NPTEL Artificial Intelligence: Knowledge Representation And Reasoning Week 4 Assignment Answers 2024

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NPTEL Artificial Intelligence: Knowledge Representation And Reasoning Week 4 Assignment Answers 2024

1. Which of the following are well formed formulas in L(P, F, C)?

  • Celebrity(Ada) ∧ ¬¬¬¬Celebrity(Ada)
  • WorksWith(Ada,x)
  • ∀x WorksWith(Ada,x)
  • ∀x (WorksWith(x,Ada) ¬⊃ Celebrity(x))
  • ∃x WorksWith(x,y)
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2. Which of the following are sentences in L(P, F, C)?

  • Celebrity(Ada) ∧ ¬¬¬¬Celebrity(Ada)
  • WorksWith(Ada,x)
  • ∀x WorksWith(Ada,x)
  • ∀x (WorksWith(x,Ada) ¬⊃ Celebrity(x))
  • ∃x WorksWith(x,y)
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3. Which of the following is an atomic formula in L(P, F, C)?

  • ¬Celebrity(Ben)
  • Celebrity(father(Ada))
  • FriendOf(x,Ada)
  • ¬(mother(Ada) = mother(Ben))
  • mother(Ada) = mother(Ben)
  • mother(father(Ada))
  • WorksWith(Ben,Ada) ∧ WorksWith(mother(Ben),mother(Ada))
  • ∃x FriendOf(x,Ada)
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4. Select the formulas that are equivalent to the Boolean function g(A,B).

NPTEL Artificial Intelligence: Knowledge Representation And Reasoning Week 4 Assignment Answers 2024
  • (A ∧ ¬B) ∨ (¬A ∧ B)
  • (B ⊃ A) ∨ (A ⊃ B)
  • (A ∨ B) ∧ (¬A ∨ ¬B)
  • B ≡ ¬A
Answer :- 

5. According to FOL semantics, which of the following are model(s) of the sentence ∃x∀y(WorksWith(x,y) ⊃ FriendOf(x,y)) in L(P, F, C)?

  • WorksWith={ (Ada,Ben), (Ada,Ema), (Ben,Don) }; FriendOf={ }
  • WorksWith={ (Ada,Ben), (Ben,Don) }; FriendOf={ (Ben,Ada), (Ben,Don) }
  • WorksWith={ (Ada,Ben) }; FriendOf={ (Ada,Ben), (Ben,Don) }
  • WorksWith={ }; FriendOf={ (Ada,Ben), (Ada,Ema), (Ben,Don) }
Answer :- 

6. According to FOL semantics, which of the following are model(s) of the sentence ∃x∀y(WorksWith(x,y) ⊃ FriendOf(x,y)) in L(P, F, C)?

  • WorksWith={ (Ada,Ben), (Ada,Ema), (Ben,Don) }; FriendOf={ }
  • WorksWith={ (Ada,Ben), (Ben,Don) }; FriendOf={ (Ben,Ada), (Ben,Don) }
  • WorksWith={ (Ada,Ben) }; FriendOf={ (Ada,Ben), (Ben,Don) }
  • WorksWith={ }; FriendOf={ (Ada,Ben), (Ada,Ema), (Ben,Don) }
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7. According to FOL semantics, which of the following is/are true about predicates in L(P, F, C)?

  • FOL derives the meaning of predicates like Celebrity, WorksWith and FriendOf from the meaning of the words used in the predicate name.
  • The logical part of FOL uses WorksWith(x,y) to infer WorksWith(y,x).
  • WorksWith is a binary predicate.
  • ¬(father(Ada) = mother(Ada)) is true in all interpretations.
Answer :- 

8. According to FOL semantics, which of the following is/are true?

  • Ada always refers to an element in the domain.
  • mother(father(Ada)) always refers to an element in the domain.
  • Celebrity(x) returns an element from the domain.
  • mother(father(Ada)) must be equal to one of the constants in L(P,F,C).
Answer :- 

9. Consider the FOL language L( {P,Q}, {f,g,h}, {A,B,C} ) with variables x, y, z. Which of the following pairs of clauses are unifiable?

  • P(?x, f(?x), g(f(?x))); P(A, B, C);
  • P(?x, f(?x), g(f(?x))); P(A, ?y, g(?y));
  • Q(?x, ?y, h(?y), ?x); Q(A, f(A), h(?z), f(B));
  • Q(?x, ?y, h(?y), ?x); Q(A, f(A), h(f(?z)), ?z);
Answer :- 

10. Match each FOL formula to a suitable Skolem form. Here, P and Q are predicates; x, y and z are variables; sk1, sk2, …, are Skolem constants and Skolem functions.

A. (∃x P(x)) ⊃ (∀y Q(y))
B. ∃x ((∃y R(x,y)) ⊃ Q(x))
C. ∀x (Q(x) ⊃ ∃y R(x,y))

Skolem forms:

  1. ¬P(sk3) ∨ Q(?y)
  2. ¬P(?x) ∨ Q(?y)
  3. ¬R(sk1,?y) ∨ Q(?y)
  4. ¬R(sk3,sk4) ∨ Q(sk4)
  5. ¬R(?x,sk2(?x)) ∨ Q(?x)
  6. R(?x,sk2(?x)) ∨ ¬Q(?x)

For the formulas A, B and C, in that order, enter the matching Skolem formula numbers as a comma separated list.

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11. Use FOL to state that WorksWith is a symmetric relation. WorksWith is symmetric because: if x works with y then it also means that y works with x.

  • WorksWith(x,y) ⊃ WorksWith(y,x)
  • ∀x∀y (WorksWith(x,y) ⊃ WorksWith(y,x))
  • ∃x∃y (WorksWith(x,y) ⊃ WorksWith(y,x))
  • All of the above
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