1. EA C461 Artificial Intelligence
Logical Agent
S.P.Vimal
BITS-Pilani
EA C461 Artificial Intelligence

2. EA C461 Artificial Intelligence
To discuss
Reasoning Patterns
Modus Ponens, AND Elimination
Inference Based on Resolution
Forward Chaining, Backward Chaining
EA C461 Artificial Intelligence

3. EA C461 Artificial Intelligence
Reasoning Patterns
Modus Ponens
Given α β, α the sentence β can be inferred
And Elimination
Given α Λ β, α the sentence β can be inferred
Sound inferences can be generated
EA C461 Artificial Intelligence

4. EA C461 Artificial Intelligence
Reasoning Patterns
R1:  P1,1
R2: B1,1
R3: B2,1
R4: B1,1  (P1,2  P2,1)
R5: B2,1  (P1,1  P2,2  P3,1)
The KB is now
R1Λ R2Λ R3Λ R4Λ R5
How do you infer there is no pit in (1,2) . That is ( P1,2)..
Sequence of derivations (proof) will help us conclude ( P1,2)
Finding proof is similar to searching
Trying to find a proof Vs Truth Table Enumeration
EA C461 Artificial Intelligence

5. Resolution Conjunctive Normal Form (CNF)
conjunction of disjunctions of literals
clauses
E.g., (A  B)  (B  C  D)
Resolution inference rule (for CNF):
li …  lk, m1  …  mn
li  …  li-1  li+1  …  lk  m1  …  mj-1  mj+1 ...  mn
where li and mj are complementary literals.
E.g., P1,3  P2,2, P2,2
P1,3
Resolution is sound and complete for propositional logic
EA C461 Artificial Intelligence

6. EA C461 Artificial Intelligence
Resolution algorithm
Inference based on resolution
Proof by contradiction, i.e., show KBα unsatisfiable
EA C461 Artificial Intelligence

7. EA C461 Artificial Intelligence
Resolution example
KB = (B1,1  (P1,2 P2,1))  B1,1
α = P1,2
EA C461 Artificial Intelligence

8. Resolution Closure--RC(S)
Set of all clauses derivable by repeated application of the resolution rule to clauses in S and their derivables
Ground Resolution Theorem
If a set of clauses are unsatisfiable, then their RC will contain an empty clause
Any complete search algorithm , applying only resolution rule can derive any conclusions entailed by any knowledge base in propositional logic
Refutation Completeness
Resolution can always be used to confirm or refute a sentence.
EA C461 Artificial Intelligence

9. Forward and backward chaining
Horn Form (restricted)
KB = conjunction of Horn clauses
Horn clause =
proposition symbol; or
(conjunction of symbols)  symbol
E.g., C  (B  A)  (C  D  B)
Modus Ponens (for Horn Form): complete for Horn KBs
α1, … ,αn, α1  …  αn  β β
Can be used with forward chaining or backward chaining.
These algorithms are very natural and run in linear time
EA C461 Artificial Intelligence

10. EA C461 Artificial Intelligence
Forward chaining
Idea: fire any rule whose premises are satisfied in the KB,
add its conclusion to the KB, until query is found
EA C461 Artificial Intelligence

11. Forward chaining algorithm
Forward chaining is sound and complete for Horn KB
EA C461 Artificial Intelligence

12. Forward chaining example
EA C461 Artificial Intelligence

13. Forward chaining example
EA C461 Artificial Intelligence

14. Forward chaining example
EA C461 Artificial Intelligence

15. Forward chaining example
EA C461 Artificial Intelligence

16. Forward chaining example
EA C461 Artificial Intelligence

17. Forward chaining example
EA C461 Artificial Intelligence

18. Forward chaining example
EA C461 Artificial Intelligence

19. Forward chaining example
EA C461 Artificial Intelligence

20. EA C461 Artificial Intelligence
Proof of completeness
FC derives every atomic sentence that is entailed by KB
FC reaches a fixed point where no new atomic sentences are derived
Consider the final state as a model m, assigning true/false to symbols
Every clause in the original KB is true in m
a1  …  ak  b
Hence m is a model of KB
If KB╞ q, q is true in every model of KB, including m
EA C461 Artificial Intelligence

21. EA C461 Artificial Intelligence
Backward chaining
Idea: work backwards from the query q:
to prove q by BC,
check if q is known already, or
prove by BC all premises of some rule concluding q
Avoid loops: check if new subgoal is already on the goal stack
Avoid repeated work: check if new subgoal
has already been proved true, or
has already failed
EA C461 Artificial Intelligence

22. Backward chaining example
EA C461 Artificial Intelligence

23. Backward chaining example
EA C461 Artificial Intelligence

24. Backward chaining example
EA C461 Artificial Intelligence

25. Backward chaining example
EA C461 Artificial Intelligence

26. Backward chaining example
EA C461 Artificial Intelligence

27. Backward chaining example
EA C461 Artificial Intelligence

28. Backward chaining example
EA C461 Artificial Intelligence

29. Backward chaining example
EA C461 Artificial Intelligence

30. Backward chaining example
EA C461 Artificial Intelligence

31. Backward chaining example
EA C461 Artificial Intelligence

32. Forward vs. backward chaining
FC is data-driven, automatic, unconscious processing,
e.g., object recognition, routine decisions
May do lots of work that is irrelevant to the goal
BC is goal-driven, appropriate for problem-solving,
e.g., Where are my keys? How do I get into a PhD program?
Complexity of BC can be much less than linear in size of KB
EA C461 Artificial Intelligence