SoftComputing.pdf

 Text Book:Neuro Fuzzy and Soft Computing
by J.S.R.Jang and C.T.Sun,Prentice Hall.
 Reference Books:Fuzzy logic with Engg
App,Timothy J Ross,Willey Pub.
 Soft Computing and Its application,Vol 1
K.S.Ray,Apple Academic Press.
 First Course on Fuzzy Theory and
App.K.H.Lee,Spinger.
 Fuzzy Set theory and its
app,H.Z.Zimmermann,Spinger Science

 The idea behind soft computing is to model cognitive
behavior of human mind.
 Soft computing is foundation of conceptual intelligence
in machines.
 Unlike hard computing , Soft computing is tolerant of
imprecision, uncertainty, partial truth, and
approximation.

∙ Hard computing
− Based on the concept of precise modeling and analyzing
to yield accurate results.
− Works well for simple problems, but is bound by the
NP-Complete set.
∙ Soft computing
− Aims to surmount NP-complete problems.
− Uses inexact methods to give useful but inexact answers
to intractable problems.
− Represents a significant paradigm shift in the aims of
computing - a shift which reflects the human mind.
− Tolerant to imprecision, uncertainty, partial truth, and
approximation.
− Well suited for real world problems where ideal models
are not available.

 Can all computational problems be solved by a computer?
 There are computational problems that can not be solved by
algorithms even with unlimited time.
 For example Turing Halting problem (Given a program and an
input, whether the program will eventually halt when run with
that input, or will run forever)
 Alan Turing proved that general algorithm to solve the halting
problem for all for all possible program-input pairs cannot
exist
 A key part of the proof is, Turing machine was used as a
mathematical definition of a computer and program (Source
Halting Problem).

 NP complete problems are problems whose status is
unknown.
 No polynomial time algorithm has yet been discovered for any
NP complete problem, nor has anybody yet been able to
prove that no polynomial-time algorithm exist for any of
them.
 The interesting part is, if any one of the NP complete
problems can be solved in polynomial time, then all of them
can be solved.

 P is set of problems that can be solved by a
deterministic Turing machine in Polynomial
time.
 NP is set of decision problems that can be
solved by a Non-deterministic Turing
Machine in Polynomial time.
 P is subset of NP (any problem that can be
solved by deterministic machine in
polynomial time can also be solved by non-
deterministic machine in polynomial time).

 NP-complete problems are the hardest problems in
NP set. A decision problem L is NP-complete if:
 1) L is in NP (Any given solution for NP-complete
problems can be verified quickly, but there is no
efficient known solution)
 2) Every problem in NP is reducible to L in
polynomial time
 A problem is NP-Hard if it follows property 2
mentioned above, doesn’t need to follow property
1. Therefore, NP-Complete set is also a subset of
NP-Hard set

Hard Computing Soft Computing
Conventional computing requires a
precisely stated analytical model.
Soft computing is tolerant of
imprecision.
Often requires a lot of computation time. Can solve some real world problems in
reasonably less time.
Not suited for real world problems for
which ideal model is not present.
Suitable for real world problems.
It requires full truth Can work with partial truth
It is precise and accurate Imprecise.
High cost for solution Low cost for solution

• Soft Computing is an approach for constructing
systems which are
− computationally intelligent,
− possess human like expertise in particular domain,
− can adapt to the changing environment and can learn
to do better
− can explain their decisions

∙ Components of soft computing include:
− Fuzzy Logic (FL)
− Evolutionary Computation (EC) - based on the
origin of the species
➢ Genetic Algorithm
➢ Swarm Intelligence
➢ Ant Colony Optimizations
− Neural Network (NN)
− Machine Learning (ML)

 AI: predicate logic and symbol
manipulation techniques
User
Interface
Inference
Engine
Explanation
Facility
Knowledge
Acquisition
KB:•Fact
•rules
Global
Database
Knowledge
Engineer
Human
Expert
Question
Response
Expert Systems
User

ANN
Learning and
adaptation
Fuzzy Set Theory
Knowledge representation
Via
Fuzzy if-then RULE
Genetic Algorithms
Systematic
Random Search
AI
Symbolic
Manipulation

cat
cut
knowledge
Animal? cat
Neural character
recognition

 Conventional AI:
◦ Focuses on attempt to mimic human
intelligent behavior by expressing it in
language forms or symbolic rules
◦ Manipulates symbols on the assumption
that such behavior can be stored in
symbolically structured knowledge bases
(physical symbol system hypothesis)

 Intelligent Systems
Sensing Devices
(Vision)
Natural
Language
Processor
Mechanical
Devices
Perceptions
Actions
Task
Generator
Knowledge
Handler
Data
Handler Knowledge
Base
Machine
Learning
Inferencing
(Reasoning)
Planning

