Fuzzy logic - Approximate reasoning

Department of Information Technology 1Soft Computing (ITC4256 )
Dr. C.V. Suresh Babu
Professor
Department of IT
Hindustan Institute of Science & Technology
Approximate reasoning

Action Plan
• Truth values and tables,
• Fuzzy propositions,
• Formation of rules decomposition of rules,
• Aggregation of fuzzy rules,
• Fuzzy reasoning‐fuzzy inference systems
• Overview of fuzzy expert system‐
• Fuzzy decision making.
• Quiz at the end of session`

Truth values and tables
Truth values have been put to quite different uses in philosophy and logic, being characterized, for
example, as:
• primitive abstract objects denoted by sentences in natural and formal languages,
• abstract entities hypostatized as the equivalence classes of sentences,
• what is aimed at in judgements,
• values indicating the degree of truth of sentences,
• entities that can be used to explain the vagueness of concepts,
• values that are preserved in valid inferences,
• values that convey information concerning a given proposition.

Fuzzy propositions
• Fuzzy Proposition:
– The proposition whose truth value is [0,1]
– Classification of Fuzzy Proposition
• Unconditional or Conditional
• Unqualified of Qualified
– Focus on how a proposition can take truth value
from fuzzy sets, or membership functions.

Fuzzy Proposition
• Unconditional and Unqualified
– Example:
)()(
.ofgradeMembership)(
predicatefuzzy,onsetFuzzy
valuessthat takeVariable
.is:
vFpT
vvF
VF
Vv
Fp





)"(is75temp"ofeTruth valu)(
]120,0[valuessthat take
.is:
vFHighpT
vAir temp.
HighAir temp.p
o
oo




))(()(
predicatefuzzy,onsetFuzzy
)tosetoffunctionais(Variable:
.is)(:
ivFpT
VF
VIVI
Fip




)())(()(
[0,1])(:
]100,0[},,{:
.is)(:
vFiAgeYoungpT
iageYoung
CBAage
youngiagep




Unconditional and Qualified Propositions
• Truth qualified and Probability qualified
– Truth qualified
“Tina is young is very true”
"is"eventsfuzzyofyProbabilit}isPr{
[0,1]onsetfuzzyAquantifieryprobabilitFuzzy
[0,1]onsetfuzzyAquantifierFuzzy
.is}isPr{:
or
.isis:
FF
P
S
PFp
SFp






0.760.87)())(()(0.87)26(26)(  SvFSpTFTinaAge

Unconditional and Qualified Propositions
– Probability qualified
– Note:
Truth quantifiers = “True, False” with hedges
Probability quantifiers =“Likely, Unlikely” with hedges
.95)80()(
.is}75isPr{temp.:
:Example
))()(()(
)()(}isPr{
,onondistributiprob.givenanyFor







.PpT
likelyaroundtp
vFvfPpT
vFvfF
Vf
o
Vv
Vv


Conditional and Unqualified Propositions
• Conditional and Unqualified
– Example with Lukaseiwicz implication
relationnImplicatioFuzzy),())(),(()),(()(
andonsetFuzzy,
setinarevalueswhoseVariables
.isthen,isIf:



yxRyBxAΙyxTpT
YXBA
YX
BAp
YX,
YX
12
11
21321
and,when0.7
and,when1)(
15.
17.
11
)1,1min(),(
/.1/5./.1/8./1.
yYxX
yYxXpT
babaR
yyBxxxA















Conditional and Qualified Propositions
• Conditional and Qualified






Yy
Y
X Y
XY
yfyB
yxfyxR
BYAX
BYAXPpT
PBYAXp
yxRSpT
SBYAXp
)()(
),(),(
}is|isPr{
})is|is(Pr{)(
.is}is|isPr{:
or
)),(()(
.isisthen,isIf:

