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?