8/6/2023 21
• The real world problems are pervasively
imprecise and uncertain
• Precision and certainty carry a cost
• Some problems may not even have any
precise solution
• may not even have any precise solutions
Premises of Soft Computing

8/6/2023 22
The guiding principle of soft computing is:
•Exploit the tolerance for imprecision,
uncertainty, partial truth, and
approximation to achieve non-conventional
solutions, tractability (easily handled,
managed, or controlled), robustness and
low costs.
Guiding Principle of Soft Computing

8/6/2023 23
Hard Computing
•Premises and guiding principles of Hard
Computing are
- Precision, Certainty, and Rigor.
• Many contemporary problems do not lend
themselves to precise solutions such as
- Recognition problems (handwriting,
speech, objects, images, texts)
- Mobile robot coordination, forecasting,
combinatorial problems etc.
- Reasoning on natural languages

 The man is about eighty to eighty five years
old(pure imprecision)
 The man is very old(imprecision and
vagueness)
 The man is probably from India(uncertainty)

8/6/2023 25
•Soft computing employs ANN, EC, FL etc, in a
complementary rather than a competitive way.
• One example of a particularly effective
combination is "neurofuzzy systems.”
• Such systems are becoming increasingly
visible
as consumer products ranging from air
conditioners and washing machines to
photocopiers, camcorders and many industrial
applications.
Implications of Soft Computing

8/6/2023 26
Unique Property of Soft computing
• Learning from experimental data →
generalization
• Soft computing techniques derive their power
of generalization from approximating or
interpolating to produce outputs from previously
unseen inputs by using outputs from previous
learned inputs
• Generalization is usually done in a high
dimensional space.

8/6/2023 27
• Handwriting recognition
• Automotive systems and manufacturing
• Image processing and data compression
• Architecture
• Decision-support systems
• Data Mining
• Power systems
• Control Systems
Current Applications using Soft
Computing

 What is fuzzy thinking
◦ Experts rely on common sense when they solve
the problems
◦ How can we represent expert knowledge that
uses vague and ambiguous terms in a computer
◦ Fuzzy logic is not logic that is fuzzy but logic that
is used to describe the fuzziness. Fuzzy logic is
the theory of fuzzy sets, set that calibrate the
vagueness.
◦ Fuzzy logic is based on the idea that all things
admit of degrees. Temperature, height, speed,
distance, beauty – all come on a sliding scale.
Jim is tall guy
It is really very hot today

 Communication of “fuzzy “ idea
This box is
too heavy.. Therefore, we
need a lighter
one…

 Boolean logic
◦ Uses sharp distinctions. It forces us to
draw a line between a members of class
and non members.
 Fuzzy logic
◦ Reflects how people think. It attempt to
model our senses of words, our decision
making and our common sense -> more
human and intelligent systems

 Classical Set vs Fuzzy set

1
0
175 Height(cm)
1
0
175 Height(cm)
Universe of discourse
Membership value Membership value






=
→
A
x
A
x
x
f
X
x
f A
A
if
,
0
if
,
1
)
(
where
},
1
,
0
{
:
)
(
Let X be the universe of discourse and its elements be denoted as x.
In the classical set theory, crisp set A of X is defined as function fA(x) called the
the characteristic function of A
In the fuzzy theory, fuzzy set A of universe of discourse X is defined by function
called the membership function of set A
)
(x
A

.
in
partly
is
if
1
)
(
0
;
in
not
is
if
0
)
(
;
in
totally
is
if
1
)
(
],
1
,
0
[
:
)
(
A
x
x
A
x
x
A
x
x
where
X
x
A
A
A
A


=
=
→





 An example:
◦ Define the seven levels of education:
36
Highly
educated (0.8)
Very highly
educated (0.5)

 Several fuzzy sets representing linguistic concepts such as low,
medium, high, and so one are often employed to define states of a
variable. Such a variable is usually called a fuzzy variable.
 For example:
37

 Given a universal set X, a fuzzy set is defined by a
function of the form
This kind of fuzzy sets are called ordinary fuzzy
sets(type 1 fuzzy set).
L-fuzzy set is ,
L is partial order set
 Interval-valued fuzzy sets:
◦ The membership functions of ordinary fuzzy sets are often
overly precise. We may be able to identify appropriate
membership functions only approximately.
◦ .
]
1
,
0
[
: →
X
A
38
Power set
:
A X L
→

 Interval-valued fuzzy sets: a fuzzy set
whose membership functions does not
assign to each element of the universal set
one real number, but a closed interval of
real numbers between the identified lower
and upper bounds.
]),
1
,
0
([
: 
→
X
A

 Fuzzy sets of type 2:
◦ : the set of all ordinary fuzzy sets that can be defined
with the universal set [0,1].
◦ is also called a fuzzy power set of [0,1].
42

 Discussions:
◦ The primary disadvantage of interval-value fuzzy sets,
compared with ordinary fuzzy sets, is computationally
more demanding.
◦ The computational demands for dealing with fuzzy sets
of type 2 are even greater then those for dealing with
interval-valued fuzzy sets.
◦ This is the primary reason why the fuzzy sets of type 2
have almost never been utilized in any applications.
43

 Let Set A=“adult”. The MF of this set maps the
entire range of ‘age’ to ‘infant’, ’young’,
’adult’ ,’senior’.
 The values of MFs for ‘infant’, ’young’etc are
FSs.Thus set ‘adult’ is type-2 FS. The sets
‘infant’, ’young’, and so on are type-1 FS.
If the values of MF of ‘infant’, ’young’ and so
on are type -2 ,the set ‘adult ‘is ……….