Fuzzy Quantifiers
• Absolute Quantifiers
– Fuzzy Numbers:
about 10, much more than 100, at least 5
|)(|)(
allfor))(()(
.")high(is))(English(influencywhose
)class(givenain)students(i)10(aboutareThere"
.is)(such thatinsi'areThere:
EQpT
IiiVFiE
FiV
IQ
FiVIQp


0.625(2.25))(2.25
0.5Flu(70)High(Eve))Flu(High
1.Flu(95)High(David))Flu(High
0.75Flu(80)High(Cathy))Flu(High
0Flu(20)High(Bob))Flu(High
0Flu(35)High(Adams))Flu(High
Eve}David,Cathy,Bob,{Adam,I
.")high(is))(English(influencywhose
:Example







QpTE
V
V
V
V
V
FiV
IQ

Fuzzy Quantifiers
– Fuzzy Number with Connectives
))])(()),((min[(|)(|)(
allfor))(()(and))(()(
.")young(are))(who(and)high(is))(English(influencywhose
.is)(andis)(such thatinsi'areThere:
221121
222111
2211
2211
iVFiVFQEEQpT
IiiVFiEiVFiE
FiVFiV
IQ
FiVFiVIQp
Ii




Fuzzy Quantifiers
• Relative Quantifier
– Example: “almost all”, “about half”, ”most”
• See Fig. 8.5
)()(
subsethoodofdegree
))((
))](()),((min[
allfor))(()(and))(()(
.")high(is))(English(influencywhoses)i'all(almostarethere
),young())(are(that)class(givenain)students(Among"
.is)(such that
insi'arethere,is)(such thatinsi'Among:
11
2211
1
21
222111
22
1
22
11
WQpT
iVF
iVFiVF
E
EE
W
IiiVFiEiVFiE
FiVQ
FiVIi
FiV
IQFiVIp
i
i








Linguistic Hedges
• Modifiers
– “very”, ”more or less”, “fairly”, “extremely”
– Interpretation
– Example: Age(John)=26 Young(26)=0.8
Very Young(26)=0.64
Fairly Young(26)=0.89
aaHlessormorefairly
aaHextremelyvery
aaHxFH



)(or
)(or
:Example
]1,0[where),())((
2

Inference from Conditional Fuzzy Propositions
• Crisp Case
)],(),(min[or},,|{
relationbyintervalfromInterval.4
},|{
relationbypointfromInterval.3
if)(
)(
functionbyintervalfromInterval.2
if)(
)(
functionbypointfromPoint.1
sup
000
yxχxχχAxRyxYyB
RyxYyB
AxAfB
xfy
xxxfy
xfy
RA
Xx
B








• Fuzzy Case
– Compositional Rule of Inference
– Modus Ponen
RAB
yxRxAyB
XAYXR
'
'
Xx
'

or
)],(),(min[)(
then,onsetfuzzyais,onrelationfuzzythe,Given
'
'
sup




)],(),(min[)(
Fact)(Newis
(Fact)is:
))()(1,1min())(),(Im(),(
(Rule).isthen,isIf:
sup'
'
'
yxRxAyB
BY
AXq
yBxAyBxAyxR
BYAXp
'
Xx



– Modus Tollen
– Hypothetical Syllogism
case.crispin:Note
)],(),(min[)(
Fact)(Newis
(Fact)is:
))()(1,1min())(),(Im(),(
(Rule).isthen,isIf:
''
'
'
'
sup
B, BAA
yxRyByA
AX
BYq
yBxAyBxAyxR
BYAXp
'
Yy




)],(),,(min[),(
))(),(Im(),(n)(Conclusio.isthen,isIf
))(),(Im(),(2)(Rule.isthen,isIf:
))(),(Im(),(1)(Rule.isthen,isIf:
213
3
2
1
sup zyRyxRzxR
zCxAzxRCZAX
zCyBzyRCZBYq
yBxAyxRBYAXp
Yy





