CS621: Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture–8: (a) Some Proofs in Formal System;(b) How to read research papers 5 th 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|WFF  WFF 6. Inference rule : only one Given A  B and A write B known asMODUS PONENS

7. Axioms : Starting structures A1: A2: A3 This formal system defines the propositional calculus

Notion of proof 1. 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) PH1 ii) H2 iii) QMP, (i), (ii) iv) H3 v) RMP, (iii), (iv)

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)

Shorthand 1. 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

A very useful theorem (Actually a meta theorem, called deduction theorem) Statement If A 1, A 2, A A n ├ B then A 1, A 2, A 3, A n-1 ├ ├ is read as 'derives' Given A1A2A3....AnBA1A2A3....AnB Picture 1 A 1 A 2 A 3. A n-1 Picture 2

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

Prove i.e. ├ F ├ (D.T) ├ Q (M.P with A3) P ├ ├

More proofs

Proof Sketch of the Deduction Theorem To show that If A 1, A 2, A 3,… A n |- B Then A 1, A 2, A 3,… A n-1 |- A n  B

Case-1: B is an axiom One is allowed to write A 1, A 2, A 3,… A n-1 |- B |- B  (A n  B) |- (A n  B); mp-rule

Case-2: B is A n A n  A n is a theorem (already proved) One is allowed to write A 1, A 2, A 3,… A n-1 |- (A n  A n ) i.e.|- (A n  B)

Case-3: B is A i where (i <>n) Since A i is one of the hypotheses One is allowed to write A 1, A 2, A 3,… A n-1 |- B |- B  (A n  B) |- (A n  B); mp-rule

Case-4: B is result of MP Suppose B comes from applying MP on E i and E j Where, E i and E j come before B in A 1, A 2, A 3,… A n |- B

B is result of MP (contd) If it can be shown that A 1, A 2, A 3,… A n-1 |- A n  E i and A 1, A 2, A 3,… A n-1 |- (A n  (E i  B)) Then by applying MP twice A 1, A 2, A 3,… A n-1 |- A n  B

B is result of MP (contd) This involves showing that If A 1, A 2, A 3,… A n |- E i Then A 1, A 2, A 3,… A n-1 |- A n  E i (similarly for A n  E j )

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 A 1, A 2, A 3,… A n |- B Which is finite. This process has to terminate. QED

Important to note Deduction Theorem is a meta-theorem (statement about the system) P  P 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 PRACTICAL From 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 pagesturning 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. Fong ng/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?