Introduction to Artificial Intelligence
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Transcript of Introduction to Artificial Intelligence
Fuzzy Logic
By Manoj Harsule
OverviewOverview
• A Little History• Fuzzy Logic – A Definition• Fuzzy set theory• Introduction to fuzzy set• Fuzzy Relations
A little History
In the 1960’s Lotfi A. Zadeh Ph.D,. University of California, Berkeley, published an obscure paper on fuzzy sets . His unconventional theory allowed for approximate information and uncertainty when generating complex solutions; a process that previously did not exist.
Fuzzy Logic has been around since the mid 60’s but was not readily excepted until the 80’s and 90’s. Although now prevalent throughout much of the world, China, Japan and Korea were the early adopters
WHAT IS FUZZY LOGIC?WHAT IS FUZZY LOGIC?
Definition of fuzzy
Fuzzy – “not clear, distinct, or not precise;
uncertain”
Definition of fuzzy logic
A form of knowledge representation suitable for
notions that cannot be defined precisely, but which
depend upon their contexts.
TRADITIONAL REPRESENTATION OF LOGIC
Slow (Low) Fast (High)
Speed = 0 Speed = 1
FUZZY LOGIC FUZZY LOGIC REPRESENTATIONREPRESENTATION
For every problem must represent in terms of fuzzy sets.
Slowest
Fastest
Slow
Fast
[ 0.0 – 0.25 ]
[ 0.25 – 0.50 ]
[ 0.50 – 0.75 ]
[ 0.75 – 1.00 ]
Introduction to Introduction to Fuzzy Set TheoryFuzzy Set Theory
Fuzzy SetsFuzzy Sets
Types of UncertaintyTypes of Uncertainty
• Stochastic uncertaintyStochastic uncertainty– E.g., rolling a diceE.g., rolling a dice
• Linguistic uncertaintyLinguistic uncertainty– E.g., low price, tall people, young ageE.g., low price, tall people, young age
• Informational uncertaintyInformational uncertainty– E.g., credit worthiness, honestyE.g., credit worthiness, honesty
Crisp or Fuzzy LogicCrisp or Fuzzy Logic
• Crisp LogicCrisp Logic– A proposition can be A proposition can be truetrue oror falsefalse only.only.
• Ajay is a student (true)Ajay is a student (true)• Smoking is healthy (false)Smoking is healthy (false)
– The degree of truth is The degree of truth is 0 or 10 or 1..
• Fuzzy LogicFuzzy Logic– The degree of truth is The degree of truth is between 0 and 1between 0 and 1..
• Raj is young (0.3 truth) Raj is young (0.3 truth) • Amol is smart (0.9 truth) Amol is smart (0.9 truth)
Crisp SetsCrisp Sets
• Classical sets are called crisp setsClassical sets are called crisp sets– either an element either an element belongsbelongs to a set or to a set or
not, i.e.,not, i.e.,
• Member Function of crisp setMember Function of crisp set
x A or x A
0( )
1A
x Ax
x A
( ) 0,1A x
P
Crisp SetsCrisp Sets
P : the set of all people.
Y : the set of all young people. YY
( ) 25,ageYoung y y x x P
1
y
( )Young y
25
Fuzzy SetsFuzzy SetsCrisp sets ( ) 0,1A x
( ) [0,1]A x
1
y
( )Young y
Example
Definition:Fuzzy Sets and Membership Functions
If U is a collection of objects denoted generically by x, then a fuzzy set A in U is defined as a set of ordered pairs:
( , ( ))AA x x x U membership
function
: [0,1]A U
U : universe of discourse.
Example (Discrete Universe)
{1,2,3,4,5,6,7,8}U # courses a student may take in a semester.
(1,0.1) (2,0.3) (3,0.8) (4,1)
(5,0.9) (6,0.5) (7,0.2) (8,0.1)A
#appropriate courses taken
0.5
1
02 4 6 8
x : # courses
( )A x
Example (Discrete Universe)
{1,2,3,4,5,6,7,8}U # courses a student may take in a semester.
(1,0.1) (2,0.3) (3,0.8) (4,1)
(5,0.9) (6,0.5) (7,0.2) (8,0.1)A
appropriate # courses taken
Alternative Representation:
1 2 3 40.1/ 0.3 / 0.8 / 1.0 / 0.9 / 0.5 / 0.2 / 0.1/5 6 7 8A
Example (Continuous Universe)
possible agesU : the set of positive real numbers
( , ( ))BB x x x U
4
1( )
501
5
B xx
about 50 years old
0
0.2
0.4
0.6
0.8
1
1.2
0 20 40 60 80 100
x : age
( )B x
4505
1
1 xRB x
Alternative Representation:
Alternative NotationAlternative Notation
( , ( ))AA x x x U
U : discrete universe
U : continuous universe
( ) /i
A i ix U
A x x
( ) /AUA x x
Note that and integral signs stand for the union of membership grades; “
/ ” stands for a marker and does not imply division.
