AI3391 Artificial intelligence Session 29 Forward and backward chaining.pdf

1. AI3391 ARTIFICAL INTELLIGENCE
(II YEAR (III Sem))
Department of Artificial Intelligence and Data Science
Session 29
by
Asst.Prof.M.Gokilavani
NIET
1/23/2024 Department of AI & DS 1

2. TEXTBOOK:
• Artificial Intelligence A modern Approach, Third Edition, Stuart
Russell and Peter Norvig, Pearson Education.
REFERENCES:
• Artificial Intelligence, 3rd Edn, E. Rich and K.Knight (TMH).
• Artificial Intelligence, 3rd Edn, Patrick Henny Winston, Pearson
Education.
• Artificial Intelligence, Shivani Goel, Pearson Education.
• Artificial Intelligence and Expert Systems- Patterson, Pearson
Education.
1/23/2024 Department of AI & DS 2

3. Topics covered in session 29
1/23/2024 Department of AI & DS 3
• Logical Reasoning: Knowledge-Based Agents
• Propositional Logic
• Propositional Theorem Proving
• Effective Propositional Model Checking
• Agents Based on Propositional Logic
• First order logic
• Syntax and semantics
• Knowledge representation and engineering
• Inference and first order logic
• Forward and backward chaining

4. First-Order Logic
Chapter 8
Examples
[04PBT 04PFT 04PLT 04PNT 04PRT]
Genap 2011-2012
[dks0638]

5. First-order logic
• Whereas propositional logic assumes the
world contains facts,
• first-order logic (like natural language)
assumes the world contains
– Objects: people, houses, numbers, colors,
baseball games, wars, …
– Relations: red, round, prime, brother of, bigger
than, part of, comes between, …
– Functions: father of, best friend, one more
than, plus, …

6. Syntax of FOL
• Constants KingJohn, 2, UofA,...
• Predicates Brother, >,...
• Functions Sqrt, LeftLegOf,...
• Variables x, y, a, b,...
• Connectives , , , , 
• Equality =
• Quantifiers , 

7. Example Knowledge Base
The law says that it is a crime for an
American to sell weapons to hostile
nations. The country Nono, an enemy of
America, has some missiles, and all of its
missiles were sold to it by Colonel West,
who is American.
Prove that Col. West is a criminal!

8. Example Knowledge Base contd.

9. 6

10. 7

11. 8

12. Properties of forward chaining
9
Sound and complete for first-order definite clauses (proof similar to
propositional proof)
Datalog = first-order definite clauses + no functions (e.g., crime KB)
FC terminates for Datalog in poly iterations: at most p ∙ nk literals
May not terminate in general if  is not entailed
This is unavoidable: entailment with definite clauses is semidecidable

13. Efficiency of forward chaining
Simple observation: no need to match a rule on iteration k
if a premise wasn't added on iteration k - 1
 match each rule whose premise contains a newly added
literal
Matching itself can be expensive
Database indexing allows O(1) retrieval of known facts
e.g., query Missile(x) retrieves Missile(M1)
Matching conjunctive premises against known facts is NP-hard
Forward chaining is widely used in deductive databases

14. 11

15. 12

16. 13

17. 14

18. 15

19. 16

20. 17

21. Properties of backward chaining
18
Depth-first recursive proof search: space is linear in size of proof
Incomplete due to infinite loops
 fix by checking current goal against every goal on stack
Inefficient due to repeated subgoals (both success and failure)
 fix using caching of previous results (extra space!)
Widely used (without improvements!) for logic programming

22. 19

24. 21

25. Resolution refutation proofs involves
the following steps:
1. Put the premises or axioms into clause form.
2. Add the negation of what is to be proved, in
clause form, to the set of axioms.
3. Resolve these clauses together, producing new
clauses that logically follow from them.
4. Produce a contradiction by generating the empty
clause.
5. The substitutions used to produce the empty
clause are those under which the opposite of the
negated goal is true.

26. A Facts in Propositional Logic
23
Given Axioms Clause Form
P P (1)
(P  Q)  R  P   Q  R (2)
(S  T)  Q  S  Q (3)
 T  Q (4)
T T (5)
Prove R!

27. Resolution in Propositional Logic
24
 P   Q  R (2)  R
 P   Q P (1)
 T  Q (4)  Q
 T T (5)

28. Prove fido will die!
1. All dogs are animal.
2. Fido is a dog.
3. All animals will die.

29. Resolution proof for the “dead dog” problem.

30. Lucky Student
1. Anyone passing his history exams and
winning the lottery is happy.
2. Anyone who studies or is lucky can
pass all his exams.
3. John did not study but he is lucky.
4. Anyone who is lucky wins the lottery.
Prove that John is happy!

34. One resolution refutation for the “happy student” problem.
[1] [6]
[5]
[3]
[5]
[7]
[10]
[9]
[9]
[8]

35. Exciting life
1. All people that are not poor and are smart
are happy.
2. Those people that read are not stupid.
3. John can read and is wealthy.
4. Happy people have exiting lives.
Can anyone be found with an exciting life?

36. [1]
[3]
[2]
[4]
[6]
[5]

37. Resolution proof for the “exciting life” problem.
[6] [5]
[1]
[2]
[4]
[3]

38. Resolution Example
1. John likes all kinds of food.
x: food(x)  likes(john, x)
2. Apples are food.
food(apple)
3. Chicken is food.
food(chicken)
4. Anything anyone eats and isn't killed by is food.
x:(y: eats(y, x)   killedby(y, x))  food(x)
5. Bill eats peanuts and is still alive.
A. eats(Bill, peanuts) B. alive(Bill)
6. Sue eats everything Bill eats.
x:eats(Bill, x) eats(Sue, x)
7. x:y: alive(x)  killedby (x,y)
35

39. Clause form:
1.  food(x1)  likes(John, x1)
2. food(apples)
3. food(chicken)
4.  eats(y4,x4)  killedby(y4 , x4)  food(x4)
5. Eats (Bill, peanuts)
6. Alive (Bill)
7.  eats(Bill,x7)  eats(Sue,x7)
8.  alive(x8)   killedby(x8, y8)
36

40. 37
Resolution proof that John likes peanuts
 likes (John, peanuts) (1)  food(x1)  likes (John, x1)
 food(peanuts) (4)  eats(y4,x4)  killedby(y4 . x4)  food(x4)
 eats(y4,peanuts)  killedby(y4. peanuts) (5) eats(Bill.peanuts)
killedby(Bill. peanuts) (8)  alive(x8)   killedby(x8 y8)
 alive(Bill) (6) alive(Bill)
peanuts/x1
peanuts/x4
Bill/x8, peanuts/y8
Bill/y4

41. Topics to be covered in next session 30
• Acting under uncertainity
Thank you!!!
1/23/2024 Department of AI & DS 1