 Find: support,core,crossover point,
alpha cut(0.7) of A
Magnitude of FS
Cardinality
Relative Cardinality

 E=Complete Relation, O=Null Relation

 Let R1 is a Tolerance Relation

 R1 can become equivalence relation through
one composition R1oR1

 Work out similarly for division

 The operation of projection decreases the
dimension of given MF

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

 Syntactic rule: refers to the way the linguistic
values in the term set T(age) are generated.
 Semantic rule: defines the MFs of each
linguistic value of the term set.


 Assume:very=too, extremely=vary very very

 A coupled with B
 A entails B

 w the degree of belief for the antecedent part
of a rule,gets propagated by the if-then rules
and the resulting degree of belief or MF for
the consequent part should be no greater
than w

 Fuzzy rule based system,Fuzzy expert
system,Fuzzy model,Fuzzy associative
memory,Fuzzy logic contoller,

 Mean-max MF(middle-of-maxima)

 Center of largest area
 First or last of maxima
 Find the crisp value using all the
defuzzification methods methods

 Mean-max MF method=(6+7)/2=6.5m

 If the out put of the fuzzy set has at least two
convex sub regions then the center of gravity
of the convex fuzzy sub region with the
largest area is used to obtain the defuzzified
value of the output z*

 Advantage of sum-product composition is that

 Consider three fuzzy sets that represent the concepts of a
young, middle-aged, and old person. The membership
functions are defined on the interval [0,80] as follows:
251
Find line passing through
(x,y) and (20,1):
1/[35-20] = y/[35-x]

 -cut and strong -cut
◦ Given a fuzzy set A defined on X and any number
the -cut and strong -cut are the crisp sets:
◦ The -cut of a fuzzy set A is the crisp set that contains
all the elements of the universal set X whose
membership grades in A are greater than or equal to
the specified value of .
◦ The strong -cut of a fuzzy set A is the crisp set that
contains all the elements of the universal set X whose
membership grades in A are only greater than the
specified value of .
 
253
],
1
,
0
[


 





 A level set of A:
◦ The set of all levels that represent distinct -cuts
of a given fuzzy set A.
◦ For example:
254
]
1
,
0
[

 

 For example: consider the discrete approximation D2 of
fuzzy set A2
255

 The standard complement of fuzzy set A with respect to the
universal set X is defined for all by the equation
◦ Elements of X for which are called equilibrium points
of A.
◦ For example, the equilibrium points of A2 in Fig. 1.7 are 27.5
and 52.5.
X
x
)
(
)
( x
A
x
A =
256
)
(
1
)
( x
A
x
A −
=

 Given two fuzzy sets, A and B, their standard intersection and
union are defined for all by the equations
where min and max denote the minimum operator and the
maximum operator, respectively.
X
x
257
)],
(
),
(
max[
)
)(
(
)],
(
),
(
min[
)
)(
(
x
B
x
A
x
B
A
x
B
x
A
x
B
A
=

=


 Another example:
◦ A1, A2, A3 are normal.
◦ B and C are subnormal.
◦ B and C are convex.
◦ are not
convex.
258
2
1 A
A
B 
=
3
2 A
A
C 
=
C
B
C
B 
 and
Normality and convexity
may be lost when we
operate on fuzzy sets by
the standard operations
of intersection and
complement.

 Discussions:
◦ Normality and convexity
may be lost when we
operate on fuzzy sets by
the standard operations
of intersection and
complement.
◦ The fuzzy intersection
and fuzzy union will
satisfies all the properties
of the Boolean lattice
listed in Table 1.1 except
the low of contradiction
and the low of excluded
middle.
259

 The law of contradiction
 To verify that the law of contradiction is violated for fuzzy
sets, we need only to show that
is violated for at least one .
◦ This is easy since the equation is obviously violated for any value
, and is satisfied only for
0
)]
(
1
),
(
min[ =
− x
A
x
A
X
x
260
)
1
,
0
(
)
( 
x
A }.
1
,
0
{
)
( 
x
A

=
 A
A

 To verify the law of absorption,
◦ This requires showing that
is satisfied for all .
◦ Consider two cases:
(1)
(2)
)
(
)
( x
B
x
A 
)
(
)
( x
B
x
A 
261
A
B
A
A =

