cs621: introduction to artificial intelligencecs621-2011/lectures... · cs621: introduction to...
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CS621: Introduction to Artificial
Intelligence
Pushpak BhattacharyyaCSE Dept.,
IIT Bombay
Lecture–5: Fuzzy Controller; start of
formal system
29th July 2010
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Fuzzy definition of subset
Measured in terms of “fit violation”, i.e. violating the
condition
Degree of subset hood S(B,A)= 1- degree of superset
=
m(B) = cardinality of B
=
)()( xx AB
)(
))()(,0max(
1Bm
xxx
AB
x
B x)(
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We can show that
Exercise 1:
Show the relationship between entropy and subset hood
Exercise 2:
Prove that
),()( cc AAAASAE
)(/)(),( BmBAmABS
Subset hood of B in A
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Fuzzy sets to fuzzy logic
Forms the foundation of fuzzy rule based system or fuzzy expert system
Expert System
Rules are of the form
If
then
Ai
Where Cis are conditions
Eg: C1=Colour of the eye yellow
C2= has fever
C3=high bilurubin
A = hepatitis
nCCC ...........21
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In fuzzy logic we have fuzzy predicates
Classical logic
P(x1,x2,x3…..xn) = 0/1
Fuzzy Logic
P(x1,x2,x3…..xn) = [0,1]
Fuzzy OR
Fuzzy AND
Fuzzy NOT
))(),(max()()( yQxPyQxP
))(),(min()()( yQxPyQxP
)(1)(~ xPxP
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Fuzzy Implication Many theories have been advanced and many
expressions exist
The most used is Lukasiewitz formula
t(P) = truth value of a proposition/predicate. In
fuzzy logic t(P) = [0,1]
t( ) = min[1,1 -t(P)+t(Q)]QP
Lukasiewitz definition of implication
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Linguistic Variables
Fuzzy sets are named by Linguistic Variables (typically adjectives).
Underlying the LV is a numerical quantity
E.g. For „tall‟ (LV), „height‟ is numerical quantity.
Profile of a LV is the plot shown in the figure shown alongside.
μtall(h)
1 2 3 4 5 60
height h
1
0.4
4.5
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Example Profiles
μrich(w)
wealth w
μpoor(w)
wealth w
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Example Profiles
μA (x)
x
μA (x)
x
Profile representingmoderate (e.g. moderately rich)
Profile representingextreme
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Concept of Hedge
Hedge is an intensifier
Example:
LV = tall, LV1 = very tall, LV2 = somewhat tall
„very‟ operation:
μvery tall(x) = μ2tall(x)
„somewhat‟ operation:
μsomewhat tall(x) = √(μtall(x))
1
0h
μtall(h)
somewhat tall tall
very tall
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Fuzzy Inferencing
Two methods of inferencing in classical logic
Modus Ponens
Given p and pq, infer q
Modus Tolens
Given ~q and pq, infer ~p
How is fuzzy inferencing done?
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A look at reasoning
Deduction: p, pq|- q
Induction: p1, p2, p3, …|- for_all p
Abduction: q, pq|- p
Default reasoning: Non-monotonic reasoning: Negation by failure
If something cannot be proven, its negation is asserted to be true
E.g., in Prolog
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Fuzzy Modus Ponens in terms of truth values
Given t(p)=1 and t(pq)=1, infer t(q)=1
In fuzzy logic,
given t(p)>=a, 0<=a<=1
and t(p>q)=c, 0<=c<=1
What is t(q)
How much of truth is transferred over the channel
p q
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Lukasiewitz formula
for Fuzzy Implication t(P) = truth value of a proposition/predicate. In
fuzzy logic t(P) = [0,1]
t( ) = min[1,1 -t(P)+t(Q)]QP
Lukasiewitz definition of implication
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Use Lukasiewitz definition
t(pq) = min[1,1 -t(p)+t(q)]
We have t(p->q)=c, i.e., min[1,1 -t(p)+t(q)]=c
Case 1:
c=1 gives 1 -t(p)+t(q)>=1, i.e., t(q)>=a
Otherwise, 1 -t(p)+t(q)=c, i.e., t(q)>=c+a-1
Combining, t(q)=max(0,a+c-1)
This is the amount of truth transferred over the channel pq
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Eg: If pressure is high then Volume is low
))(),(min()( QtPtQPt
))()(( volumelowpressurehight
Pressure/
Volume
High
Pressure
ANDING of Clauses on the LHS of implication
Low
Volume
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An Example
Controlling an inverted pendulum:
θ dtd /.
= angular velocity
Motor i=current
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The goal: To keep the pendulum in vertical position (θ=0)
in dynamic equilibrium. Whenever the pendulum departs
from vertical, a torque is produced by sending a current „i‟
Controlling factors for appropriate current
Angle θ, Angular velocity θ.
Some intuitive rules
If θ is +ve small and θ. is –ve small
then current is zero
If θ is +ve small and θ. is +ve small
then current is –ve medium
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-ve
med
-ve
small
Zero
+ve
small
+ve
med
-ve
med-ve
smallZero
+ve
small
+ve
med
+ve
med
+ve
small
-ve
small
-ve
med-ve
small
+ve
smallZero
Zero
Zero
Region of
interest
Control Matrix
θ. θ
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Each cell is a rule of the form
If θ is <> and θ. is <>
then i is <>
4 “Centre rules”
1. if θ = = Zero and θ. = = Zero then i = Zero
2. if θ is +ve small and θ. = = Zero then i is –ve small
3. if θ is –ve small and θ.= = Zero then i is +ve small
4. if θ = = Zero and θ. is +ve small then i is –ve small
5. if θ = = Zero and θ. is –ve small then i is +ve small
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Linguistic variables
1. Zero
2. +ve small
3. -ve small
Profiles
-ε +εε2-ε2
-ε3 ε3
+ve small-ve small
1
Quantity (θ, θ., i)
zero
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Inference procedure
1. Read actual numerical values of θ and θ.
2. Get the corresponding μ values μZero, μ(+ve small),
μ(-ve small). This is called FUZZIFICATION
3. For different rules, get the fuzzy I-values from
the R.H.S of the rules.
