CS344: Artificial Intelligence

1. CS344: Artificial Intelligence Pushpak BhattacharyyaCSE Dept., IIT Bombay Lecture 12, 13– Predicate Calculus and Knowledge Representation

2. Logic and inferencing Vision NLP • Search • Reasoning • Learning • Knowledge Expert Systems Robotics Planning Obtaining implication of given facts and rules -- Hallmark of intelligence

3. Inferencing through • Deduction (General to specific) • Induction (Specific to General) • Abduction (Conclusion to hypothesis in absence of any other evidence to contrary) Deduction Given: All men are mortal (rule) Shakespeare is a man (fact) To prove: Shakespeare is mortal (inference) Induction Given: Shakespeare is mortal Newton is mortal (Observation) Dijkstra is mortal To prove: All men are mortal (Generalization)

4. If there is rain, then there will be no picnic Fact1: There was rain Conclude: There was no picnic Deduction Fact2: There was no picnic Conclude: There was no rain (?) Induction and abduction are fallible forms of reasoning. Their conclusions are susceptible to retraction Two systems of logic 1) Propositional calculus 2) Predicate calculus

5. 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

6. 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

7. Semantic Tree/Tableau method of proving tautology Start with the negation of the formula - α - formula α-formula β-formula - β - formula α-formula - α - formula

8. Example 2: X (α - formula) (α - formulae) α-formula (β - formulae) B C B C Contradictions in all paths

9. Exercise: Prove the backward implication in the previous example

10. Inferencing in PC Backward chaining Forward chaining Resolution

11. Knowledge Procedural Declarative • Declarative knowledge deals with factoid questions (what is the capital of India? Who won the Wimbledon in 2005? Etc.) • Procedural knowledge deals with “How” • Procedural knowledge can be embedded in declarative knowledge

12. Example: Employee knowledge base Employee record Emp id : 1124 Age : 27 Salary : 10L / annum Tax : Procedure to calculate tax from basic salary, Loans, medical factors, and # of children

13. Text Knowledge Representation

14. past tense bought time agent object student June computer the: definite in: modifier a: indefinite modifier new A Semantic Graph The student bought a new computer in June.

15. UNL representation Representation of Knowledge Ram is reading the newspaper

16. UNL: a United Nations project Dave, Parikh and Bhattacharyya, Journal of Machine Translation, 2002 • Started in 1996 • 10 year program • 15 research groups across continents • First goal: generators • Next goal: analysers (needs solving various ambiguity problems) • Current active language groups • UNL_French (GETA-CLIPS, IMAG) • UNL_Hindi (IIT Bombay with additional work on UNL_English) • UNL_Italian (Univ. of Pisa) • UNL_Portugese (Univ of Sao Paolo, Brazil) • UNL_Russian (Institute of Linguistics, Moscow) • UNL_Spanish (UPM, Madrid)

17. Knowledge Representation UNL Graph - relations read agt obj Ram newspaper

18. Knowledge Representation UNL Graph - UWs read(icl>interpret) obj agt newspaper(icl>print_media) Ram(iof>person)

19. Knowledge Representation UNL graph - attributes @entry @present @progress read(icl>interpret) obj agt @def newspaper(icl>print_media) Ram(iof>person) Ram is reading the newspaper

20. go(icl>move) @ entry @ past agt plt boy(icl>person) @ entry school(icl>institution) here agt plc :01 work(icl>do) The boy who works here went to school Another Example

21. Predicate Calculus

22. Predicate Calculus: well known examples • Man is mortal : rule ∀x[man(x) → mortal(x)] • shakespeare is a man man(shakespeare) • To infer shakespeare is mortal mortal(shakespeare)

23. Forward Chaining/ Inferencing • man(x) → mortal(x) • Dropping the quantifier, implicitly Universal quantification assumed • man(shakespeare) • Goal mortal(shakespeare) • Found in one step • x = shakespeare, unification

24. Backward Chaining/ Inferencing • man(x) → mortal(x) • Goal mortal(shakespeare) • x = shakespeare • Travel back over and hit the fact asserted • man(shakespeare)

25. Resolution - Refutation • man(x) → mortal(x) • Convert to clausal form • ~man(shakespeare) mortal(x) • Clauses in the knowledge base • ~man(shakespeare) mortal(x) • man(shakespeare) • mortal(shakespeare)