 )
(
)
(
)]]
(
),
(
min[
),
(
max[ x
A
x
B
x
A
x
A =
X
x
)
(
)]
(
),
(
max[
)]]
(
),
(
min[
),
(
max[ x
A
x
A
x
A
x
B
x
A
x
A =
=
)
(
)]
(
),
(
max[
)]]
(
),
(
min[
),
(
max[ x
A
x
B
x
A
x
B
x
A
x
A =
=
)
(
)]]
(
),
(
min[
),
(
max[ x
A
x
B
x
A
x
A =

 Given two fuzzy set
we say that A is a subset of B and write iff
for all .
◦
)
(
)
( x
B
x
A 
X
x
262
B
A
any
for
and
iff B
B
A
A
B
A
B
A =

=



263
0 10 20 30 40 50 60 70 80
0 0 0.1 0.2 0.3 0.4 0.5 0.7 0.9 1
10 0 0 0.1 0.2 0.3 0.4 0.5 0.7 0.9
20 0 0 0 0.1 0.2 0.3 0.4 0.5 0.7
30 0 0 0 0 0.1 0.2 0.3 0.4 0.5
40 0 0 0 0 0 0.1 0.2 0.3 0.4
50 0 0 0 0 0 0 0.1 0.2 0.3
60 0 0 0 0 0 0 0 0.1 0.2
70 0 0 0 0 0 0 0 0 0.1
80 0 0 0 0 0 0 0 0 0
Example: Fuzzy Relation R [LESS_THAN] on U1  U2,
where U1=U2={0,10,20,…}
Any fuzzy set R on U= U1 U2  …  Un is called fuzzy relation on U
Fuzzy Relations

264
Let s = [i(1),i(2),..,i(k)] be a subsequence of [1,2,…,n] and let
s* = [i(k+1), i(k+2),…, i(n)] be the sequence complementary to
[i(1),i(2),..,i(k)].
The projection of n-ary fuzzy relation R on U(s) = U(i1)  U(i2)  ..  U(ik)
denoted Proj[U(s)](R) is k-ary fuzzy relation
{((u(i(1)),u(i(2)),…u(i(k))), sup [R](u(1),u(2),…u(n))}
u(i(k+1), u(i(k+2)), … u(i(n))
Example: Let’s take relation R – less than (previous page).
Proj[U1](R) = {(0,1),(10, 0.9), (20, 0.7), (30, 0.5),…..}
The converse of the projection of n-ary relation is called a cylindrical
extension.
Let R be k-ary fuzzy relation on U(s) = U(i1)  U(i2)  ..  U(ik).
A cylindrical extension of R in U = U(1) U(2)  …  U(n) is
C(R)= {(u(1),u(2),..u(n)): [R](u(i1),u(i2),…u(i(n)))}.

265
Example: Fuzzy set Fast1 on U1, Fast 2 on U2.
U1= U2 ={0,10,20,30,40,50,60,70,80}.
Fast1 = Fast2 ={(0,0), (10,0.01), (20, 0.02), (30, 0.05), (40, 0.1), (50, 0.4),
(60, 0.8), (70, 0.9), (80, 1)}.
C(Fast2) – cylindrical extension on U1
0 10 20 30 40 50 60 70 80
0 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
10 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
20 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
30 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
40 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
50 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
60 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
70 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
80 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1

266
Example: Fuzzy set Fast1 on U1, Fast 2 on U2.
U1= U2 ={0,10,20,30,40,50,60,70,80}.
Fast1 = Fast2 ={(0,0), (10,0.01), (20, 0.02), (30, 0.05), (40, 0.1), (50, 0.4),
(60, 0.8), (70, 0.9), (80, 1)}.
C(Fast2) – cylindrical extension on U1
0 10 20 30 40 50 60 70 80
0 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
10 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
20 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
30 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
40 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
50 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
60 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
70 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
80 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1

267
The join of c(Fast1) and c(Fast2)
0 10 20 30 40 50 60 70 80
0 0 0 0 0 0 0 0 0 0
10 0 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
20 0 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02
30 0 0.01 0.02 0.05 0.05 0.05 0.05 0.05 0.05
40 0 0.01 0.02 0.05 0.1 0.1 0.1 0.1 0.1
50 0 0.01 0.02 0.05 0.1 0.4 0.4 0.4 0.4
60 0 0.01 0.02 0.05 0.1 0.4 0.8 0.8 0.8
70 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 0.9
80 0 0.01 0.02 0.05 0.1 0.4 0.8 0.9 1
Different versions of composition exist.

