Post on 13-Dec-2015
CS621: Artificial Intelligence
Pushpak BhattacharyyaCSE Dept., IIT Bombay
Lecture–8: (a) Some Proofs in Formal System;(b) How to read research papers
5th August, 2010
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
7. Axioms : Starting structuresA1:
A2:
A3
This formal system defines the propositional calculus
))(( ABA
)))()(())((( CABACBA
)))((( AFFA
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
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
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
Shorthand1. is written as and called 'NOT
P'
2. is written as and
called
'P OR Q’
3. is written as
and called
'P AND Q'
Exercise: (Challenge)
- Prove that
¬P FP
))(( QFP )( QP
)))((( FFQP )( QP
))(( AA
A very useful theorem (Actually a meta theorem, called deduction theorem)StatementIf
A1, A
2, A
3 ............. A
n ├ B
thenA
1, A
2, A
3, ...............A
n-1├
├ is read as 'derives'
GivenA
1
A2
A3
.
.
.
.A
n
B Picture 1
A1
A2
A3
.
.
.
.A
n-1
Picture 2
BAn
BAn
Use of Deduction Theorem Prove
i.e.,
├ F (M.P)
A├ (D.T)
├ (D.T)
Very difficult to prove from first principles, i.e., using axioms and inference rules only
))(( AA
))(( FFAA
FAA ,
FFA )(
))(( FFAA
Prove
i.e.
├ F
├ (D.T)
├ Q (M.P with A3)
P├
├
)( QPP
))(( QFPP
FQFPP ,,
FPP , FFQ )(
QFP )(
))(( QFPP
More proofs
))(()(.3
)()(.2
)()(.1
QPQQP
PQQP
QPQP
Proof Sketch of the Deduction Theorem
To show that
If A1, A2, A3,… An |- B
ThenA1, A2, A3,… An-1 |- An B
Case-1: B is an axiom
One is allowed to writeA1, A2, A3,… An-1 |- B
|- B(AnB)
|- (AnB); mp-rule
Case-2: B is An
AnAn is a theorem (already proved)
One is allowed to writeA1, A2, A3,… An-1 |- (AnAn)
i.e. |- (AnB)
Case-3: B is Ai where (i <>n)
Since Ai is one of the hypotheses
One is allowed to writeA1, A2, A3,… An-1 |- B
|- B(AnB)
|- (AnB); mp-rule
Case-4: B is result of MP
SupposeB comes from applying MP on
Ei and Ej
Where, Ei and Ej come before B in
A1, A2, A3,… An |- B
B is result of MP (contd)
If it can be shown thatA1, A2, A3,… An-1 |- An Ei
andA1, A2, A3,… An-1 |- (An (EiB))
Then by applying MP twiceA1, A2, A3,… An-1 |- An B
B is result of MP (contd)
This involves showing thatIf
A1, A2, A3,… An |- Ei
ThenA1, A2, A3,… An-1 |- An Ei
(similarly for AnEj)
B is result of MP (contd)
Adopting a case by case analysis as before,
We come to shorter and shorter length proof segments eating into the body of
A1, A2, A3,… An |- B
Which is finite. This process has to terminate. QED
Important to note Deduction Theorem is a meta-
theorem (statement about the system)
PP is a theorem (statement belonging to the system)
The distinction is crucial in AI Self reference, diagonalization Foundation of Halting Theorem,
Godel Theorem etc.
Example of ‘of-about’ confusion
“This statement is false” Truth of falsity cannot be decided
HOW TO READ RESEARCH PAPERS
Before that: How to read a book
1940 classic by Mortimer Adler Revised and coauthored by Charles
Van Doren in 1972 Guidelines for critically reading
good and great books of any tradition
Three types of Knowledge• Practical
– though teachable, cannot be truly mastered without experience
• Informational– that only informational knowledge can be gained
by one whose understanding equals the author's• Comprehensive
– comprehension (insight) is best learned from who first achieved said understanding — an "original communication
Three Approaches to Reading (non-fiction)• Structural
– Understanding the structure and purpose of the book– Determining the basic topic and type of the book– Distinguish between practical and theoretical books, as well as determining
the field of study that the book addresses.– Divisions in the book, and that these are not restricted to the divisions laid
out in the table of contents. – Lastly, What problems the author is trying to solve.
