Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

) p→ (q→ r) ⇔ (p Λ q) → r
c) (p → q) Λ (q → p) ⇔ (p Λ q) \/ (¬p Λ ¬q)
[Note: For (b) and (c ), prove the theorems first from left to right and then
from right to left.]
3. Apply resolution theorem to prove the following theorem:
p \/ q, p → r, q → r ⇒ r .
[Hints: Here, goal is r ; so resolve ¬ r with the CNF form of premise
clauses to prove a resulting null clause.]
4. For a triangle ABC, it is given that the sum of the interior angles: ∠A +
∠B + ∠C = 180 degrees. Show by resolution theorem that the exterior
angle is the sum of the opposite interior angles.
[Hints: We denote the exterior angle ∠A by EXT∠A. Use the following
predicates:
Equal (sum (∠A , ∠B , ∠C) ,180)
Equal (sum (∠A, EXT(∠A), 180)
and rules
Equal (X, Y), Equal(Z, Y) →Equal (X, Z)
Equal (sum (X, Y), sum (X, Z)) →Equal (Y, Z).
Equal (Y, Z) → Equal (Z, Y).
The rest of the proof is obvious.]
5. Represent the following sentences into clause form:
a) On (X, Y) Λ (Above (Y, Z) ∨ On (Y, Z)) Λ On (Z, W) → On (X, W)
b) ∀ X Fly (X) Λ ∃ X Has-wings (X) →Bird(X) ∨ Kite(X)
c) ∀ X Man (X) Λ ∀ Y (Child (Y) ∨ Woman (Y)) → ¬Dislikes (X, Y)
6. Prove that Dog (fido) follows from the following statements by the
resolution theorem.
a) ∀ X Barks (X) → Dog (X).
b) ∀ X ∀ Y ∃ Z Has-master (X, Y) Λ Likes (X, Y) Λ Unprecedentedsituation
(Z) → Barks (X).
c) Unprecedented-situation (noise).
d) Likes (fido, jim).
e) Has-master (fido, jim).