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📗 [4 points] Given there are states in a search tree with levels (max depth, the root is at depth 0 or level 0 but counts as one level) and no goal states, what is the imum possible number of states expanded during an IDS (Iterative Deepening Search)? The same state may be expanded multiple times during different iterations (with different depth limits) of the search.
📗 Note: the number of levels is fixed in this question, a tree with more or fewer levels is not valid. The answer obtained from not counting the root as one of the levels will be accepted too.
📗 Answer: .
📗 [3 points] Given the following two admissible heuristic for A* search, write down another admissible heuristic that (weakly) dominates the two. Write \(\left(h\left(s_{1}\right), h\left(s_{2}\right), h\left(s_{3}\right), h\left(s_{4}\right), h\left(s_{5}\right)\right)\) as a comma separate list. If there are multiple possible values, write one of them, if there are none, write \(-1, -1, -1, -1, -1\).
State
\(s_{1}\)
\(s_{2}\)
\(s_{3}\)
\(s_{4}\)
\(s_{5}\)
\(h_{1}\)
\(h_{2}\)
\(h\)
\(h\left(s_{1}\right)\)
\(h\left(s_{2}\right)\)
\(h\left(s_{3}\right)\)
\(h\left(s_{4}\right)\)
\(h\left(s_{5}\right)\)
📗 Answer (comma separated vector): .
📗 [4 points] Given the following BoS (Battle of the Sexes) game, consider a sequential move game with the same payoffs (values), in which with probability \(p\) = , Romeo (row player) moves first, and with probability \(q\) = , Juliet (column player) moves first. What is the expected (game theoretic) value of the game (two values one for Romeo and one for Juliet)?
Actions
B
S
B
\(\left(0, 0\right)\)
S
\(\left(0, 0\right)\)
📗 Note: this is a game with Chance moving first, and Romeo second Juliet third in one branch, and Juliet second Romeo third in the other branch. In case of ties, the players will choose B instead of S.
📗 Answer (comma separated vector): .
📗 [3 points] Consider the following zero-sum game, in a Nash equilibrium, the row player uses actions \(U, M, D\) with probabilities , and the column player uses actions \(L, C, R\) with probabilities \(q_{1}, q_{2}, q_{3}\). Write down \(q\) as a vector (probabilities that sum up to 1).
Actions
L
C
R
U
M
D
📗 Answer (comma separated vector): .
📗 [2 points] Suppose scaled dot-product attention function is used. Given two vectors \(q\) = , \(k\) = , calculate the attention score of \(q\) to \(k\).
📗 Answer: .
📗 [3 points] Assume tokenization rule is using whitespaces between words as separator, input one sentence \(s_{1}\) into decoder stack during training time. Write down the attention mask of self-attention block in decoder, where \(1\) = attented, \(0\) = masked.
Sentence: \(s_{1}\) = "". (Note: "< s >" is one token, not three).
📗 Answer (matrix with multiple lines, each line is a comma separated vector):
📗 [4 points] There are 3 states \(s_{0}, s_{1}, s_{2}\) and 3 actions \(a_{0}, a_{1}, a_{2}\). We start from , choose , we get the reward and then move to , choose . Update the Q value for (, ) based on the current Q table and the movement above, using SARSA and Q-learning (enter two numbers, comma separated)? The reward decay (discount rate) is \(\gamma\) = , and the step size (learning rate) is \(\alpha\) = .
State \ Action
\(a_{0}\)
\(a_{1}\)
\(a_{2}\)
\(s_{0}\)
\(s_{1}\)
\(s_{2}\)
📗 Answer (comma separated vector): .
📗 [4 points] Let the states be 3D integer points with integer coordinates \(\left(i, j, k\right)\) with boundary constrains and and . Each state \(\left(i, j, k\right)\) has six successors \(\left(i - 1, j, k\right), \left(i + 1, j, k\right), \left(i, j - 1, k\right), \left(i, j + 1, k\right), \left(i, j, k - 1\right), \left(i, j, k + 1\right)\) or a subset thereof subject to the boundary constraints. The score of state \(\left(i, j, k\right)\) is . Which local minimum will be reached if hill climbing is used starting from ? Enter the state, not the score.
📗 Answer (comma separated vector): .
📗 [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?
The goal state is:
\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(7\)
\(8\)
\(0\)
The initial state is:
📗 Answer: .
📗 [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:
(1)
(2)
(3)
(4)
(5)
(6)
(7) None of the above
📗 Answer (comma separated vector): .
📗 [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: .
📗 [4 points] The Nash equilibrium of the following simultaneous move zero-sum game is (U, L): the entry marked by \(x\). What is the smallest and largest possible values of \(x\)? Enter two numbers. (U, L) can be one of possibly many Nash equilibria.
📗 Note: if there is only one possible value, enter the same value twice; and if no values are possible, enter \(0, 0\).
MAX \ MIN
L
C
R
U
\(x\)
M
R
📗 Answer (comma separated vector): .
📗 [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.
Index
1
2
3
4
State
📗 Answer: .
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📗 Answer: .
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