# zero-sum game tree

Task to Do: Complete the following:

1. Given the zero-sum game tree below.
2. What is the value of that game?
3. Using alpha-beta pruning, list all the edges that will be pruned.
4. Given the zero-sum game tree below. What is the value of that game? Show your steps.
1. Why can’t we prune Expectimax trees using alpha-beta pruning?
1. Given the non-zero-sum multi-agent game tree below.
2. What is the value of that game? Show your steps.
3. If we change the green node to be a chance node with p(a)=0.6, will the value of the game change? If so, what is the new value of the game? Show your steps.
1. Assume a zero-sum Board Game for two players; each has one playing piece. Player A uses a random number generator to get either “1” or “2”, then she chooses the best movement of her piece over 2 possible movements “a” or “b” depending on the random number she got. Then Player B takes her turn in the same way. Show an example game tree of depth 2 for that game. What type of game tree is it?

6.

1. Prove by resolution that (P Þ Q) Ù (ØP Þ R) entails (ØQ Þ R)
2. Given a knowledge base represented by the sentence:

(A Ú B) Ù (ØB Ú C) Ù (ØD Ú ØA) Ù(ØA Ú ØB) Ù (ØC Ú ØD) Ù (D Ú E)
Prove using resolution that the knowledge base entails E.

1. Represent the following sentences in First-Order Logic. Whenever necessary, define all predicates, functions, and constants you use. Do not forget to use quantifiers whenever needed.
2. Doha is a beautiful city.
3. Qatar hosts World Cup in 2022.
4. Some students who study hard succeed.
5. None of QU students is Brazilian.
6. Every cat who hates any of its sisters is not happy.
7. If an apartment is more expensive than all apartments in Wakra, it must be in Doha.
1. Given the following knowledge base:

1 “x P(F(x)) Þ P(x)
2 “x Q(x) Þ P(F(x))
3 “x,y P(x) Ù P(y) Ù (x ¹ y) Þ R(x, y)
4 P(A)
5 Q(B)

Prove using Forward Chaining and GMP that the knowledge base entails R(A, B). Show your steps.

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