Fuzzy Logic

Content…

• Truth Values and Tables in Fuzzy Logic• Fuzzy Propositions• Formation of Rules (Fuzzy Rule Based System)• Decomposition of Rules (Compound Rules)• Aggregation of Fuzzy Rules• Fuzzy Reasoning (Approximate Reasoning)• Fuzzy Inference Systems (FIS)– Mamdani FIS– Sugeno FIS

Truth Values and Tables in Fuzzy Logic

• Fuzzy Logic uses linguistic variables.• A linguistic variable is a variable whose values

are sentences in a natural or artificial language.

i.e., height is a linguistic variable if it takes values such as tall, medium, short, very tall, and so..

It provides approximate characterization of a complex problem.

• A part from the linguistic variables there exists what are called as linguistic hedges (linguistic modifiers).

i.e., in the fuzzy set “very tall”, the word “very” is a linguistic hedge.

A popular linguistic hedges include : very, highly, slightly, moderately, plus, minus, fairly, rather.

• Reasoning has logic as its basis, whereas propositions are text sentences expressed in any language and are generally expressed in an canonical form as

Z is P,• Where

Z = symbol of the subjectP = predicate designing the characteristics of the

subject

Fuzzy Propositions• For extending the reasoning capability fuzzy logic uses

following propositions:1) Fuzzy Predicate: In fuzzy logic, the predicate can be fuzzy. For

example short, tall, high..2) Fuzzy-predicate modifiers: It acts as hedges, for example very

high, very fairly, moderately,.. These are necessary for generating the values of a linguistic variable.

3) Fuzzy quantifiers: such as most, several, many,..4) Fuzzy qualifiers:

1) Fuzzy truth qualificationNOT very true

2) Fuzzy probability qualificationLikely, very likely, unlikely

3) Fuzzy possibility qualificationpossible, quite possible, Almost Impossible

4) Fuzzy usually qualificationHigh probability of occurrence, usually

Formation of Rules(Fuzzy Rule Based System)

• The general way of representing human knowledge is by forming natural language expressions given by

IF antecedent THEN consequent.• It is referred to as the IF-THEN rule based form.• There are three general forms that exist for any linguistic

variable. They are:– Assignment statements;– Conditional statements;– Unconditional statements;

• Statements are connected by linguistic connectives such as “and”, “or”, or “else”.

Decomposition of Rules(Compound Rules)

• It is a collection of many simple rules combined together.

• Any compound rule structure may be decomposed and reduced to a number of simple canonical rule forms.

• The following are the methods used for decomposition of compound linguistic rules into simple canonical rules.1. Multiple conjunctive antecedents2. Multiple disjunctive antecedents3. Conditional statements (with ELSE and UNLESS)4. Nested IF-THEN rules

• 1. Multiple conjunctive antecedentsIF x is A1, A2, . . . . . . , An THEN y is Bm.

Assume a new fuzzy subset Am defined as

Am = A1 ∩ A2∩ . . . . . ∩ An

and expressed by means of membership function μAm(x) = min [μA1(x), μA2(x), . . . , μAn(x)] .

In view of the fuzzy intersection operation, the compound rule may be rewritten as IF Am THEN Bm.

• 2. Multiple disjunctive antecedentsIF x is A1, OR x is A2, . . . . . . , OR x is An THEN y is Bm.

Assume a new fuzzy subset Am defined as

Am = A1 U A2 U . . . . . U An

and expressed by means of membership function μAm(x) = max [μA1(x), μA2(x), . . . , μAn(x)] .

In view of the fuzzy union operation, the compound rule may be rewritten as IF Am THEN Bm.

• 3. Conditional statements (with ELSE and UNLESS)IF A1 THEN ( B1 ELSE B2 )

Can be decomposed into two simple canonical rule forms, connected by “OR”:

IF A1 THEN B1

OR

IF NOT A1 THEN B2

IF A1 ( THEN B1 ) UNLESS A2

Can be decomposed asIF A1 THEN B1

ORIF A2 THEN NOT B1

• 4. Nested IF - THEN rules:

The rule “IF A1 THEN [ IF A2 THEN ( B1 ) ]”

Can be form as IF A1 AND IF A2 THEN B1.

Aggregation of Fuzzy Rules

• The Rule based system involves more than one rule. Aggregation of rules is the process of obtaining the overall consequents from the individual consequents provided by each rule.

