) THEN speed is medium (0.4)

• Inference

Example: Air ConditionerExample: Air Conditioner• Composition

speed is slow (0.3) speed is medium (0.4)+

Example: Air ConditionerExample: Air Conditioner• Defuzzification

COG = 0.125(12.5) + 0.25(15) + 0.3(17.5+20+…+40+42.5) + 0.4(45+47.5+…+52.5+55) + 0.25(57.5) 0.125 + 0.25 + 0.3(11) + 0.4(5) + 0.25

= 45.54rpm

Summary

Example: Cart Pole ProblemExample: Cart Pole Problem

The problem is to balance an upright pole, with a mass m at its head and mass M at its base. A weightless shaft connects these two masses. The base can be moved on a horizontal axis. The task is to determine the FORCE (F) necessary to balance the pole. The calculation of the force F involves the measurement of the angle θ and the angular velocity, of the pole .

nnbb nnmm nnss aazz ppss ppmm ppbb

nnbb ppss ppbb

nnmm ppmm

nnss nnmm nnss ppss

aazz nnbb nnmm nnss aazz ppss ppmm ppbb

ppss nnss ppss ppmm

ppmm nnmm

ppbb nnbb nnss

nb: negative big, nm: negative medium, ns: negative small az: approximately zerops: positive small, pm: positive medium, pb: positive big

IF is negative medium and is approximately zero THEN F is negative medium

Example: Cart Pole ProblemExample: Cart Pole ProblemThe fuzzy sets for θ, and F are based on the linear equation μ(x)=ax + b, and are defined based on the following table:

(b) Consider the case when the input variables are: θ = 50, = -5.

Use the rule base, execute each of the four tasks to compute the force F necessary to balance the pole using the Centre of Gravity in the Defuzzification task.

(a) Based on the fuzzy sets table draw three graphs showing the fuzzy sets (nb, nm, ns, az, ps, pm, pb) for each θ, and F individually.

Fuzzification

i) Determine where θ and the angular velocity fall in the table θ: pm, pbaz

ii) Formulate possible rules from linguistic values obtained

IF θ is pm AND is az THEN F is pmIF θ is pb AND is az THEN F is pb

Example: Cart Pole ProblemExample: Cart Pole ProblemFuzzification

iii) Compute membership functions

-11.25 -5 0 10

22.5 45 50 67.5

45 50 67.5

1 1 1/22.5 = /17.5

0.78θ: pm

θ: pb

67.55045

11/22.5 = /5

: az 1/11.25 = /6.25

Example: Cart Pole ProblemExample: Cart Pole ProblemInference

The two premises in RULE 1 are conjunctive minimum of the two: min{0.78, 0.56}=0.56

1 IF θ is pm AND is az THEN F is pm

2 IF θ is pb AND is az THEN F is pb

The two premises in RULE 2 are conjunctive minimum of the two: min{0.22, 0.56}=0.22

Example: Cart Pole ProblemExample: Cart Pole ProblemComposition

Defuzzification

3 0.2 (4 5 6) 0.56 (7 8) 0.225.30

0.2 0.56 0.56 0.56 0.22 0.22SoG

Summary

A service centre keeps spare parts and repairs failed ones. A customer A service centre keeps spare parts and repairs failed ones. A customer brings a failed item and receives a spare of the same type. Failed brings a failed item and receives a spare of the same type. Failed parts are repaired, placed on the shelf, and thus become spares. parts are repaired, placed on the shelf, and thus become spares.

The objective is to advise a manager of the service centre on certain The objective is to advise a manager of the service centre on certain decision policies to keep the customers satisfied.decision policies to keep the customers satisfied.

Case Study: Building a Fuzzy Case Study: Building a Fuzzy Expert SystemExpert System

Step 1: Specify the problem and define linguistic variables

There are four main There are four main linguistic variableslinguistic variables: average waiting time (mean : average waiting time (mean delay) delay) mm, repair utilisation factor of the service centre , repair utilisation factor of the service centre , number of , number of servers servers ss, and initial number of spare parts , and initial number of spare parts nn..

