Preparation for Second Midterm

Practice Questions

1. Convert the following knowledge base to conjunctive normal form:

   A AND B
   B => (A OR C)
   

2. Consider the following knowledge base:

   A
   B
   NOT A
   

   Does this knowledge base entail the following sentence:

   B OR C
   

   Justify your answer.

3. Given a propositional-logic vocabulary with only four symbols, A, B, C, D, how many models are there for the following sentences? In other words, for each of those statements, determine how many models that statement is true in. Note that each model is defined by assigning boolean values to all four symbols A, B, C, D.

   (a) (A and B) or (B and C)
   (b) A or B
   (c) A <=> B <=>C

4. Textbook exercise 7.6 (second edition), exercise 7.8 (third edition)

5. Textbook exercise 7.8 (second edition), exercise 7.10 (third edition)

6. For some sentence S involving literals A, B, C, here is the truth table:

   A	B	C	Sentence
   false	false	false	true
   false	false	true	false
   false	true	false	false
   false	true	true	true
   true	false	false	false
   true	false	true	false
   true	true	false	false
   true	true	true	true

   Give a conjunctive normal form for sentence S.

7. What is the negation of each of the following sentences?

   1. for-every x, exists y: son(x) = y
   2. for-every x, for-every y: son(x) = y <=> father(y) = x
   

   In your answers, any "not" may only appear after the last appearance of a universal or existential quantifier.

8. Textbook exercise 8.8 (in both second and third edition).

9. Consider the technique of propositionalization. For each of the following two knowledge bases, decide if propositionalization can be applied successfully. If not, why not?

   KB 1:
   
   for-every x: king(x) and greedy(x) => evil(x)
   king(John)
   greedy(John)
   brother(Richard, John)
   
   
   KB 2:
   for-every x: king(x) and greedy(x) => evil(x)
   king(John)
   greedy(John)
   brother(Richard, John)
   king(father(John))
   

10. What is the most general unifier for each of the following pairs of expressions:

    1. major(John, x), major(y, mathematics)
    2. major(John, x), major(y, z)
    3. major(John, x), major(y, x)
    4. major(John, x), major(x, y)
    

11. Textbook exercise 9.3 (in both second and third edition).

12. Textbook exercise 9.4 (in both second and third edition).

13. Textbook exercise 9.19, parts (a), (b), (c) (second edition), exercise 9.24, parts (a), (b), (c) (third edition).