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

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

3. Inferencing through Deduction Induction
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: Contradictions in all paths X (α - formula) (α - formulae)B C
Contradictions in all paths B C

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

10. Inferencing in PC
Backward chaining
Resolution
Forward chaining

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

15. Representation of Knowledge
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
Ram(iof>person)
newspaper(icl>print_media)

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

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

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. From predicate calculus
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 tableQ
S’s Answer R T
Yes F No

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?

51. From Propositional Calculus
Problem-4
From Propositional Calculus

52. Another tourist example: this time in a restaurant setting in a different country (Manna, 1974)
Facts: A tourist is in a restaurant in a country when the waiter tells him:
“do you see the three men in the table yonder? One of them is X who always speaks the truth, another is Y who always lies and the third is Z who sometimes speaks the truth and sometimes lies, i.e., answers yes/no randomly without regard to the question.
Question: Can you (the tourist) ask three yes/no questions to these men, always indicating who should answer the question, and determine who of them is X, who y and who Z?

53. Solution: Most of the steps are doable by humans only
Number the persons: 1, 2, 3
1 can be X/Y/Z
2 can be X/Y/Z
3 can be X/Y/Z
Let the first question be to 1
One of 2 and 3 has to be NOT Z.
Critical step in the solution: only humans can do?

54. Now cast the problem in the same setting as the tourist and the capital example
Solving by analogy
Use of previously solved problems
Hallmark of intelligence

55. Analogy with the tourist and the capital problem
Find the direction to the capital
 Find Z; who amongst 1, 2 and 3 is Z?
Ask a single yes/no question to S (the person standing at the junction)
 Ask a single yes/no question to 1
Answer forced to reveal the direction of the capital
 Answer forced to reveal who from 1,2,3 is Z

56. Question to 1
Ask “Is R true” and the answer is yes if and only if 2 is not Z
Propositions
P: 2 is not Z
Q: 1 always speaks the truth, i.e., 1 is X

57. Use of truth table as beforeP Q
1’s Answer R T
Yes F No

58. Question to 1: the first question
Is it true that 2 is not Z if and only if you are X?

59. Analysis of 1’s answer Ans= yes
Case 1: 1 is X/Y (always speaks the truth or always lies)
2 is indeed not Z (we can trust 1’s answer)
Case 2: 1 is Z
2 is indeed not Z (we cannot trust 1’s answer; but that does not affect us)

60. Analysis of 1’s answer (contd)
Ans= no
Case 1: 1 is X/Y (always speaks the truth or always lies)
2 is Z; hence 3 is not Z
Case 2: 1 is Z
3 is not Z
Note carefully: how cleverly Z is identified.
Can a machine do it?

61. Next steps: ask the 2nd question to determine X/Y
Once “Not Z” is identified- say 2, ask him a tautology
Is P≡P
If yes, 2 is X
If no, 2 is Y

62. Ask the 3rd Question Ask 2 “is 1 Z” If 2 is X If 2 is Y (always lies)
Ans=yes, 1 is Z
Ans=no, 1 is Y
If 2 is Y (always lies)
Ans=yes, 1 is X
Ans=no, 1 is Z
3 is the remaining person

63. What do these examples show?
Logic systematizes the reasoning process
Helps identify what is mechanical/routine/automatable
Brings to light the steps that only human intelligence can perform
These are especially of foundational and structural nature (e.g., deciding what propositions to start with)
Algorithmizing reasoning is not trivial