CS344: Introduction to Artificial Intelligence
(associated lab: CS386)
Pushpak Bhattacharyya
CSE Dept., IIT Bombay
Lecture–1 to 6: Introduction; Fuzzy sets and logic; inverted pendulum
7th, 8th, 9th 14th, 15th , 21st Jan, 2013
Basic Facts
Faculty instructor: Dr. Pushpak Bhattacharyya (www.cse.iitb.ac.in/~pb)
TAship: Kashyap, Bibek, Samiulla, Lahari, Jayaprakash, Nikhil, Kritika and Shuvam
kan.pop@gmail.com, bibek.iitkgp@gmail.com, samiulla@cse.iitb.ac.in, lahari@cse.iitb.ac.in,
jayaprakash@cse.iitb.ac.in, nikhilkumar@cse.iitb.ac.in, kritika@cse.iitb.ac.in, shubhamg@cse.iitb.ac.in
Course home page
www.cse.iitb.ac.in/~cs344-2013 (will be up soon)
Venue: LCC 02, opp KR bldg
1 hour lectures 3 times a week: Mon-10.30, Tue-11.30, Thu- 8.30 (slot 2)
Perspective
AI Perspective (post-web)
Planning
Computer Vision
NLP
Expert Systems
Robotics
Search, Reasoning,
Learning IR
From Wikipedia
Artificial intelligence (AI) is the intelligence of machines and the branch of
computer science that aims to create it. Textbooks define the field as "the study and design of intelligent agents"[1] where an intelligent agent is a system that perceives its environment and takes actions that maximize its chances of
success.[2] John McCarthy, who coined the term in 1956,[3] defines it as "the science and engineering of making intelligent machines."[4]
The field was founded on the claim that a central property of humans, intelligence—
the sapience of Homo sapiens—can be so precisely described that it can be simulated by a machine.[5] This raises philosophical issues about the nature of the mind and limits of scientific hubris, issues which have been addressed by myth, fiction and philosophy since antiquity.[6] Artificial intelligence has been the subject of optimism,[7] but has also suffered setbacks[8] and, today, has become an essential part of the technology industry, providing the heavy lifting for many of the most difficult problems in computer science.[9]
AI research is highly technical and specialized, deeply divided into subfields that often fail to communicate with each other.[10] Subfields have grown up around particular institutions, the work of individual researchers, the solution of specific problems, longstanding differences of opinion about how AI should be done and the application of widely differing tools. The central problems of AI include such traits as reasoning, knowledge, planning, learning, communication, perception and the ability to move and manipulate objects.[11] General intelligence (or
"strong AI") is still a long-term goal of (some) research.[12]
Topics to be covered (1/2)
Search
General Graph Search, A*, Admissibility, Monotonicity
Iterative Deepening, α-β pruning, Application in game playing
Logic
Formal System, axioms, inference rules, completeness, soundness and consistency
Propositional Calculus, Predicate Calculus, Fuzzy Logic, Description Logic, Web Ontology Language
Knowledge Representation
Semantic Net, Frame, Script, Conceptual Dependency
Machine Learning
Decision Trees, Neural Networks, Support Vector Machines, Self Organization or Unsupervised Learning
Topics to be covered (2/2)
Evolutionary Computation
Genetic Algorithm, Swarm Intelligence
Probabilistic Methods
Hidden Markov Model, Maximum Entropy Markov Model, Conditional Random Field
IR and AI
Modeling User Intention, Ranking of Documents, Query Expansion, Personalization, User Click Study
Planning
Deterministic Planning, Stochastic Methods
Man and Machine
Natural Language Processing, Computer Vision, Expert Systems
Philosophical Issues
Is AI possible, Cognition, AI and Rationality, Computability and AI, Creativity
Foundational Points
Church Turing Hypothesis
Anything that is computable is computable by a Turing Machine
Conversely, the set of functions computed by a Turing Machine is the set of ALL and ONLY computable functions
Turing Machine
Finite State Head (CPU)
Infinite Tape (Memory)
Foundational Points (contd)
Physical Symbol System Hypothesis (Newel and Simon)
For Intelligence to emerge it is enough to manipulate symbols
Foundational Points (contd)
Society of Mind (Marvin Minsky)
Intelligence emerges from the interaction of very simple information processing units
Whole is larger than the sum of parts!