268
Let R be fuzzy relation on U(1) U(2) …  U(r), and S be fuzzy
relation on U(s)  U(s+1) …  U(n).
Let {i1, i2,.., ik}= ({1,2…,r}- {s, s+1,…,n})  ({s, s+1,…,n}- {1,2,…,r})
Symmetric difference
The composition of R and S denoted by RS is defined as:
Proj[U(i1), U(i2), …, U(ik)](c(R)c(S)).
Example: R = Fast  Less_Than

269
u _Fast
0 0
10 0.01
20 0.02
30 0.05
40 0.1
50 0.4
60 0.8
70 0.9
80 1
0 10 20 30 40 50 60 70 80
0 0 0.1 0.2 0.3 0.4 0.5 0.7 0.9 1
10 0 0 0.1 0.2 0.3 0.4 0.5 0.7 0.9
20 0 0 0 0.1 0.2 0.3 0.4 0.5 0.7
30 0 0 0 0 0.1 0.2 0.3 0.4 0.5
40 0 0 0 0 0 0.1 0.2 0.3 0.4
50 0 0 0 0 0 0 0.1 0.2 0.3
60 0 0 0 0 0 0 0 0.1 0.2
70 0 0 0 0 0 0 0 0 0.1
80 0 0 0 0 0 0 0 0 0
= R
S =
Find composition R  S = ?
Need to be extended

Conception of Fuzzy Logic
⚫ Many decision-making and problem-solving
tasks are too complex to be defined precisely
⚫ however, people succeed by using imprecise
knowledge
⚫ Fuzzy logic resembles human reasoning in its
use of approximate information and
uncertainty to generate decisions.

271
Natural Language
⚫ Consider:
⚫ Joe is tall -- what is tall?
⚫ Joe is very tall -- what does this differ from tall?
⚫ Natural language (like most other activities in
life and indeed the universe) is not easily
translated into the absolute terms of 0 and 1.
“false” “true”

272
Fuzzy Logic
⚫ An approach to uncertainty that combines
real values [0…1] and logic operations
⚫ Fuzzy logic is based on the ideas of fuzzy set
theory and fuzzy set membership often found
in natural (e.g., spoken) language.

273
Example: “Young”
⚫ Example:
⚫ Ann is 28, 0.8 in set “Young”
⚫ Bob is 35, 0.1 in set “Young”
⚫ Charlie is 23, 1.0 in set “Young”
⚫ Unlike statistics and probabilities, the degree
is not describing probabilities that the item is
in the set, but instead describes to what
extent the item is the set.

274
Membership function of fuzzy logic
Age
25 40 55
Young Old
1
Middle
0.5
DOM
Degree of
Membership
Fuzzy values
Fuzzy values have associated degrees of membership in the set.
0

275
Crisp set vs. Fuzzy set
A traditional crisp set A fuzzy set

Benefits of fuzzy logic
⚫ You want the value to switch gradually as
Young becomes Middle and Middle becomes
Old. This is the idea of fuzzy logic.

278
Fuzzy Set Operations
⚫ Fuzzy union (): the union of two fuzzy sets
is the maximum (MAX) of each element from
two sets.
⚫ E.g.
⚫ A = {1.0, 0.20, 0.75}
⚫ B = {0.2, 0.45, 0.50}
⚫ A  B = {MAX(1.0, 0.2), MAX(0.20, 0.45), MAX(0.75, 0.50)}
= {1.0, 0.45, 0.75}

279
⚫ Fuzzy intersection (): the intersection of two
fuzzy sets is just the MIN of each element
from the two sets.
⚫ E.g.
⚫ A  B = {MIN(1.0, 0.2), MIN(0.20, 0.45), MIN(0.75,
0.50)} = {0.2, 0.20, 0.50}

280
Fuzzy Set Operations
⚫ The complement of a fuzzy variable with
DOM x is (1-x).
⚫ Complement ( _c): The complement of a
fuzzy set is composed of all elements’
complement.
⚫ Example.
⚫ Ac = {1 – 1.0, 1 – 0.2, 1 – 0.75} = {0.0, 0.8, 0.25}

281
Crisp Relations
⚫ Ordered pairs showing connection between two
sets:
(a,b): a is related to b
(2,3) are related with the relation “<“
⚫ Relations are set themselves
< = {(1,2), (2, 3), (2, 4), ….}
⚫ Relations can be expressed as matrices
…

282
Fuzzy Relations
⚫ Triples showing connection between two sets:
(a,b,#): a is related to b with degree #
⚫ Fuzzy relations are set themselves
⚫ Fuzzy relations can be expressed as matrices
…

283
Fuzzy Relations Matrices
⚫ Example: Color-Ripeness relation for tomatoes

284
Where is Fuzzy Logic used?
⚫ Fuzzy logic is used directly in very few
applications.
⚫ Most applications of fuzzy logic use it as the
underlying logic system for decision support
systems.