• Interpretative– Constructing the author's arguments– Requires the reader to note and understand any special phrases and terms– Find and work to understand each proposition that the author advances, as
well as the author's support for those propositions.• Syntopical
– Judge the book's merit and accuracy• AKA, Structure-Proposition-Evaluation (SPE) method
VERY PRACTICALFrom Wikihow!
Steps Find a book Buy/rent it and take it home Settle into a comfortable chair
or get comfortable on the couch
Be calm and alert Start the book by
turning the pages Read and enjoy it Close book
Warnings Do not forget about your daily life.
Check the time and take a break every once in a while.
If the book is rented, then be very careful to not damage it, and return it on time.
You will pay for lateness, and is not fun.
If you read the book in a bus/subway, then be careful to not miss the station where you should go off.
Reading research papers
From Philip W. Fonghttp://www2.cs.uregina.ca/~pwlfong/CS499/reading-
paper.pdf
Comprehension: what does the paper say
A common pitfall for a beginner is to focus solely on the technicalities
Technical content is no way the only focus of a careful reading
Question-1: What is the research problem the paper attempts to address? What is the motivation of the research
work? Is there a crisis in the research field
that the paper attempts to resolve? Is the research work attempting to
overcome the weaknesses of existing approaches?
Is an existing research paradigm challenged?
In short, what is the niche of the paper?
How do the authors substantiate their claims?
What is the methodology adopted to substantiate the claims?
What is the argument of the paper? What are the major theorems? What experiments are conducted?
Data analyses? Simulations? Benchmarks? User studies? Case studies? Examples?
In short, what makes the claims scientific (as opposed to being mere opinions (science as opposed to science fiction)
What are the conclusions? What have we learned from the
paper? Shall the standard practice of the
field be changed as a result of the new findings?
Is the result generalizable? Can the result be applied to other
areas of the field? What are the open problems? In short, what are the lessons one
can learn from the paper?
VVIMP
Look first to the abstract for answers to previous questions The paper should be an elaboration of
the abstract. Every good paper tells a story
ask yourself, “What is the plot?” The four questions listed above make
up a plot structure
Evaluation An integral component of
scholarship: critical of scientific claims
Fancy claims are usually easy to make but difficult to substantiate]
Solid scholarship involves careful validation of scientific claims
Reading research paper is therefore an exercise of critical thinking
Evaluation question-1: Is the research problem significant
Is the work scratching minor itches?
Are the authors solving artificial problems
Does the work enable practical applications, deepen understanding, or explore new design space?
Are the contributions significant?
Is the paper worth reading? Are the authors simply repeating
the state of the art? Are there real surprises? Are the authors aware of the
relation of their work to existing literature?
Is the paper addressing a well-known open problem?
Are the claims valid? Have the authors been cutting
corners (intentionally or unintentionally)?
Has the right theorem been proven? Errors in proofs? Problematic experimental setup? Confounding factors? Unrealistic, artificial benchmarks? Comparing apples and oranges? Methodological misunderstanding?
Do the numbers add up? Are the generalizations valid? Are the claims modest enough?
Synthesis: your own research agenda coming from the reading of the paper
Creativity does not arise from the void.
Interacting with the scholarly community through reading research papers is one of the most effective way for generating novel research agendas
When you read a research paper, you should see it as an opportunity for you to come up with new research projects
Cautionary note Be very skeptical of work that is so
“novel” that it bears no relation to any existing work, builds upon no existing paradigm, and
yet addresses a research problem so
significant that it promises to transform the world
Such are the signs that the author might not be aware of existing literature on the topic
Repeat of work done decades ago?
Questions to help formulate research agenda
What is the crux of the research problem? What are some alternative approaches to
address the research problem? What is a better way to substantiate the claim
of the authors?
Questions to help formulate research agenda
What is a good argument against the case made by the authors?
How can the research results be improved? Can the research results be applied to another
context? What are the open problems raised by this
work? Bottomline: Can we do better than the
authors?