- Fuzzy Logic Tutorial
- Fuzzy Logic - Home
- Fuzzy Logic - Introduction
- Fuzzy Logic - Classical Set Theory
- Fuzzy Logic - Set Theory
- Fuzzy Logic - Membership Function
- Traditional Fuzzy Refresher
- Approximate Reasoning
- Fuzzy Logic - Inference System
- Fuzzy Logic - Database and Queries
- Fuzzy Logic - Quantification
- Fuzzy Logic - Decision Making
- Fuzzy Logic - Control System
- Adaptive Fuzzy Controller
- Fuzziness in Neural Networks
- Fuzzy Logic - Applications
- Fuzzy Logic Useful Resources
- Fuzzy Logic - Quick Guide
- Fuzzy Logic - Useful Resources
- Fuzzy Logic - Discussion
Fuzzy Logic - Approximate Reasoning
Following are the different modes of approximate reasoning −
Categorical Reasoning
In this mode of approximate reasoning, the antecedents, containing no fuzzy quantifiers and fuzzy probabilities, are assumed to be in canonical form.
Qualitative Reasoning
In this mode of approximate reasoning, the antecedents and consequents have fuzzy linguistic variables; the input-output relationship of a system is expressed as a collection of fuzzy IF-THEN rules. This reasoning is mainly used in control system analysis.
Syllogistic Reasoning
In this mode of approximation reasoning, antecedents with fuzzy quantifiers are related to inference rules. This is expressed as −
x = S1A′s are B′s
y = S2C′s are D′s
------------------------
z = S3E′s are F′s
Here A,B,C,D,E,F are fuzzy predicates.
S1 and S2 are given fuzzy quantifiers.
S3 is the fuzzy quantifier which has to be decided.
Dispositional Reasoning
In this mode of approximation reasoning, the antecedents are dispositions that may contain the fuzzy quantifier “usually”. The quantifier Usually links together the dispositional and syllogistic reasoning; hence it pays an important role.
For example, the projection rule of inference in dispositional reasoning can be given as follows −
usually( (L,M) is R ) ⇒ usually (L is [R ↓ L])
Here [R ↓ L] is the projection of fuzzy relation R on L
Fuzzy Logic Rule Base
It is a known fact that a human being is always comfortable making conversations in natural language. The representation of human knowledge can be done with the help of following natural language expression −
IF antecedent THEN consequent
The expression as stated above is referred to as the Fuzzy IF-THEN rule base.
Canonical Form
Following is the canonical form of Fuzzy Logic Rule Base −
Rule 1 − If condition C1, then restriction R1
Rule 2 − If condition C1, then restriction R2
.
.
.
Rule n − If condition C1, then restriction Rn
Interpretations of Fuzzy IF-THEN Rules
Fuzzy IF-THEN Rules can be interpreted in the following four forms −
Assignment Statements
These kinds of statements use “=” (equal to sign) for the purpose of assignment. They are of the following form −
a = hello
climate = summer
Conditional Statements
These kinds of statements use the “IF-THEN” rule base form for the purpose of condition. They are of the following form −
IF temperature is high THEN Climate is hot
IF food is fresh THEN eat.
Unconditional Statements
They are of the following form −
GOTO 10
turn the Fan off
Linguistic Variable
We have studied that fuzzy logic uses linguistic variables which are the words or sentences in a natural language. For example, if we say temperature, it is a linguistic variable; the values of which are very hot or cold, slightly hot or cold, very warm, slightly warm, etc. The words very, slightly are the linguistic hedges.
Characterization of Linguistic Variable
Following four terms characterize the linguistic variable −
- Name of the variable, generally represented by x.
- Term set of the variable, generally represented by t(x).
- Syntactic rules for generating the values of the variable x.
- Semantic rules for linking every value of x and its significance.
Propositions in Fuzzy Logic
As we know that propositions are sentences expressed in any language which are generally expressed in the following canonical form −
s as P
Here, s is the Subject and P is Predicate.
For example, “Delhi is the capital of India”, this is a proposition where “Delhi” is the subject and “is the capital of India” is the predicate which shows the property of subject.
We know that logic is the basis of reasoning and fuzzy logic extends the capability of reasoning by using fuzzy predicates, fuzzy-predicate modifiers, fuzzy quantifiers and fuzzy qualifiers in fuzzy propositions which creates the difference from classical logic.
Propositions in fuzzy logic include the following −
Fuzzy Predicate
Almost every predicate in natural language is fuzzy in nature hence, fuzzy logic has the predicates like tall, short, warm, hot, fast, etc.
Fuzzy-predicate Modifiers
We discussed linguistic hedges above; we also have many fuzzy-predicate modifiers which act as hedges. They are very essential for producing the values of a linguistic variable. For example, the words very, slightly are modifiers and the propositions can be like “water is slightly hot.”
Fuzzy Quantifiers
It can be defined as a fuzzy number which gives a vague classification of the cardinality of one or more fuzzy or non-fuzzy sets. It can be used to influence probability within fuzzy logic. For example, the words many, most, frequently are used as fuzzy quantifiers and the propositions can be like “most people are allergic to it.”
Fuzzy Qualifiers
Let us now understand Fuzzy Qualifiers. A Fuzzy Qualifier is also a proposition of Fuzzy Logic. Fuzzy qualification has the following forms −
Fuzzy Qualification Based on Truth
It claims the degree of truth of a fuzzy proposition.
Expression − It is expressed as x is t. Here, t is a fuzzy truth value.
Example − (Car is black) is NOT VERY True.
Fuzzy Qualification Based on Probability
It claims the probability, either numerical or an interval, of fuzzy proposition.
Expression − It is expressed as x is λ. Here, λ is a fuzzy probability.
Example − (Car is black) is Likely.
Fuzzy Qualification Based on Possibility
It claims the possibility of fuzzy proposition.
Expression − It is expressed as x is π. Here, π is a fuzzy possibility.
Example − (Car is black) is Almost Impossible.