Foundational Points (contd)
Limits to computability
Halting problem: It is impossible to
construct a Universal Turing Machine that given any given pair <M, I> of Turing
Machine M and input I, will decide if M halts on I
What this has to do with intelligent computation? Think!
Foundational Points (contd)
Limits to Automation
Godel Theorem: A “sufficiently powerful”
formal system cannot be BOTH complete and consistent
“Sufficiently powerful”: at least as powerful as to be able to capture Peano’s Arithmetic
Sets limits to automation of reasoning
Foundational Points (contd)
Limits in terms of time and Space
NP-complete and NP-hard problems: Time for computation becomes extremely large as the length of input increases
PSPACE complete: Space requirement becomes extremely large
Sets limits in terms of resources
Two broad divisions of Theoretical CS
Theory A
Algorithms and Complexity
Theory B
Formal Systems and Logic
AI as the forcing function
Time sharing system in OS
Machine giving the illusion of attending simultaneously with several people
Compilers
Raising the level of the machine for better man machine interface
Arose from Natural Language Processing (NLP)
NLP in turn called the forcing function for AI
Allied Disciplines
Philosophy Knowledge Rep., Logic, Foundation of AI (is AI possible?)
Maths Search, Analysis of search algos, logic Economics Expert Systems, Decision Theory,
Principles of Rational Behavior
Psychology Behavioristic insights into AI programs Brain Science Learning, Neural Nets
Physics Learning, Information Theory & AI, Entropy, Robotics
Computer Sc. & Engg. Systems for AI
Goal of Teaching the course
Concept building: firm grip on foundations, clear ideas
Coverage: grasp of good amount of material, advances
Inspiration: get the spirit of AI,
motivation to take up further work
Resources
Main Text:
Artificial Intelligence: A Modern Approach by Russell & Norvik, Pearson, 2003.
Other Main References:
Principles of AI - Nilsson
AI - Rich & Knight
Knowledge Based Systems – Mark Stefik
Journals
AI, AI Magazine, IEEE Expert,
Area Specific Journals e.g, Computational Linguistics
Conferences
IJCAI, AAAI
Positively attend lectures!
Grading
Midsem
Endsem
Paper reading (possibly seminar)
Quizzes
Modeling Human Reasoning
Fuzzy Logic
Fuzzy Logic tries to capture the human ability of reasoning with imprecise information
Works with imprecise statements such as:
In a process control situation, “If the
temperature is moderate and the pressure is high, then turn the knob slightly right”
The rules have “Linguistic Variables”, typically adjectives qualified by adverbs (adverbs are hedges).
Alternatives to fuzzy logic model human reasoning (1/2)
Non-numerical
Non monotonic Logic
Negation by failure (“innocent unless proven guilty”)
Abduction (PQ AND Q gives P)
Modal Logic
New operators beyond AND, OR, IMPLIES, Quantification etc.
Naïve Physics
Abduction Example
If
there is rain (P)
Then
there will be no picnic (Q)
Abductive reasoning:
Observation: There was no picnic(Q)
Conclude : There was rain(P); in absence
of any other evidence
Alternatives to fuzzy logic model human reasoning (2/2)
Numerical
Fuzzy Logic
Probability Theory
Bayesian Decision Theory
Possibility Theory
Uncertainty Factor based on Dempster Shafer Evidence
Theory
(e.g. yellow_eyesjaundice; 0.3)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 6 0
height h 1
0.4 4.5
Example Profiles
µrich(w)
wealth w
µpoor(w)
wealth w
Example Profiles
µA (x)
x
µA (x)
x Profile representing
moderate (e.g. moderately rich)
Profile representing extreme
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
0 h
µtall(h)
somewhat tall tall
very tall
An Example
Controlling an inverted pendulum:
θ
d / dt
.