Mem
bers
hip
valu
e
height
1
0
Membership Functions (MF’s)
• A fuzzy set is completely characterized A fuzzy set is completely characterized by a membership function.by a membership function.– a a subjectivesubjective measure. measure.– notnot a a probabilityprobability measure. measure.
“tall” in Asia
“tall” in USA“tall” in Aus
5’10”
Fuzzy Partition
• Fuzzy partitions formed by the Fuzzy partitions formed by the linguisticlinguistic values “values “youngyoung”, “”, “middle agedmiddle aged”, and “”, and “oldold”:”:
Introduction to Fuzzy Set Theory
Set-Theoretic Operations
Set-Theoretic OperationsSet-Theoretic Operations
• SubsetSubset
• ComplementComplement
• UnionUnion
• IntersectionIntersection
( ) ( ), A BA B x x x U
( ) max( ( ), ( )) ( ) ( )C A B A BC A B x x x x x
( ) min( ( ), ( )) ( ) ( )C A B A BC A B x x x x x
( ) 1 ( )AAA U A x x
Set-Theoretic Operations
A BA B
A BA B
AA
A BA B
PropertiesProperties
A AInvolutionA B B A
Commutativity A B B A
A B C A B C Associativity A B C A B C
A B C A B A C Distributivity A B C A B A C
A A A Idempotence A A A
A A B A Absorption A A B A
A B A B De Morgan’s laws
A B A B
Properties
• The following properties are The following properties are invalidinvalid for fuzzy sets:for fuzzy sets:
– The laws of contradictionThe laws of contradiction
– The laws of excluded middleThe laws of excluded middle
A A
A A U
Other Definitions for Set Operations
• UnionUnion
• IntersectionIntersection
( ) min 1, ( ) ( )A B A Bx x x
( ) ( ) ( )A B A Bx x x
Other Definitions for Set Operations
• UnionUnion
• IntersectionIntersection
( ) ( ) ( ) ( ) ( )A B A B A Bx x x x x
( ) ( ) ( )A B A Bx x x
Generalized Union/Intersection
• Generalized IntersectionGeneralized Intersection
• Generalized UnionGeneralized Union
t-norm
t-conorm
T-Norm
:[0,1] [0,1] [0,1]T
Or called triangular norm.
1.1. SymmetrySymmetry
2.2. AssociativityAssociativity
3.3. MonotonicityMonotonicity
4.4. Border ConditionBorder Condition
( , ) ( , )T x y T y x
( ( , ), ) ( , ( , ))T T x y z T x T y z
1 2 1 2 1 1 2 2, ( , ) ( , )x x y y T x y T x y
( ,1)T x x
T-ConormT-Conorm
:[0,1] [0,1] [0,1]S
Or called s-norm.
1.1. SymmetrySymmetry
2.2. AssociativityAssociativity
3.3. MonotonicityMonotonicity
4.4. Border ConditionBorder Condition
( , ) ( , )S x y S y x
( ( , ), ) ( , ( , ))S S x y z S x S y z
1 2 1 2 1 1 2 2, ( , ) ( , )x x y y S x y S x y
( ,0)S x x
Fuzzy Relations
Review Fuzzy Relations
Aa1
a2
a3
a4
B
b1
b2
b3
b4
b5
Binary Relation (Binary Relation (RR))R A B
1 1 1 3 2 5
3 1 3 4 4 2
( , ), ( , ), ( , )
( , ), ( , ), ( , )
a b a b a bR
a b a b a b
1 0 1 0 0
0 0 0 0 1
1 0 0 1 0
0 1 0 0 0
RM
1 1a Rb 1 3a Rb 2 5a Rb
3 1a Rb 3 4a Rb 4 2a Rb
The Real-Life Relation
• x is close to y – x and y are numbers
• x depends on y– x and y are events
• x and y look alike– x and y are persons or objects
• If x is large, then y is small– x is an observed reading and y is a corresponding
action
Fuzzy RelationsFuzzy Relations
A fuzzy relation R is a 2D MF:
( ,,( , ) ( , )) |R x yx y x yR X Y
Example (Approximate Equal)
( ,,( , ) ( , )) |R x yx y x yR X Y
{1,2,3,4,5}X Y U
1 0.8 0.3 0 0
0.8 1 0.8 0.3 0
0.3 0.8 1 0.8 0.3
0 0.3 0.8 1 0.8
0 0 0.3 0.8 1
RM
1 0
0.8 1( , )
0.3 2
0
R
u v
u vu v
u v
otherwise
A fuzzy relation defined on X an Z.