4. “Collate” by some method and get ONE current
value. This is called DEFUZZIFICATION
5. Result is one numerical value of „i‟.
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if θ is Zero and dθ/dt is Zero then i is Zeroif θ is Zero and dθ/dt is +ve small then i is –ve smallif θ is +ve small and dθ/dt is Zero then i is –ve smallif θ +ve small and dθ/dt is +ve small then i is -ve medium
-ε +εε2-ε2
-ε3 ε3
+ve small-ve small
1
Quantity (θ, θ., i)
zero
Rules Involved
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Suppose θ is 1 radian and dθ/dt is 1 rad/secμzero(θ =1)=0.8 (say)Μ+ve-small(θ =1)=0.4 (say)μzero(dθ/dt =1)=0.3 (say)μ+ve-small(dθ/dt =1)=0.7 (say)
-ε +εε2-ε2
-ε3 ε3
+ve small-ve small
1
Quantity (θ, θ., i)
zero
Fuzzification
1rad
1 rad/sec
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Suppose θ is 1 radian and dθ/dt is 1 rad/secμzero(θ =1)=0.8 (say)μ +ve-small(θ =1)=0.4 (say)μzero(dθ/dt =1)=0.3 (say)μ+ve-small(dθ/dt =1)=0.7 (say)
Fuzzification
if θ is Zero and dθ/dt is Zero then i is Zeromin(0.8, 0.3)=0.3
hence μzero(i)=0.3if θ is Zero and dθ/dt is +ve small then i is –ve small
min(0.8, 0.7)=0.7hence μ-ve-small(i)=0.7
if θ is +ve small and dθ/dt is Zero then i is –ve smallmin(0.4, 0.3)=0.3
hence μ-ve-small(i)=0.3if θ +ve small and dθ/dt is +ve small then i is -ve medium
min(0.4, 0.7)=0.4hence μ-ve-medium(i)=0.4
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-ε +ε-ε2
-ε3
-ve small
1zero
Finding i
0.4
0.3
Possible candidates:i=0.5 and -0.5 from the “zero” profile and μ=0.3i=-0.1 and -2.5 from the “-ve-small” profile and μ=0.3i=-1.7 and -4.1 from the “-ve-small” profile and μ=0.3
-4.1-2.5
-ve small-ve medium
0.7
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-ε +ε
-ve small
zero
Defuzzification: Finding iby the centroid method
Possible candidates:i is the x-coord of the centroid of the areas given by the
blue trapezium, the green trapeziums and the black trapezium
-4.1-2.5
-ve medium
Required i valueCentroid of three trapezoids
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Start of Formal System
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Logic and inferencing
Vision NLP
Expert
Systems
Planning
Robotics
Search
Reasoning
Learning
Knowledge
Obtaining implication of given facts and rules -- Hallmark of
intelligence
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Propositions
− Stand for facts/assertions
− Declarative statements
− As opposed to interrogative statements (questions) or imperative
statements (request, order)
Operators
=> and ¬ form a minimal set (can express other operations)
- Prove it.
Tautologies are formulae whose truth value is always T, whatever the
assignment is
)((~),),(),( NIMPLICATIONOTORAND
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Model
In propositional calculus any formula with n propositions has 2n models
(assignments)
- Tautologies evaluate to T in all models.
Examples:
1)
2)
-e Morgan with AND
PP
)()( QPQP
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Formal Systems
Rule governed
Strict description of structure and rule application
Constituents
Symbols
Well formed formulae
Inference rules
Assignment of semantics
Notion of proof
Notion of soundness, completeness, consistency,
decidability etc.
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Hilbert's formalization of propositional calculus
1. Elements are propositions : Capital letters
2. Operator is only one : (called implies)
3. Special symbol F (called 'false')
4. Two other symbols : '(' and ')'
5. Well formed formula is constructed according to the grammar
WFF P|F|WFFWFF
6. Inference rule : only one
Given AB and
A
write B
known as MODUS PONENS
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7. Axioms : Starting structuresA1:
A2:
A3
This formal system defines the propositional calculus
))(( ABA
)))()(())((( CABACBA
)))((( AFFA
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Notion of proof1. Sequence of well formed formulae
2. Start with a set of hypotheses
3. The expression to be proved should be the last line in the
sequence
4. Each intermediate expression is either one of the hypotheses
or one of the axioms or the result of modus ponens
5. An expression which is proved only from the axioms and
inference rules is called a THEOREM within the system
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Example of proof
From P and and prove R
H1: P
H2:
H3:
i) P H1
ii) H2
iii) Q MP, (i), (ii)
iv) H3
v) R MP, (iii), (iv)
QP
QP
QP
RQ
RQ
RQ
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Prove that is a THEOREM
i) A1 : P for A and B
ii) A1: P for A and for B
iii)
A2: with P for A, for B and P for C
iv) MP, (ii), (iii)
v) MP, (i), (iv)
)( PP
))(( PPPP
)( PPP
))]())((()))(([( PPPPPPPPP
)( PP
))()(( PPPPP
)( PP
)( PP