Aggregation of fuzzy rules
• The process of obtaining the overall conclusion from the individually mentioned consequents
contributed by each rule in the fuzzy rule this is known as aggregation of rule.
• (1) Conjunctive system of rules
• The rules that are connected by “AND” connectives satisfy the connective system of rules. In this
case, the aggregated output may be found by the fuzzy intersection of all individual rule
consequents.
• (2) Disjunctive system of rules
• The rules that are connected by “OR” connectives satisfies the disjunctive system of rules. In this
case, the aggregated output may be found by the fuzzy union of all individual rule consequents

Properties of set of rules
The properties for the sets of rules are
– Completeness,
– Consistency,
– Continuity, and
– Interaction.
(a) Completeness
A set of IF–THEN rules is complete if any combination of input values result in an appropriate output value.
(b) Consistency
A set of IF–THEN rules is inconsistent if there are two rules with the same rules-antecedent but different rule-
consequents.
(c) Continuity
A set of IF–THEN rules is continuous if it does not have neighbouring rules with output fuzzy sets that have empty
intersection.
(d) Interaction
In the interaction property, suppose that is a rule, “IF x is A THEN y is B,” this meaning is represented by a fuzzy relation
R2, then the composition of A and R does not deliver B

Fuzzy reasoning‐fuzzy inference systems
• Fuzzy Inference Systems(FIS)
• •Known as fuzzy rule-based systems, fuzzy model, fuzzy expert system, and fuzzy associative
memory.
• •The FIS formulates suitable rules and based upon the rules the decision is made.
• •Mainly based on the concepts of the fuzzy set theory, fuzzy IF–THEN rules, and fuzzy reasoning.

Fuzzy Inference Systems (FIS)
Fuzzy Inference Methods
Mamdani Fuzzy Inference Model
-Commonly used
-Introduced by Mamdani and Assilian in 1975
-Uses fuzzy sets as rule consequent
Sugeno or Takagi-Sugeno-Kang method
-Introduced by Sugeno in 1985
-Employs linear functions of input variables as rule consequent
All the existing results on fuzzy systems as universal approximators deal with Mamdani fuzzy sys-
tems only and no result is available for TS fuzzy systems with linear rule consequent.

Construction and Working Of FIS
Construction and Working of Inference System
-Consists of a fuzzification interface, a rule base, a database, a decision-making unit, and finally a defuzzification
interface.
The function of each block is as follows:
– a rule base containing a number of fuzzy IF–THEN rules;
– a database which defines the membership functions of the fuzzy sets used in the fuzzy rules;
– a decision-making unit which performs the inference operations on the rules;
– a fuzzification interface which transforms the crisp inputs into degrees of match with linguistic values;
– a defuzzification interface which transforms the fuzzy results of the inference into a crisp output.

Working Of FIS
Working of FIS:
Conversion of crisp input to fuzzy by fuzzification
Formation of rule base
(Rule base and database are referred jointly as knowledge base) Defuzzification-Conversion of fuzzy
value to real world values
Exact steps:
1. Compare the input variables with the membership functions on the antecedent part to obtain
the membership values of each linguistic label. (this step is often called fuzzification.)
2. Combine (through a specific t-norm operator, usually multiplication or min) the membership
values on the premise part to get firing strength (weight) of each rule.
3. Generate the qualified consequents (either fuzzy or crisp) or each rule depending on the firing
strength.
4. Aggregate the qualified consequents to produce a crisp output. (This step is called
defuzzification.)

Overview of fuzzy expert system
Meta KB
Knowledge
Base
Knowledge
Aq. Module
Expert User
Explanatory
Interface
Inference
Engine
Data Base
(Fact)
Expert System

Expert System
– Knowledge Base (Long-Term Memory)
• Fuzzy Production Rules (If-Then)
– Data Base (Short-Term Memory)
• Fact from user or Parameters
– Inference Engine
• Data Driven (Forward Chaining, Modus Ponen)
• Goal Driven (Backward Chaining, Modus Tollen)
– Meta-Knowledge Base
– Explanatory Interface
– Knowledge Acquisition Module