285
Fuzzy Expert System
⚫ Fuzzy expert system is a collection of
membership functions and rules that are
used to reason about data.
⚫ Usually, the rules in a fuzzy expert system
are have the following form:
“if x is low and y is high then z is medium”

286
Operation of Fuzzy System
Crisp Input
Fuzzy Input
Fuzzy Output
Crisp Output
Fuzzification
Rule Evaluation
Defuzzification
Input Membership Functions
Rules / Inferences
Output Membership Functions

287
Building Fuzzy Systems
⚫ Fuzzification
⚫ Inference
⚫ Composition
⚫ Defuzzification

288
Fuzzification
⚫ Establishes the fact base of the fuzzy system. It identifies the
input and output of the system, defines appropriate IF THEN
rules, and uses raw data to derive a membership function.
⚫ Consider an air conditioning system that determine the best
circulation level by sampling temperature and moisture levels.
The inputs are the current temperature and moisture level.
The fuzzy system outputs the best air circulation level: “none”,
“low”, or “high”. The following fuzzy rules are used:
1. If the room is hot, circulate the air a lot.
2. If the room is cool, do not circulate the air.
3. If the room is cool and moist, circulate the air slightly.
⚫ A knowledge engineer determines membership functions that map
temperatures to fuzzy values and map moisture measurements to fuzzy
values.

289
Inference
⚫ Evaluates all rules and determines their truth values.
If an input does not precisely correspond to an IF
THEN rule, partial matching of the input data is used
to interpolate an answer.
⚫ Continuing the example, suppose that the system has
measured temperature and moisture levels and mapped them
to the fuzzy values of .7 and .1 respectively. The system now
infers the truth of each fuzzy rule. To do this a simple method
called MAX-MIN is used. This method sets the fuzzy value of
the THEN clause to the fuzzy value of the IF clause. Thus, the
method infers fuzzy values of 0.7, 0.1, and 0.1 for rules 1, 2,
and 3 respectively.

290
Composition
⚫ Combines all fuzzy conclusions obtained by inference
into a single conclusion. Since different fuzzy rules
might have different conclusions, consider all rules.
⚫ Continuing the example, each inference suggests a different
action
⚫ rule 1 suggests a "high" circulation level
⚫ rule 2 suggests turning off air circulation
⚫ rule 3 suggests a "low" circulation level.
⚫ A simple MAX-MIN method of selection is used where the
maximum fuzzy value of the inferences is used as the final
conclusion. So, composition selects a fuzzy value of 0.7 since
this was the highest fuzzy value associated with the inference
conclusions.

291
Defuzzification
⚫ Convert the fuzzy value obtained from composition
into a “crisp” value. This process is often complex
since the fuzzy set might not translate directly into a
crisp value.Defuzzification is necessary, since
controllers of physical systems require discrete
signals.
⚫ Continuing the example, composition outputs a fuzzy value of
0.7. This imprecise value is not directly useful since the air
circulation levels are “none”, “low”, and “high”. The
defuzzification process converts the fuzzy output of 0.7 into
one of the air circulation levels. In this case it is clear that a
fuzzy output of 0.7 indicates that the circulation should be set
to “high”.

292
Defuzzification
⚫ There are many defuzzification methods. Two of the
more common techniques are the centroid and
maximum methods.
⚫ In the centroid method, the crisp value of the output
variable is computed by finding the variable value of
the center of gravity of the membership function for
the fuzzy value.
⚫ In the maximum method, one of the variable values
at which the fuzzy subset has its maximum truth
value is chosen as the crisp value for the output
variable.

293
Fuzzification
⚫ Two Inputs (x, y) and one output (z)
⚫ Membership functions:
low(t) = 1 - ( t / 10 )
high(t) = t / 10
Low High
1
0
t
X=0.32 Y=0.61
0.32
0.68
Low(x) = 0.68, High(x) = 0.32, Low(y) = 0.39, High(y) = 0.61
Crisp Inputs

294
Create rule base
⚫ Rule 1: If x is low AND y is low Then z is high
⚫ Rule 2: If x is low AND y is high Then z is low
⚫ Rule 3: If x is high AND y is low Then z is low
⚫ Rule 4: If x is high AND y is high Then z is high

295
Inference
⚫ Rule1: low(x)=0.68, low(y)=0.39 =>
high(z)=MIN(0.68,0.39)=0.39
⚫ Rule2: low(x)=0.68, high(y)=0.61 =>
low(z)=MIN(0.68,0.61)=0.61
⚫ Rule3: high(x)=0.32, low(y)=0.39 =>
low(z)=MIN(0.32,0.39)=0.32
⚫ Rule4: high(x)=0.32, high(y)=0.61 =>
high(z)=MIN(0.32,0.61)=0.32
Rule strength

296
Composition
Low High
1
0
t
•Low(z) = MAX(rule2, rule3) = MAX(0.61, 0.32) = 0.61
•High(z) = MAX(rule1, rule4) = MAX(0.39, 0.32) = 0.39
0.61
0.39

