📗 [3 points] The initial state and goal state of an 8-puzzle are given below. If the heuristic is the sum of Manhattan distances between the current position of each tile and the goal position, what is the heuristic of the initial state?

📗 Answer: . Calculate

📗 [3 points] There are two players and \(k\) coins on the table. Players move sequentially with player 1 moving first. Each player chooses to take either one or two coins from the table. The player who takes the last coin wins. For which of the following values of \(k\) = {} does the first player has a winning strategy? Enter the values of \(k\), not the indices.

📗 Answer (comma separated vector): . Calculate

📗 [4 points] For a zero-sum game in which moves first and if the action Left is chosen, then Chance (Chn) moves Left with probability \(p\) and Right with probability \(1 - p\), and if the action Right is chosen, then Chance moves Left with probability and Right with probability . Suppose the player who moves first uses a mixed strategy \(\dfrac{1}{2}\) Left and \(\dfrac{1}{2}\) Right in a solution, what is the value of \(p\)? If it's impossible, enter \(-1\).

📗 Note: in case the diagram is not clear, the values on the leafs (each sub-branch is a row): .

📗 Answer: . Calculate

📗 [4 points] Suppose the score (fitness) of a state \(\left(d_{1}, d_{2}, d_{3}, d_{4}\right)\) is \(d_{1} + d_{2} + d_{3} + d_{4}\), and only 1-point crossover with the cross-over point between \(d_{2}\) and \(d_{3}\) is used in a genetic algorithm (i.e. mutation probabilities are 0). Two states are chosen as parents at random according to the reproduction probabilities, what is the probability that one of their children is the optimal state (i.e. \(\left(1, 1, 1, 1\right)\)? Enter a number between 0 and 1.

📗 Note: the two parents are sampled with replacement, meaning the probability that two states are chosen as parents is the product of their reproduction probabilities.

📗 Answer: . Calculate

📗 [3 points] Given the variance matrix \(\hat{\Sigma}\) = , what is the first principal component? Enter a unit vector.

📗 Answer (comma separated vector): . Calculate

📗 [3 points] There are \(n\) = cookies. The brother first proposes a division of these cookies into two piles (two integers adding up to \(n\)) and then the sister take one of the two piles. Both the brother and the sister want to maximize the number of cookies they take. What is the value of the game to the brother (measured by the number of cookies he gets)? Enter an integer.

📗 Answer: . Calculate

📗 [3 points] Given the variance matrix \(\hat{\Sigma}\) is a diagonal matrix, what is the smallest value of \(K\) so that the Manhattan distance between the vector \(\begin{bmatrix} 1 \\ 1 \\ ... \\ 1 \end{bmatrix}\) with ones (\(1\)'s) and its reconstruction using the first \(K\) principal components is less than or equal to ?

📗 Answer: . Calculate

📗 [4 points] In a by grid, Tom is located at (, ) and Jerry is located at (, ). Tom uses to find Jerry and the successors of a state (one cell in the grid) are the four neighboring states on the grid (the cells above, below, to the left and to the right). What is the imum number of states that need to be expanded to find (and expand) the goal state? The order in which the successors are added can be arbitrary. Do not count repeated expansion of the same state. Include both the initial and the goal states.

📗 Answer: . Calculate

📗 [3 points] There are \(n\) = students in CS540, for simplicity, assume student \(0\) gets grade \(g = 0\), student \(1\) gets grade \(g = 1\), ..., student \(n - 1\) gets grade \(g = n - 1\). The payoff for each student who drop the course is \(0\), the payoff for the students who stay is if the student has the lowest grade among all students who decide to stay in the class, and the otherwise. If each student only uses actions that are rationalizable (i.e. survive the iterated elimination of strictly dominated actions), how many students will stay in the course? If there are multiple correct answers, enter one of them.

📗 Answer: . Calculate

📗 [4 points] Given the dataset , the cluster centers are computed by k-means clustering algorithm with \(k = 2\). The first cluster center is \(x\) and the second cluster center is . What is the imum value of \(x\) such that the second cluster is empty (contains 0 instances). In case of a tie in distance, the point belongs to cluster 1.

📗 Answer: . Calculate

📗 [3 points] Let \(x\) = and \(v\) = . The projection of \(x\) onto \(v\) is the point \(y\) on the direction of \(v\) such that the line connecting \(x, y\) is perpendicular to \(v\). Compute \(y\).

📗 Answer (comma separated vector): . Calculate

📗 [3 points] Given three clusters, \(A\) = {, }, \(B\) = {\(x\)}, \(C\) = {, }. Find a value of \(x\) so that \(A\) and \(B\) will be merged in the next iteration of single linkage hierarchical clustering, and \(B\) and \(C\) will be merged in the next iteration of complete linkage hierarchical clustering. Break ties by merging with the cluster with the smaller index (i.e. \(A\), then \(B\), then \(C\)).

📗 Note: there can be multiple answers, including non-integer answers, enter one of them. If there are none, enter 0.

📗 Answer: . Calculate

📗 [4 points] Suppose the states are integers between and . The initial state is , and the goal state is . The successors of a state \(i\) are \(2 i\) and \(2 i + 1\), if exist. How many states are expanded using a Depth First Search? Include both the initial and goal states.

📗 Note: use the convention used in the lectures, push the states with larger index into the stack first (i.e. expand the states with the smaller index first).

📗 Answer: . Calculate

📗 [4 points] When using the Genetic Algorithm, suppose the states are \(\begin{bmatrix} x_{1} & x_{2} & ... & x_{T} \end{bmatrix}\) = , , , . Let \(T\) = , the fitness function (not the cost) is \(\mathop{\mathrm{argmax}}_{t \in \left\{0, ..., T\right\}} x_{t} = 1\) with \(x_{0} = 1\) (i.e. the index of the last feature that is 1). What is the reproduction probability of the first state: ?