Advantages of rule-based expert system
Natural knowledge representation – an expert usually explains the problem-solving procedure with “In
such-and-such situation, I do so-and-so”. represented quite naturally as IF-THEN production rules.
• Uniform structure: production rules have uniform IF-THEN structure. Each rule is an independent
piece of knowledge (self-documented)
• Separation of knowledge from its process
The structure provides an effective separation of the knowledge base from the inference engine. This
makes it possible to develop different applications using the same expert system shell.
• Dealing with incomplete and uncertain knowledge
Capable of representing and reasoning with incomplete and uncertain knowledge

Disadvantages of rule-based expert systems
Opaque relations between rules.
Although individual production rules are relatively simple and self-documented, their logical interactions
within large set of rules may be opaque. Rule-based systems make it difficult to observe how individual
rules serve the overall strategy.
• Ineffective search strategy
The inference engine applies an exhaustive search through all the production rules during each cycle with
a large set of rules (over 100 rules) can be slow, and thus large rule-based systems can be unsuitable for
real-time applications
•Inability to learn
In general, rule-based expert systems do not have an ability to learn from experience.
Unlike a human expert, who knows when to “break the rules”, an expert system cannot automatically
modify its knowledge base, or adjust existing rules or add new ones.

Fuzzy decision making
By decision-making in a fuzzy environment is meant a decision process in which the goals and/or the
constraints, but not necessarily the system under control, are fuzzy in nature. This means that the
goals and/or the constraints constitute classes of alternatives whose boundaries are not sharply
defined.
Steps for Decision Making
Let us now discuss the steps involved in the decision making process −
Determining the Set of Alternatives − In this step, the alternatives from which the decision has to be
taken must be determined.
Evaluating Alternative − Here, the alternatives must be evaluated so that the decision can be taken
about one of the alternatives.
Comparison between Alternatives − In this step, a comparison between the evaluated alternatives is
done.

Test Yourself
1. What is the Fuzzy Approximation Theorem(FAT) ?
a) fuzzy system can model any continuous system
B. The conversion of fuzzy logic to probability.
C. A continuous system can model any fuzzy system.
D. Fuzzy patches covering a series of fuzzy rules.
2. Fuzzy logic is usually represented as ___________
a) IF-THEN-ELSE rules
b) IF-THEN rules
c) Both IF-THEN-ELSE rules & IF-THEN rules
d) None of the mentioned
3. The values of the set membership is represented by ___________
a) Discrete Set
b) Degree of truth
c) Probabilities
d) Both Degree of truth & Probabilities
4. When capturing tacit knowledge, which of the following technologies would not be used?
a) Fuzzy logic systems
b) Expert systems.
c) Case-based reasoning.
d) Virtual reality
5. The inference engine is:
a) A method of organizing expert system knowledge into chunks.
b) A strategy for searching the rule base in an expert system that begins with information entered by the user.
c) The programming environment of an expert system.
d) A strategy used to search through the rule base in an expert system.

Answers
1. What is the Fuzzy Approximation Theorem(FAT) ?
a) fuzzy system can model any continuous system
B. The conversion of fuzzy logic to probability.
C. A continuous system can model any fuzzy system.
D. Fuzzy patches covering a series of fuzzy rules.
2. Fuzzy logic is usually represented as ___________
a) IF-THEN-ELSE rules
b) IF-THEN rules
c) Both IF-THEN-ELSE rules & IF-THEN rules
d) None of the mentioned
3. The values of the set membership is represented by ___________
a) Discrete Set
b) Degree of truth
c) Probabilities
d) Both Degree of truth & Probabilities
4. When capturing tacit knowledge, which of the following technologies would not be used?
a) Fuzzy logic systems.
b) Expert systems.
c) Case-based reasoning.
d) Virtual reality.
5. The inference engine is:
a) A method of organizing expert system knowledge into chunks.
b) A strategy for searching the rule base in an expert system that begins with information entered by the user.
c) The programming environment of an expert system.
d) A strategy used to search through the rule base in an expert system.

Fuzzy logic - Approximate reasoning

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