• The following two methods exists that aid in determining the aggregation of fuzzy rules:1. Conjunctive System of Rules2. Disjunctive System of Rules

Aggregation of Fuzzy Rules

1. Conjunctive System of Rules– Rules are connected by “and”– y = y1 ∩ y2∩ . . . . . ∩ yn

– μy(y) = min [μy1(y), μy2(y), . . . , μyn(y)] for y ϵ Y.

1. Disjunctive System of Rules– Rules are connected by “or”– y = y1 U y2 U . . . . . U yn

– μy(y) = max [μy1(y), μy2(y), . . . , μyn(y)] for y ϵ Y.

Fuzzy Reasoning (Approximate Reasoning)

• Fuzzy reasoning is the collection of Formation of Rules, Decomposition of Rules and Aggregation of Rules.

• There exists four modes of fuzzy approximate reasoning, which include:1) Categorical reasoning2) Qualitative reasoning3) Syllogistic reasoning4) Dispositional reasoning

1. Categorical Reasoning– In this type of Reasoning, the antecedents contain

no fuzzy quantifiers and fuzzy probabilities.– The antecedents are assumed to be in canonical

form.– Consider, L, M, N, ….. = fuzzy variable taking in

the universes U, V, W;– A, B, C = fuzzy predicates.

1) The Projection rule of inference is defined by

L, M, is R

L is [R L]

Where [R L] denotes the projection of fuzzy relation R on L.

2) The conjunction rule of inference is given by

L is A, L is B => L is A ∩ B

(L, M) is A, L is B => (L, M) is A ∩ (B x V)

(L,M) is A, (M,N) is B=>(L,M,N) is (A x W)∩(U x B)

3) The disjunction rule of inference is given by

L is A OR L is B => L is A x B

L is A, M is B => (L, M) is A x B

4) The negative rule of inference is given by

NOT (L is A) => L is Ᾱ

5) The compositional rule of inference is given by

L is A, (L, M) is R => M is A ◦ R

6) The extension principle is defined as

L is A => f(L) is f(A)

Where “f” is a mapping from u to v so that L is mapped into f(L); and based on the extension principle, the membership function of f(A) is defined as

μf(A) (v) = Sup μA(u) u ϵ U, v ϵ V.v=1 f(u)

2. Qualitative Reasoning• In qualitative reasoning the input-output relationship of a system

is expressed as a collection of fuzzy IF THEN rules.• Qualitative reasoning is widely used in control system analysis.• Let A and B is the fuzzy input variables and C is the fuzzy output

variable;

If A is x1 AND B is y1 , THEN C is z1.

If A is x2 AND B is y2 , THEN C is z2.

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If A is xn AND B is yn , THEN C is zn.

3. Syllogistic Reasoning• In syllogistic reasoning, antecedents with fuzzy quantifiers are related to

inference rules. x = k1 A’s are B’s

y = k2 C’s are D’s

z = k3 E’s are F’s

• In the above A, B, C, D, E and F are fuzzy predicates; k1 and k2 are given fuzzy quantifiers and k3 is the fuzzy quantifier which has to be decided.

• All the fuzzy predicates provide a collection of fuzzy syllogisms. These syllogisms create a set of inference rules, which combines evidence through conjunction and disjunction.

i. Produce syllogism: C ◦ A Λ B, F = C Λ Dii. Chaining syllogism: C = B, F = D, E = Aiii. Consequent conjunction syllogism: F = B Λ D, A = C = Eiv. Consequent disjunction syllogism: F = B v D, A = C = Ev. Precondition conjunction syllogism: E = A Λ C, B = D = Fvi. Precondition disjunction syllogism: E = A v C, B = D = F

4. Dispositional Reasoning• In this kind of reasoning, the antecedents are dispositions that may contain,

implicitly or explicitly, the fuzzy quantifier “usually”. Usuality plays a major role in dispositional reasoning and it links together the dispositional and syllogistic modes of reasoning.

1) Dispositional projection rule of inference:

Usually ((L, M) is R) => usually ( L is [R ↓ L])

2) Dispositional chaining hypersyllogism:

k1 A’s are B’s, k2 B’s are C’s, usually (B subset A)

Usually ( → (k1 (◦) k2 ) A’s are C’s are).

3) Dispositional consequent conjunction syllogism:

Usually ( A’s are B’s), usually (A’s are C’s) =>

( 2 usually (-) 1 ( A’s are (B and C)’s )

is a specific case of dispositional reasoning.

4) Dispositional entailment rule of inference:

Usually (x is A), A subset B => usually ( x is B)

X is A, usually (A subset B) => usually ( x is B)

Usually (x is A), usually (A subset B) => usually2 ( x is B)

is the dispositional entailment rule of inference. Here “usually2 “ is less specific than “usually”.