Max-Min Composition
X Y ZR: fuzzy relation defined on X and Y.
S: fuzzy relation defined on Y and Z.
R 。 S: the composition of R and S.
( , ) max min ( , ), ( , )R S y R Sx z x y y z
( , ) ( , )y R Sx y y z
Example
1 0.1 0.2 0.0 1.0
2 0.3 0.3 0.0 0.2
3 0.8 0.9 1.0 0.4
R a b c d0.9 0.0 0.3
0.2 1.0 0.8
0.8 0.0 0.7
0.4 0.2 0.3
S
a
b
c
d
1 0.4 0.2 0.3
2 0.3 0.3 0.3
3 0.8 0.9 0.8
R S
0.1 0.2 0.0 1.00.9 0.2 0.8 0.4min0.1 0.2 0.0 0.4max
( , ) max min ( , ), ( , )S R v R Sx y x v v y
Max-Product CompositionMax-Product Composition
( , ) max ( , ) ( , )R S y R Sx z x y y z
A fuzzy relation defined on X an Z.
X Y ZR: fuzzy relation defined on X and Y.
S: fuzzy relation defined on Y and Z.
R 。 S: the composition of R and S.
Max-min composition is not mathematically tractable, therefore other compositions such as max-product composition have been suggested.
ProjectionProjectionR
XR R X YR R Y
Dimension Reduction
ProjectionProjectionR
XR R X YR R Y
R
XR R X YR R Y
XR R X YR R Y max ( , ) /RX y
x y xmax ( , ) /RY xx y y
( ) max ( , )YR R
xy x y ( ) max ( , )
XR Ry
x x y
Dimension Reduction
Cylindrical Extension Cylindrical Extension Dimension Expansion
A : a fuzzy set in X.
C(A) = [AXY] : cylindrical extension of A.
( ) ( ) | ( , )AX YC A x x y
( ) ( , ) ( )C A Ax y x
Types of Fuzzy Relations
• Reflexive– Irreflexive– Antireflexive– Epsilon Reflexive
• Symmetric– Asymmetric– Antisymmetric
XxxxR allfor 1),(
XxxxR somefor 1),(
XxxxR allfor 1),(
XxxxR allfor ),(
XxxyRyxR allfor ),(),(
XxxyRyxR somefor ),(),(
XyxyxxyRyxR , allfor 0 ),( and 0),(
Types of Fuzzy Relations
• Transitive (max-min transitive)
– Non-transitive: For some (x,z), the above do not satisfy.– Antitransitive:
• Example: X = Set of cities, R=“very far” Reflexive, symmetric, non-transitive
X allfor )],(),,(min[max),(
x,zzyRyxRzxRYy
X allfor )],(),,(min[max),(
x,zzyRyxRzxRYy
Types of Fuzzy Relations
• Transitive Closure– Crisp: Transitive relation that contains R(X,X)
with fewest possible members– Fuzzy: Transitive relation that contains
R(X,X) with smallest possible membership– Algorithm:
TRR
RRRR
RRRR
'
''
'
:Stop 3.
1 step togo and make , If .2
).( .1
Types of Fuzzy Relations
• Fuzzy Equivalence or Similarity Relation– Reflexive, symmetric, and transitive– Decomposition:
– Partition Tree[0,1]}|) ({(R)
:partitions ofSet
relation. eequivalenc crisp a is
]1,0[
R
R
RR
Types of Fuzzy Relations
• Fuzzy Compatibility or Tolerance Relation– Reflexive and symmetric– Maximal compatibility class and complete cover
• Compatibility class• Maximal compatibility class: largest compatibility class• Complete cover: Set of maximal compatibility classes
– Maximal alpha-compatibility class– Complete alpha-covers– Note: Relation from distance metrics forms tolerance
relation in clustering.
Ryx,such that of Subset XA
BibliographyBibliography
• J. R. Jang, C. Sun, E. Mizutani, “Neuro-J. R. Jang, C. Sun, E. Mizutani, “Neuro-Fuzzy and Soft Computing: A Fuzzy and Soft Computing: A Computational Approach to Learning and Computational Approach to Learning and Machine Intelligence, Prentice HallMachine Intelligence, Prentice Hall
• Slides and notes: Slides and notes: http://equipe.nce.ufrj.br/adriano/fuzzy/bibliogr-ic.htm