1. CS621 : Artificial Intelligence
Pushpak Bhattacharyya CSE Dept., IIT Bombay
Lecture 34
Nonmonotonic reasoning

2. Predicate Calculus deals with complete and consistent knowledge
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

3. 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.

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

5. Characteristic of classical reasoning
Monotonicity: every inference is valid, since the axioms and inference rules are valid
Cannot deal with pieces of assertion which can be both true and false in the current state of knowledge

6. Questions
How can the knowledge base be extended to allow inferences to be made on the basis of lack of knowledge as well as on the presence of it?
How can the knowledge base be updated properly when a new fact is added to the system or when an old one is removed?
How can knowledge be used to help resolve conflicts when there are several inconsistent nonmonotonic inferences that couldbe drawn?

7. “Interpretation of WFFs”
Fix a domain D
Assign for each constant, an element of D
Assign for each n-ary function, a mapping from Dn  D
Assign for each m-ary predicate, a mapping from Dm  {T, F}

8. Interpretation in Logic
Logical expressions or formulae are “FORMS” (placeholders) for whom contents are created through interpretation.
Example:
This is a Second Order Predicate Calculus formula.
Quantification on ‘F’ which is a function.

9. Examples Interpretation:1 D=N (natural numbers) a = 0 and b = 1 x ∈ N
P(x) stands for x > 0
g(m,n) stands for (m x n)
h(x) stands for (x – 1)
Above interpretation defines Factorial

10. Examples (contd.) Interpretation:2 D={strings) a = b = λ
P(x) stands for “x is a non empty string”
g(m, n) stands for “append head of m to n”
h(x) stands for tail(x)
Above interpretation defines “reversing a string”

11. More Examples ∀x [ P(x) → Q(x)]
Following interpretations conform to above expression:
man(x) → mortal(x)
dog(x) → mammal(x)
prime(x) → 2_or_odd(x)
CS(x) → bad_hand_writing(x)

12. Structure of Interpretation
All interpretations begin with a domain ‘D’, constants (0-order functions) and functions pick values from there.
With respect to -
∀x [ P(x) → Q(x)]
D = {living beings}
P: D → {T, F}
Q: D → {T, F}
P can be looked upon as a table shown here:
Elements of D i.e. x
P(x)
Ram T
Pushpak
Virus1201 F

13. Interpretation of Factorial in the structure
D = {0, 1, 2, …, ∞}
a = 0, b = 1
g(m, n) = m x n and g: D x D → D
h(x) = (x – 1) and h(0) = 0, h: D →D
P(x) is x > 0 and P: D → {T, F}

14. Multiple worlds in NMR For the given set of WFFs, have a model
When a new set of WFFS arrive, have a new/modified model and examine which existing WFFs need to be retracted

15. Operator M For all x and y, if x and y are related and if
the fact that x gets along with y is consistent with
everything else that is believed, then conclude that
x will defend y

16. How to prove consistency: Negation by Failure
Used in Prolog
We attempt to prove ~p
If we fail, we assume ~p to be false
Then we call p consistent
Decidable in Prolog, because prolog works with Horn clauses (only one literal on the RHS of a rule)