Linguistic variables and their ranges

Step 2: Determine fuzzy sets

Fuzzy sets can have a variety of shapes. Fuzzy sets can have a variety of shapes. However, a triangle or a trapezoid can often However, a triangle or a trapezoid can often provide an adequate representation of the expert provide an adequate representation of the expert knowledge, and at the same time, significantly knowledge, and at the same time, significantly simplifies the process of computation.simplifies the process of computation.

Fuzzy sets of Fuzzy sets of Mean Delay mMean Delay m

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Mean Delay (normalised)

Fuzzy sets of Fuzzy sets of Number of Servers sNumber of Servers s

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Number of Servers (normalised)

Fuzzy sets of Fuzzy sets of Repair Utilisation Factor Repair Utilisation Factor

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Repair Utilisation Factor

Fuzzy sets of Fuzzy sets of Number of Spares nNumber of Spares n

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S RSVS M RL L VL

Number of Spares (normalised)

Step 3: Elicit and construct fuzzy rules

To accomplish this task, we might ask the expert To accomplish this task, we might ask the expert to describe how the problem can be solved using to describe how the problem can be solved using the fuzzy linguistic variables defined previously.the fuzzy linguistic variables defined previously.

Required knowledge also can be collected from Required knowledge also can be collected from other sources such as books, computer other sources such as books, computer databases, flow diagrams and observed human databases, flow diagrams and observed human behaviour. behaviour.

The square FAM representationThe square FAM representation

The rule tableThe rule table

Rule Base 1Rule Base 1

Fuzzy If-Then RulesFuzzy If-Then Rules

• Mamdani style

If pressure is high then volume is small

high small

• Sugeno style

If speed is medium then resistance = 5*speed

mediumresistance = 5*speed

1996 Asian Fuzzy Systems Symposium

By using fuzzy sets, we can formulate fuzzy if-then rules that are commonly

used in our daily expressions. Basically, we have two types of fuzzy rules.

For Mamdani style, for instance, if pressure is high then volume is small,

where high? and small are described by fuzzy sets

For Sugeno style, if the speed of a moving object is medium then the

resistance due to atmosphere is 5 times the speed. The basic difference

between these two rules is in their THEN part, where Madman style has a

fuzzy but Surgeon style has a linear equation. Madman style fuzzy rules

were first proposed in the literature; they are more appealing to human

intuition. Surgeon style fuzzy rules are proposed later, but they are more

suited for mathematical design and analysis.

In this, we concentrate on Surgeon style fuzzy if-then rules.

Mamdani Fuzzy SystemMamdani Fuzzy System

Graphics representation:A1 B1

T-norm

X Yx is 4.5 y is 56.8 z is zCOA

Fuzzy Inference System (FIS)Fuzzy Inference System (FIS)

If speed is low then resistance = 2If speed is medium then resistance = 4*speedIf speed is high then resistance = 8*speed

Rule 1: w1 = .3; r1 = 2Rule 2: w2 = .8; r2 = 4*2Rule 3: w3 = .1; r3 = 8*2

Speed2

low medium high

Resistance = (wi*ri) / wi = 7.12

TSK Fuzzy System page 81TSK Fuzzy System page 81

• Rule baseIf X is A1 and Y is B1 then Z = p1*x + q1*y + r1

If X is A2 and Y is B2 then Z = p2*x + q2*y + r2

• Fuzzy reasoning

z1 =p1*x+q1*y+r1

z2 =p2*x+q2*y+r2

z =w1+w2

w1*z1+w2*z2

Tsukamoto Fuzzy SystemTsukamoto Fuzzy System

T-norm

X Yx is 4.5 y is 56.8

z =w1+w2

w1*z1+w2*z2

Zhang-Kandel Fuzzy SystemZhang-Kandel Fuzzy System

T-norm

X Yx is 4.5 y is 56.8

z =w1+w2

w1*z1+w2*z2

Z1={Za, Zb} Z2={Zc, Zd}

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