= angular velocityMotor i=current
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
-ve med -ve small Zero
+ve small +ve med
-ve med
-ve
small Zero +ve small
+ve med
+ve med
+ve small
-ve small
-ve med -ve
small +ve
small Zero
Zero
Zero
Region of interest
Control Matrix
θ. θ
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
Linguistic variables 1. Zero
2. +ve small 3. -ve small
Profiles
-ε +ε
ε2 -ε2
-ε3 ε3
+ve small -ve small
1
Quantity (θ, θ., i) zero
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’.
if θ is Zero and dθ/dt is Zero then i is Zero
if θ is Zero and dθ/dt is +ve small then i is –ve small if θ is +ve small and dθ/dt is Zero then i is –ve small
if θ +ve small and dθ/dt is +ve small then i is -ve medium
-ε +ε
ε2 -ε2
-ε3 ε3
+ve small -ve small
1
Quantity (θ, θ., i) zero
Rules Involved
Suppose θ is 1 radian and dθ/dt is 1 rad/sec µzero(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 +ε
ε2 -ε2
-ε3 ε3
+ve small -ve small
1
Quantity (θ, θ., i) zero
Fuzzification
1rad
1 rad/sec
Suppose θ is 1 radian and dθ/dt is 1 rad/sec µzero(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 Zero min(0.8, 0.3)=0.3
hence µzero(i)=0.3
if θ is Zero and dθ/dt is +ve small then i is –ve small min(0.8, 0.7)=0.7
hence µ-ve-small(i)=0.7
if θ is +ve small and dθ/dt is Zero then i is –ve small min(0.4, 0.3)=0.3
hence µ-ve-small(i)=0.3
if θ +ve small and dθ/dt is +ve small then i is -ve medium min(0.4, 0.7)=0.4
hence µ-ve-medium(i)=0.4
-ε +ε -ε2
-ε3
-ve small
1
zero
Finding i
0.4
0.3 Possible candidates:
i=0.5 and -0.5 from the “zero” profile and µ=0.3
i=-0.1 and -2.5 from the “-ve-small” profile and µ=0.3 i=-1.7 and -4.1 from the “-ve-small” profile and µ=0.3 -4.1
-2.5
-ve small -ve medium
0.7
-ε +ε
-ve small
zero
Defuzzification: Finding i by 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 value Centroid of three trapezoids
Fuzzy Sets
Theory of Fuzzy Sets
Intimate connection between logic and set theory.
Given any set ‘S’ and an element ‘e’, there is a very natural predicate, µs(e) called as the belongingness predicate.
The predicate is such that,
µs(e) = 1, iff e ∈ S
= 0, otherwise
For example, S = {1, 2, 3, 4}, µs(1) = 1 and µs(5) = 0
A predicate P(x) also defines a set naturally.
S = {x | P(x) is true}
For example, even(x) defines S = {x | x is even}
Fuzzy Set Theory (contd.)
Fuzzy set theory starts by questioning the fundamental assumptions of set theory viz., the belongingness
predicate, µ, value is 0 or 1.
Instead in Fuzzy theory it is assumed that, µs(e) = [0, 1]
Fuzzy set theory is a generalization of classical set theory aka called Crisp Set Theory.
In real life, belongingness is a fuzzy concept.
Example: Let, T = “tallness”
µT (height=6.0ft ) = 1.0 µT (height=3.5ft) = 0.2
An individual with height 3.5ft is “tall” with a degree 0.2
Representation of Fuzzy sets
Let U = {x1,x2,…..,xn}
|U| = n
The various sets composed of elements from U are presented as points on and inside the n-dimensional hypercube. The crisp sets are the corners of the hypercube.
