1. CS 621 Artificial Intelligence Lecture 4 – 05/08/05
Prof. Pushpak Bhattacharyya
KNOWLEDGE REPRESENTATION &
INFERENCING
USING PREDICATE CALCULUS
Prof. Pushpak Bhattacharyya, IIT Bombay

2. Prof. Pushpak Bhattacharyya, IIT Bombay
Example
Example:
John, Jack & Jill are members of Alpine club. Every member of the club is either a mountain climber or a skier. All skiers like snow. No mountain climber likes rain. Jack dislikes whatever John likes, and likes whatever John dislikes. John likes rain and snow.
Is there a member who is a mountain climber but not a skier.
Prof. Pushpak Bhattacharyya, IIT Bombay

3. Knowledge Representation
member (John, Alpine)
member (Jack, Alpine)
member (Jill, Alpine)
x [member(x, Alpine) → mc(x) sk(x)]
x [sk(x) → like(x, snow)]
x [mc(x) → ~like(x, rain)]
x [like(John, x) → ~like(John, x)]
x [~like(John, x) → like(Jack, x)]
like(John, rain)
like(John, snow)
Ques: x [member(x, Alpine) mc(x) ~sk(x)]
Prof. Pushpak Bhattacharyya, IIT Bombay

4. Inference Strategy - RESOLUTION
Basic Idea: REFUTATION of the goal
Proof by contradiction
Suppose the goal is false. Then show contradiction in the knowledge base.
Prof. Pushpak Bhattacharyya, IIT Bombay

5. Prof. Pushpak Bhattacharyya, IIT Bombay
Steps in Inferencing
Convert all expressions, including the falsified goal, into clauses.
member (John, Alpine)
member (Jack, Alpine)
member (Jill, Alpine)
~ member(x1, Alpine) ν mc(x1) ν sk(x1)
~ sk(x2) ν like(x2,snow)
~ mc(x3) ν ~ like(x3,rain)
~ like(John, x4) ν ~ like(Jack, x4)
like(John, x5) ν like(Jack, x5)
like(John, rain)
like(John, snow)
~ member(x6, Alpine) ν ~ mc(x6) ν sk(x6)
Prof. Pushpak Bhattacharyya, IIT Bombay

6. Prof. Pushpak Bhattacharyya, IIT Bombay
Run Resolution
By unification find value for x6
Theory of resolution: given P & ~P ν Q
Resolvents
we can obtain Q (Resolute)
Prof. Pushpak Bhattacharyya, IIT Bombay

7. Prof. Pushpak Bhattacharyya, IIT Bombay
Inverted Tree Diagram P
~P ν Q Q
Aim C1 C2
Indicates contradiction
null clause
Prof. Pushpak Bhattacharyya, IIT Bombay

8. Prof. Pushpak Bhattacharyya, IIT Bombay
Goal with Negation
~[ x{(member (x, Alpine) Λ mc(x) Λ ~ sk(x))}]
x[~ member (x, Alpine) ν ~ mc(x) ν sk(x)]
11. ~ member(x6, Alpine) ν ~ mc(x6) ν sk(x6)
Prof. Pushpak Bhattacharyya, IIT Bombay

9. Prof. Pushpak Bhattacharyya, IIT Bombay
Monotonic Logic
Every step of resolution increases the KB monotonically.
Non-monotonic logic which used default reasoning.
NEGATION BY FAILURE
Prof. Pushpak Bhattacharyya, IIT Bombay

10. Which Pair of Clauses for Resolvents?10 7
unification of x4 with snow
snow/x4
snow/x4
12. ~ like(Jack, snow) 5
Jack/x2
13. ~ sk(Jack) 4
Jack/x1
14. ~ member(Jack, Alpine) ν mc(Jack)
Prof. Pushpak Bhattacharyya, IIT Bombay

11. Prof. Pushpak Bhattacharyya, IIT Bombay
Resolvents (Contd) 2 14
15. mc(Jack) 11
16. ~ member(Jack) ν sk(Jack) 13
~ member(Jack) 2
Prof. Pushpak Bhattacharyya, IIT Bombay

12. Prof. Pushpak Bhattacharyya, IIT Bombay
Resolution Strategy
Start from negated goal
Use the derived clause as one of the pairs always
- set of support strategy
If the  is not reached, then
The knowledge base is not complete.
The inference rules are not adequate (modus ponens)
Wrong inference path
Prof. Pushpak Bhattacharyya, IIT Bombay

13. Modus Ponens & Modus Tolens
P & P →Q
gives Q
Modus Tolens:
~Q and P →Q
gives ~P
Prof. Pushpak Bhattacharyya, IIT Bombay

14. Prof. Pushpak Bhattacharyya, IIT Bombay
Prolog
Predicate calculus (HORN Clause) +
Resolution
HORN Clauses: All the implications have single literal as consequent.
A(antecedent) → B(consequent)
B is a single literal, never contain any operator.
Moreover B has to be a positive literal.
Prof. Pushpak Bhattacharyya, IIT Bombay