Assignment 2

Game Search, CSPs and Propositional Logic

Problem 1

Max: [4308: 15 Points, 5360: 10 Points]

MinimaxSearchTree

Figure 1. A game search tree.

a. (4308: 10 points, 5360: 5 points) In the game search tree of Figure 1, indicate what nodes will be pruned using alpha-beta search, and what the estimated utility values are for the rest of the nodes. Assume that, when given a choice, alpha-beta search expands nodes in a left-to-right order. Also, assume the MAX player plays first. Finally incidcate which action the Minmax algorithm will pick to exectute.

b. (4308: 5 points, 5360: 5 points) This question is also on the game search tree of Figure 1. Suppose we are given some additional knowledge about the game: the maximum utility value is 10, i.e., it is not mathematically possible for the MAX player to get an outcome greater than 10. How can this knowledge be used to further improve the efficiency of alpha-beta search? Indicate the nodes that will be pruned using this improvement. Again, assume that, when given a choice, alpha-beta search expands nodes in a left-to-right order, and that the MAX player plays first.

Problem 2

Max: [4308: 10 Points, 5360: 10 Points]

Figure 3: Yet another game search tree
Figure 2: Yet another game search tree

Consider the MINIMAX tree above. Suppose that we are the MAX player, and we follow the MINIMAX algorithm to play a full game against an opponent. However, we do not know what algorithm the opponent uses.

Under these conditions, what is the best possible outcome of playing the full game for the MAX player? What is the worst possible outcome for the MAX player? Justify your answer.

NOTE: the question is not asking you about what MINIMAX will compute for the start node. It is asking you what is the best and worst outcome of a complete game under the assumptions stated above.

Problem 3

Max: [4308: 15 Points, 5360: 10 Points]

Expectiminmax Tree
Figure 3: An Expectiminmax tree.

Find the value of every non-terminal node in the expectiminmax tree given above. Also indicate which action will be performed by the algoirithm.
What does the MinMax value obtained by the root node represent. For a particular game, what is the maximum and minmum actual payoff the MAX player can get?

Problem 4

Max: [4308: 15 Points + 5 Points EC, 5360: 15 Points + 5 Points EC]

The following outline map needs to be colored. Your job is to color the various sections such that no two sections sharing a border have the same color. You are allowed to use the colors (Red, Green, Blue).

Map Outline
Figure 5: Map to be colored.


Part a: Draw the Constraint Graph for this problem. Can you use this information to simplify the problem?

Part b: Assuming you are using Backtracking search to solve this problem and that you are using both MRV and Degree heuristic to select the variable, Which variable will be selected at each level of the search tree [You do not need to draw the tree. Just let me know which variable will be selected and why (MRV and degree values)]. Note: Multiple possible answers. You only have to give one.

Part c: EC (5 points): Give one valid solution to this problem.

Problem 5

Max: [4308: 10 Points, 5360: 10 Points]

A B C KB S1
   True       True       True       True       True   
True True False False True
True False True True True
True False False False True
False True True False False
False True False False False
False False True False False
False False False False False

KB and S1 are two propositional logic statements, that are constructed using symbols A, B, C, and using various connectives. The above truth table shows, for each combination of values of A, B, C, whether KB and S1 are true or false.

Part a: Given the above information, does KB entail S1? Justify your answer.

Part b: Given the above information, does statement NOT(KB) entail statement NOT(S1)? Justify your answer.


Problem 6

Max: [4308: 10 Points, 5360: 10 Points]

Suppose that some  knowledge base contains various propositional-logic sentences that utilize symbols A, B, C, D (connected with various connectives). There are only two cases when the knowledge base is false:
- First case: when A is false, B is true, C is true, D is true.
- Second case: when A is true, B is false, C is true, D is false.

In all other cases, the knowledge base is true. Write a conjunctive normal form (CNF)  for the knowledge base.

Problem 7

Max: [4308: 15 Points, 5360: 15 Points]

Consider the KB

(A <=> B) AND (B => C) AND (D => A) AND (C AND E => F) AND E AND D

Show that this entails F by

i. Forward Chaining
ii. Backward Chaining
iii. Resolution

Problem 8

Max: [4308: 10 Points, 5360: 10 Points]

On April 20, 2019, John and Mary sign the following contract:

- If it rains on May 1, 2019, then John must give Mary a check for $10,000 on May 2, 2019
- Mary must mow the lawn on May 3, 2019 if and only if John gives Mary a check for $10,000 on May 2, 2019.

What truly happened those days is the following:
- It did not rain on May 1, 2019
- John gave Mary a check for $10,000 on May 2, 2019
- Mary mowed the lawn on May 3, 2019.

Part a: Write a propositional-logic statement to express the contract. Make sure that, for each symbol that you use, you clearly define what that symbol stands for.


Part b: Write a logical statement to express what truly happened. When possible, use the same symbols as in question 6a. If you need to define any new symbols, clearly define what those new symbols stand for.


Part c: Was the contract violated or not, Justify your answer

Problem 9 (Extra Credit for 4308, Required for 5360)

Max: [4308: 10 Points EC, 5360: 10 Points]

Suppose that you want to implement an algorithm tht will compete on a two-player deterministic game of perfect information. Your opponent is a supercomputer called DeepGreen. DeepGreen does not use Minimax. You are given a library function DeepGreenMove(S), that takes any state S as an argument, and returns the move that DeepGreen will choose for that state S (more precisely, DeepGreenMove (S) returns the state resulting from the opponent's move).

Write an algorithm in pseudocode (following the style of the Minimax pseudocode) that will always make an optimal decision given the knowledge we have about DeepGreen. You are free to use the library function DeepGreenMove(S) in your pseudocode. How does this compare to Minimax wrt optimality of solution and the number of states explored.