297
Defuzzification
⚫ Center of Gravity
Low High
1
0
0.61
0.39
t
Crisp output


= Max
Min
Max
Min
dt
t
f
dt
t
tf
C
)
(
)
(
Center of Gravity

298
u _Fast
0 0
10 0.01
20 0.02
30 0.05
40 0.1
50 0.4
60 0.8
70 0.9
80 1
Fuzzy set Fast
u _Dangerous
0 0
10 0.05
20 0.1
30 0.15
40 0.2
50 0.3
60 0.7
70 1
80 1
Fuzzy set Dangerous
Fuzzy Logic
Interpretation Domain → Fuzzy Sets

299
0 10 20 30 40 50 60 70 80
0 0 0.05 0.1 0.15 0.2 0.3 0.7 1 1
10 0.01 0.05 0.1 0.15 0.2 0.3 0.7 1 1
20 0.02 0.05 0.1 0.15 0.2 0.3 0.7 1 1
30 0.05 0.05 0.1 0.15 0.2 0.3 0.7 1 1
40 0.1 0.1 0.1 0.15 0.2 0.3 0.7 1 1
50 0.4 0.4 0.4 0.4 0.4 0.4 0.7 1 1
60 0.8 0.8 0.8 0.8 0.8 0.8 0.8 1 1
70 0.9 0.9 0.9 0.9 0.9 0.9 0.9 1 1
80 1 1 1 1 1 1 1 1 1
Fuzzy logic proposition: X is fast or Y is dangerous

300
Homework:
Find the following fuzzy logic propositions:
- X is fast and Y is dangerous
- If X is fast then Y is dangerous

301
Example II
if temperature is cold and oil is cheap
then heating is high

302
Example II
if temperature is cold and oil is cheap
then heating is high
Linguistic
Variable
Linguistic
Variable
Linguistic
Variable
Linguistic
Value
Linguistic
Value
Linguistic
Value
cold cheap
high

303
Definition [Zadeh 1973]
A linguistic variable is characterized by a quintuple
( )
, ( ), , ,
x T x U G M
Name
Term Set
Universe
Syntactic Rule
Semantic Rule

304
Example
A linguistic variable is characterized by a quintuple
( )
, ( ), , ,
x T x U G M
age
old, very old, not so old,
(age) more or less young,
quite young, very young
G
 
 
=  
 
 
[0, 100]
( )
 
old
(old) , ( ) [0,100]
M u u u

= 
1
2
old
0 [0,50]
( ) 50
1 [50,100]
5
u
u u
u

−
−




=  
 −
 
+ 
 
 

 
 
 

Example semantic rule:

305
Example II
Linguistic Variable : temperature
Linguistics Terms (Fuzzy Sets) : {cold, warm, hot}
(x)
cold warm hot
20 60
1
x

306
Classical Implication
A → B
A  B A B A  B
T
T
F
F
T
F
T
F
T
F
T
T
A B A  B
1
1
0
0
1
0
1
0
1
0
1
1

307
A → B
A  B
A B A → B
1
1
0
0
1
0
1
0
1
0
1
1
A B A  B
1
1
0
0
1
0
1
0
1
0
1
1
1 ( ) ( )
( , )
( ) otherwise
A B
A B
B
x y
x y
y
 


→


= 

( )
( , ) max 1 ( ), ( )
A B A B
x y x x
  
  = −

308
A → B If A then B
A A is true
B is true
B

A  B
A
B
 
Modus Ponens
A B A → B
1
1
0
0
1
0
1
0
1
0
1
1

309
If x is A then y is B.
antecedent
or
premise
consequence
or
conclusion
A → B 

310
Examples

A → B
⚫ 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.

311
Fuzzy Rules as Relations

( ) ( )
, ,
R A B
x y x y
  →
=
R
A fuzzy rule can be defined
as a binary relation with MF
Depends on how
to interpret A → B
A → B

312
Interpretations of A → B
A
B
A entails B
x
x
y
A coupled with B
A
B
x
x
y
( ) ( )
, , ?
R A B
x y x y
  →
= =

314
A
B
A entails B
x
x
y
A coupled with B
A
B
x
x
y
( ) ( )
, , ?
R A B
x y x y
  →
= =
A entails B (not A or B)
• Material implication
• Propositional calculus
• Extended propositional calculus
• Generalization of modus ponens
R A B A B

= →  
( )
R A B A A B


= →  
( )
R A B A B B
= →   

1 ( ) ( )
( , )
( ) otherwise
A B
R
B
x y
x y
y
 




= 


315
( ) ( )
, , ?
R A B
x y x y
  →
= =
A entails B (not A or B)
• Material implication
• Propositional calculus
• Extended propositional calculus
• Generalization of modus ponens
R A B A B