(1,0) (0,0)
(0,1) (1,1)
x1 x2
x1 x2
(x1,x2)
A(0.3,0.4)
µA(x1)=0.3 µA(x2)=0.4
Φ
U={x1,x2}
A fuzzy set A is represented by a point in the n-dimensional space as the point {µA(x1), µA(x2),……µA(xn)}
Degree of fuzziness
The centre of the hypercube is the most fuzzy set. Fuzziness decreases as one nears the
corners
Measure of fuzziness
Called the entropy of a fuzzy set
) ,
( /
) ,
( )
( S d S nearest d S farthest
E
Entropy
Fuzzy set Farthest corner
Nearest corner
(1,0) (0,0)
(0,1) (1,1)
x1 x2
d(A, nearest)
d(A, farthest) (0.5,0.5)
A
Definition
Distance between two fuzzy sets
| ) (
) (
| )
,
(
1 21 2
1 s i
n
i
i
s
x x
S S
d
L1 - norm
Let C = fuzzy set represented by the centre point d(c,nearest) = |0.5-1.0| + |0.5 – 0.0|
= 1
= d(C,farthest)
=> E(C) = 1
Definition
Cardinality of a fuzzy set
n
i
i
s x
s m
1
) ( )
( (generalization of cardinality of classical sets)
Union, Intersection, complementation, subset hood
) ( 1
)
(x s x
sc
U x
x x
x s s
s
s ( ) max( ( ), ( )),
2 1
2
1
U x
x x
x s s
s
s ( ) min( ( ), ( )),
2 1
2
1
Example of Operations on Fuzzy Set
Let us define the following:
Universe U={X1 ,X2 ,X3}
Fuzzy sets
A={0.2/X1 , 0.7/X2 , 0.6/X3} and
B={0.7/X1 ,0.3/X2 ,0.5/X3}
Then Cardinality of A and B are computed as follows:
Cardinality of A=|A|=0.2+0.7+0.6=1.5 Cardinality of B=|B|=0.7+0.3+0.5=1.5 While distance between A and B
d(A,B)=|0.2-0.7)+|0.7-0.3|+|0.6-0.5|=1.0
What does the cardinality of a fuzzy set mean? In crisp sets it means the number of elements in the set.
Example of Operations on Fuzzy Set (cntd.)
Universe U={X1 ,X2 ,X3}
Fuzzy sets A={0.2/X1 ,0.7/X2 ,0.6/X3} and B={0.7/X1 ,0.3/X2 ,0.5/X3}
A U B= {0.7/X1, 0.7/X2, 0.6/X3} A ∩ B= {0.2/X1, 0.3/X2, 0.5/X3} Ac = {0.8/X1, 0.3/X2, 0.4/X3}
Laws of Set Theory
• The laws of Crisp set theory also holds for fuzzy set theory (verify them)
• These laws are listed below:
– Commutativity: A U B = B U A
– Associativity: A U ( B U C )=( A U B ) U C
– Distributivity: A U ( B ∩ C )=( A ∩ C ) U ( B ∩ C) A ∩ ( B U C)=( A U C) ∩( B U C)
– De Morgan’s Law: (A U B) C= AC ∩ BC
(A ∩ B) C= AC U BC
Distributivity Property Proof
Let Universe U={x1,x2,…xn} pi =µAU(B∩C)(xi)
=max[µA(xi), µ(B∩C)(xi)]
= max[µA(xi), min(µB(xi),µC(xi))]
qi =µ(AUB) ∩(AUC)(xi)
=min[max(µA(xi), µB(xi)), max(µA(xi), µC(xi))]
Distributivity Property Proof
Case I: 0<µC<µB<µA<1
pi = max[µA(xi), min(µB(xi),µC(xi))]
= max[µA(xi), µC(xi)]=µA(xi)
qi =min[max(µA(xi), µB(xi)), max(µA(xi), µC(xi))]
= min[µA(xi), µA(xi)]=µA(xi)
Case II: 0<µC<µA<µB<1
pi = max[µA(xi), min(µB(xi),µC(xi))]
= max[µA(xi), µC(xi)]=µA(xi)
qi =min[max(µA(xi), µB(xi)), max(µA(xi), µC(xi))]
= min[µB(xi), µA(xi)]=µA(xi) Prove it for rest of the 4 cases.
Note on definition by extension and intension S1 = {xi|xi mod 2 = 0 } – Intension
S2 = {0,2,4,6,8,10,………..} – extension
How to define subset hood?
Meaning of fuzzy subset
Suppose, following classical set theory we say
if
Consider the n-hyperspace representation of A and B A
B
x x
x A
B( ) ( )
(1,1)
(1,0) (0,0)
(0,1)
x1 x2
A .B1
.B2 .B3
Region where B(x) A(x)
This effectively means CRISPLY
P(A) = Power set of A Eg: Suppose
A = {0,1,0,1,0,1,……….,0,1} – 104 elements B = {0,0,0,1,0,1,……….,0,1} – 104 elements
Isn’t with a degree? (only differs in the 2nd element) )
(A P B
A B
Subset operator is the “odd man” out
AUB, A∩B, Ac are all “Set Constructors” while A B is a Boolean Expression or predicate.