📗 Answer: . Calculate

📗 [4 points] Run search algorithm on the following graph, starting from state 0 with the goal state being . Write down the expansion path (in the order of the states expanded). The heuristic function \(h\) is shown as subscripts. Break tie by expanding the state with a smaller index.

📗 In case the diagram is not clear: the weights are (with heuristic values on the diagonal entries): .

📗 Answer (comma separated vector): .

📗 [4 points] Which order of goal check is possible with , without specifying the order of successors when putting them in the queue (i.e. you can rearrange the order of the branches)? Enter the correct choices as a list, comma separated, without parentheses, for example, "1, 2, 4".

📗 Choices:

📗 Answer (comma separated vector):

📗 [4 points] Given the following BoS (Battle of Sexes) game, what is the column (Juliet) player's (expected) value (i.e. payoff) in the mixed strategy Nash equilibrium?

📗 Answer: . Calculate

📗 [4 points] Consider the following zero-sum game tree. MIN player moves first. Draw a new game tree by re-ordering the children of each internal node (including the root), such that the new game is equivalent to the tree above, but alpha-beta pruning will prune as many nodes as possible. (You do not have to submit the drawing.) Enter the number of nodes pruned.

📗 Answer: .

📗 [4 points] You are given the distance table. Consider the next iteration of hierarchical clustering using linkage. What will the new values be in the resulting distance table corresponding to the new clusters? If you merge two columns (rows), put the new distances in the column (row) with the smaller index. For example, if you merge columns 2 and 4, the new column 2 should contain the new distances and column 4 should be removed, i.e. the columns and rows should be in the order (1), (2 and 4), (3).

📗 Answer (matrix with multiple lines, each line is a comma separated vector): .

📗 [4 points] For a zero-sum game in which moves first and the value to the MAX player is given in the diagram below, consider the static board evaluation (heuristic function) at the internal states provided in the table below. What are the smallest and largest possible values of \(x\) above and below which IDS (iterative deepening search) with depth limit \(1\) will find the correct solution for the game? You can assume all values are between \(-100\) and \(100\). Enter two numbers between \(-100\) and \(100\) (possibly including \(-100\), \(100\)).

📗 Note: for example, if you think \(10 < x < 20\), enter \(10, 20\); if you think any \(x > 10\) works, enter \(10, 100\); if you think any \(x < 20\) works, enter \(-100, 20\); if you think every \(x\) is okay, enter \(-100, 100\); if you think no such \(x\) exist, enter \(-100, -100\) or \(100, 100\).

📗 Answer (comma separated vector): . Calculate

📗 [2 points] Consider \(n + 1\) = + \(1\) states. The initial state is \(1\), the goal state is \(n\). State \(0\) is a dead-end state with no successors. For each non-\(0\) state \(i\), it has two successors: \(i + 1\) and \(0\). We may expand the same states many times, because we do not keep track of which states are checked previously. How many states (including repeated ones) will be expanded by ? Break ties by expanding the state with the index first.

📗 Note: the tie-breaking rule may be different from the convention used during the lectures, please read the question carefully.

📗 Answer: . Calculate

📗 [3 points] Consider Depth First Search on a tree, where the nodes are denoted by numbers. Write down the sequence depth first search visited in the order they are expanded (i.e. expansion path). \(0\) is the initial state and is the goal state.

📗 Note: use the convention used in the lectures, push the rightmost successor into the stack first (i.e. expand the leftmost successor first).

📗 Answer (comma separated vector): .

📗 [3 points] If \(h_{1}\) and \(h_{2}\) are both admissible heuristic functions, which ones of following are also admissible heuristic functions? Enter the correct choices as a list, comma separated, without parentheses, for example, "1, 2, 4".

📗 Choices:

📗 Answer (comma separated vector): .

📗 [3 points] Consider a state space where the states are positive integers between 1 and . State \(i\) has two neighbors \(i - 1\) and \(i + 1\) (subject to the boundary constraints). State \(i\) has score . If one runs the hill climbing algorithm, how many initial states can reach the global minimum? Break tie by moving towards the global minimum. If there are multiple global minima, count all of them.

📗 Answer: .

📗 [4 points] What is the row player's value in a Nash equilibrium of the following zero-sum normal form game? A (row) is the max player, B (col) is the min player. If there are multiple Nash equilibria, use the one with the largest value (to the max player).

📗 Answer: .

📗 [4 points] Enter the smallest integer value of \(A\) such that \(B\) will be alpha-beta pruned? Max player moves first. In the case alpha = beta, prune the node. Enter -100 if you think the answer is negative infinity.

📗 Answer: .

📗 [4 points] Given the following game payoff table, suppose the row player uses a pure strategy, and column player uses a mixed strategy playing L with probability \(q\). What is the smallest and largest value of \(q\) in a mixed strategy Nash equilibrium?

📗 Answer (comma separated vector): . Calculate

📗 [4 points] You will receive 4 points for this question and you can choose to donate x points (a number between 0 and 4). Your final grade for this question is the points you keep plus twice the average donation (sum of the donations from everyone in your section divided by the number of people in your section, combining both versions). Enter the points you want to donate (an integer between 0 and 4).

📗 Answer: (The grade for this question will be updated later).

📗 [1 points] Give an estimate of the number of previous questions on this exam you think you answered correctly. Please enter an integer between 0 and the total number of questions on the exam (minus 2): do not leave it blank.

📗 Answer: .

📗 [1 points] Please enter any comments including possible mistakes and bugs with the questions or your answers. If you have no comments, please enter "None": do not leave it blank.

📗 Answer: .