26. Resolution – Refutation contd • Negate the goal • ~man(shakespeare) • Get a pair of resolvents

27. Resolution Tree

28. Search in resolution • Heuristics for Resolution Search • Goal Supported Strategy • Always start with the negated goal • Set of support strategy • Always one of the resolvents is the most recently produced resolute

29. Inferencing in Predicate Calculus • Forward chaining • Given P, , to infer Q • P, match L.H.S of • Assert Q from R.H.S • Backward chaining • Q, Match R.H.S of • assert P • Check if P exists • Resolution – Refutation • Negate goal • Convert all pieces of knowledge into clausal form (disjunction of literals) • See if contradiction indicated by null clause can be derived

30. P • converted to Draw the resolution tree (actually an inverted tree). Every node is a clausal form and branches are intermediate inference steps.

31. Terminology • Pair of clauses being resolved is called the Resolvents. The resulting clause is called the Resolute. • Choosing the correct pair of resolvents is a matter of search.

32. Predicate Calculus • Introduction through an example (Zohar Manna, 1974): • Problem: A, B and C belong to the Himalayan club. Every member in the club is either a mountain climber or a skier or both. A likes whatever B dislikes and dislikes whatever B likes. A likes rain and snow. No mountain climber likes rain. Every skier likes snow. Is there a member who is a mountain climber and not a skier? • Given knowledge has: • Facts • Rules

33. Predicate Calculus: Example contd. • Let mc denote mountain climber and sk denotes skier. Knowledge representation in the given problem is as follows: • member(A) • member(B) • member(C) • ∀x[member(x) → (mc(x) ∨ sk(x))] • ∀x[mc(x) → ~like(x,rain)] • ∀x[sk(x) → like(x, snow)] • ∀x[like(B, x) → ~like(A, x)] • ∀x[~like(B, x) → like(A, x)] • like(A, rain) • like(A, snow) • Question: ∃x[member(x) ∧ mc(x) ∧ ~sk(x)] • We have to infer the 11th expression from the given 10. • Done through Resolution Refutation.

34. Club example: Inferencing • member(A) • member(B) • member(C) • Can be written as

35. Negate–

36. member(A) • member(B) • member(C) • Now standardize the variables apart which results in the following

37. 10 7 12 5 4 13 14 2 11 15 16 13 2 17

38. Assignment • Prove the inferencing in the Himalayan club example with different starting points, producing different resolution trees. • Think of a Prolog implementation of the problem • Prolog Reference (Prolog by Chockshin & Melish)

39. Problem-2 From predicate calculus

40. A “department” environment • Dr. X is the HoD of CSE • Y and Z work in CSE • Dr. P is the HoD of ME • Q and R work in ME • Y is married to Q • By Institute policy staffs of the same department cannot marry • All married staff of CSE are insured by LIC • HoD is the boss of all staff in the department

41. Diagrammatic representation CSE ME Dr. P Dr. X Z Y R Q married

42. Questions on “department” • Who works in CSE? • Is there a married person in ME? • Is there somebody insured by LIC?

43. Problem-3 (Zohar Manna, Mathematical Theory of Computation, 1974) From Propositional Calculus

44. Tourist in a country of truth-sayers and liers • Facts and Rules: In a certain country, people either always speak the truth or always lie. A tourist T comes to a junction in the country and finds an inhabitant S of the country standing there. One of the roads at the junction leads to the capital of the country and the other does not. S can be asked only yes/no questions. • Question: What single yes/no question can T ask of S, so that the direction of the capital is revealed?

45. Diagrammatic representation Capital S (either always says the truth Or always lies) T (tourist)

46. Deciding the Propositions: a very difficult step- needs human intelligence • P: Left road leads to capital • Q: S always speaks the truth

47. Meta Question: What question should the tourist ask • The form of the question • Very difficult: needs human intelligence • The tourist should ask • Is R true? • The answer is “yes” if and only if the left road leads to the capital • The structure of R to be found as a function of P and Q

48. A more mechanical part: use of truth table

49. Get form of R: quite mechanical • From the truth table • R is of the form (P x-nor Q) or (P ≡ Q)

50. Get R in English/Hindi/Hebrew… • Natural Language Generation: non-trivial • The question the tourist will ask is • Is it true that the left road leads to the capital if and only if you speak the truth? • Exercise: A more well known form of this question asked by the tourist uses the X-OR operator instead of the X-Nor. What changes do you have to incorporate to the solution, to get that answer?