According to classical logic
In Crisp Set theory A B is defined as
x xA xB
So, in fuzzy set theory A B can be defined as
x µA(x) µB(x)
Zadeh’s definition of subsethood goes against the grain of fuzziness theory
Another way of defining A B is as follows:
x µA(x) µB(x)
But, these two definitions imply that µP(B)(A)=1 where P(B) is the power set of B
Thus, these two definitions violate the fuzzy principle that every belongingness except Universe is fuzzy
Fuzzy definition of subset
Measured in terms of “fit violation”, i.e. violating the condition
Degree of subset hood S(A,B)= 1- degree of superset
=
m(B) = cardinality of B
=
) ( )
(x A x
B
) (
)) (
) ( ,
0 max(
1 m B
x x
x
A
B
x
B(x)
We can show that Exercise 1:
Show the relationship between entropy and subset hood Exercise 2:
Prove that
) ,
( )
(A S A Ac A Ac
E
) ( /
) (
) ,
(B A m A B m B
S
Subset hood of B in A
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
Cn
C
C1 2 ....
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(
) ( )
(x Q y P x Q y
P
)) ( ), ( min(
) ( )
(x Q y P x Q y
P
) ( 1
) (
~ P x P x
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)]
P QLukasiewitz definition of implication
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?
A look at reasoning
Deduction: p, p q|- q
Induction: p
1, p
2, p
3, …|- for_all p
Abduction: q, p q|- 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
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
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)]
P Q Lukasiewitz definition of implicationUse 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
Two equations consistent
These two equations are consistent with each other
1 2
( , ) 1 ( , )
max(0, ( ) ( ))
1 where { , ,..., }
( )
( ( ) ( )) min(1,1 ( ( )) ( ( )))
i
i
B i A i
x U
n
B i
x U
B i A i B i A i
Sub B A Sup B A
x x
U x x x
x
t x x t x t x
Proof
Let us consider two crisp sets A and B
1 U
A B
2 U
B A
3 U
A B
4 U
A B
Proof (contd…)
Case I:
So,
( ) 1 only when ( ) 1 , ( ) ( ) 0
A xi B xi So B xi A xi
max(0, ( ) ( )) ( , ) 1
( )
1 0 1
( )
i
i
i
B i A i
x U
B i
x U
B i
x U
x x
Sub B A
x
x
Proof (contd…)
Thus, in case I these two equations are
consistent with each other (prove for other cases)
( ) ( ) 0
( ( ) ( )) min(1,1 ( ( ( )) ( ( )))) min(1,1 ( )) 1
B i A i
B i A i B i A i
Since x x
L t x x t x t x
ve
Proof of the fact that S(B,A) is consistent with crisp set theory
Case II (of the figure):
So,
( ) 1 if, ( ) 1 , ( ) ( ) 0
B xi A xi So B xi A xi
max(0, ( ) ( )) ( , ) 1
( )
i
i
B i A i
x U
B i
x U
x x
Sub B A
x
( ) ( ) 0
( , ) 1 1 ( ) , ( )
0 ( , ) 1
B i A i
A i
B i
x x
Sub B A x
x Sub B A
Proof (contd…)
Thus, in case II also, these two equations are consistent with each other.
Proof of case III
Case III:
So,
In This case, ( ) ( ) 0 for ( ) and ( ) ( ) 0 otherwise
B i A i i
B i A i
x x x B A
x x
max(0, ( ) ( )) ( , ) 1
( )
i
i
B i A i
x U
B i
x U
x x
Sub B A
x
Proof (contd…)
( ) ( ) 0 only for ( )
| | | |
( , ) 1
| | | |
B
x
i Ax
ix
iB A
B A A B
S B A
B B
Hence case III is also consistent across classical set theory and fuzzy set theory
Proof of case IV
Case IV:
So,
In This case, ( ) ( ) 1 for , ( ) ( ) 1 for
( ) ( ) 0 otherwise
B i A i i
B i A i i
B i A i
x x x A
x x x B
and x x
max(0, ( ) ( )) ( , ) 1
( )
i
i
B i A i
x U
B i
x U
x x
Sub B A
x
Proof (contd…)
( ) ( ) 1 only for <=0 otherwise,
| |
( , ) 1 0
| |
B
x
i Ax
ix
iB
and
S B A B
B
Thus, in case IV also, these two equations are consistent with each other.
Hence we can say that these two equations are consistent with each other in general.