= →  
( )
R A B A A B


= →  
( )
R A B A B B
= →   

1 ( ) ( )
( , )
( ) otherwise
A B
R
B
x y
x y
y
 




= 

( )
( , ) max 1 ( ), ( )
R A B
x y x x
  
= −
( )
( )
( , ) max 1 ( ),min ( ), ( )
R A A B
x y x x x
   
= −
( )
( )
( , ) max 1 max ( ), ( ) , ( )
R A B B
x y x x x
   
= −

316
Single rule with single antecedent
Rule:
Fact:
Conclusion:
if x is A then y is B
x is A’
y is B’

317
Fuzzy Reasoning−
Single Rule with Single Antecedent
Rule:
Fact:
Conclusion:
x is A’
y is B’
( )
x

x
A A’
y
( )
y
 B

318
Fuzzy Reasoning−
Rule:
Fact:
Conclusion:
x is A’
y is B’
( )
x

x
A A’
y
( )
y
 B
( )
( ) max min ( ), ( , )
B x A R
y x x y
  
 
=
( )
( ) ( , )
x A R
x x y
 

=  
( , ) ( ) ( )
R A B
x y x y
  
= 
( )
( ) ( ) ( )
x A A B
x x y
  

=   
( )
( ) ( ) ( )
x A A B
x x y
  

=   
 
 
B
Firing
Strength Firing Strength
Max-Min Composition

319
Fuzzy Reasoning−
Rule:
Fact:
Conclusion:
x is A’
y is B’
( )
x

x
A A’
y
( )
y
 B
( )
( ) max min ( ), ( , )
B x A R
y x x y
  
 
=
( )
( ) ( , )
x A R
x x y
 

=  
( )
B A A B
 
= →
( , ) ( ) ( )
R A B
x y x y
  
= 
( )
( ) ( ) ( )
x A A B
x x y
  

=   
( )
( ) ( ) ( )
x A A B
x x y
  

=   
 
 
B
Max-Min Composition

320
Fuzzy Reasoning−
Single Rule with Multiple Antecedents
Rule:
Fact:
Conclusion:
if x is A and y is B then z is C
x is A and y is B
z is C

321
Fuzzy Reasoning−
Rule:
Fact:
Conclusion:
x is A’ and y is B’
z is C’
( )
x

x
A A’
y
( )
y

B
B’
z
( )
z

C

322
Fuzzy Reasoning−
( )
x

x
A A’
y
( )
y

B
B’
z
( )
z

C
Rule:
Fact:
Conclusion:
z is C’
( )
, ,
( ) max min ( , ), ( , , )
C x y A B R
y x y x y z
  
  
=
R A B C
=  →
( )
( , , ) ( , , )
R A B C
x y z x y z
   
=
( ) ( ) ( )
A B C
x y z
  
=  
( )
, , ( , ) ( , , )
x y A B R
x y x y z
 
 
=  
( )
, ( ) ( ) ( ) ( ) ( )
x y A B A B C
x y x y z
    
 
=     
( ) ( )
( ) ( ) ( ) ( ) ( )
x A A y B B C
x x y y z
    
 
 
=      
 
   
Firing Strength
C
Max-Min Composition

323
323
Fuzzy Reasoning−
( )
x

x
A A’
y
( )
y

B
B’
z
( )
z

C
Rule:
Fact:
Conclusion:
z is C’
( )
, ,
( ) max min ( , ), ( , , )
C x y A B R
y x y x y z
  
  
=
R A B C
=  →
( )
( , , ) ( , , )
R A B C
x y z x y z
   
=
( ) ( ) ( )
A B C
x y z
  
=  
( )
, , ( , ) ( , , )
x y A B R
x y x y z
 
 
=  
( )
, ( ) ( ) ( ) ( ) ( )
x y A B A B C
x y x y z
    
 
=     
( ) ( )
( ) ( ) ( ) ( ) ( )
x A A y B B C
x x y y z
    
 
 
=      
 
   
Firing Strength
C
Max-Min Composition
( ) ( )
C A B A B C
  
=   →

324
Fuzzy Reasoning−
Multiple Rules with Multiple Antecedents
Rule1:
Fact:
Conclusion:
if x is A1 and y is B1 then z is C1
z is C’
Rule2: if x is A2 and y is B2 then z is C2

325
Fuzzy Reasoning−
Rule1:
Fact:
Conclusion:
z is C’
( )
x

x
A1
A’
( )
z

z
C1
( )
y

y
B1
( )
x

x
A2
( )
y

y
B2
( )
z

z
C2
A’
B’
B’

326
Fuzzy Reasoning−
Rule1:
Fact:
Conclusion:
z is C’
( )
x

x
A1
A’
( )
z

z
C1
( )
y

y
B1
( )
x

x
A2
( )
y

y
B2
( )
z

z
C2
A’
B’
B’
( )
z

z
Max
1
C
2
C
1 2
C C C
  
= 
( ) ( )
1 2
C A B R R
  
=  
( ) ( )
1 2
A B R A B R
   
=   
   
   
1 2
C C
 
= 
Max-Min Composition

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