Preparation for Second Midterm

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

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

12. Textbook exercise 9.19, parts (a), (b), (c) (second edition), exercise 9.24, parts (a), (b), (c) (third edition).
    
    
13. Perform minimax search with alpha-beta pruning for the following game tree. Indicate which nodes are never visited and which branches are pruned (assuming that ties are broken strictly from left to right ). Also indicate next to each node its computed value or an upper/lower bound for that value, as computed during the search.. The utilities of terminal nodes are indicated below the leaf nodes.

    

14. Determine the values of all nodes in the following game tree with chance nodes using Expectiminimax. The utilities of terminal nodes are indicated below the leaf nodes and the probabilities of chance nodes are next to the corresponding branches.