Solutions. 1) Is the following Boolean function linearly separable? Explain.

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1 CS/ES 480, CES 510 Mid-term # 1 Practice Problems Solutions 1) Is the following Boolean function linearly separable? Explain. f ( x, y,z ) ( x y z ) ( x y z ) ( x y z ) Solution: Geometric representation of the function is as follows: The dark circles represents the points at which f takes value = 1. From the picture, it is clear Z that f is not a linear threshold function. Y X We can prove this fact as follows: Suppose f is linear threshold. Then, there exist A, B, C and D such that Ax + By + Cz > D if and only if f(x,y,z) = 1. This gives A + B + C > D, B + C < D. Thus, A > 0. However, since f(1,0,0) = 0, A < D and since f(0,0,0) = 1, we have: D <0. Combining the last two inequalities, A < 0, which contradicts A > 0 that we derived above. 2) Consider the search problem illustrated by the graph of Figure 1. Each node of the graph represents a state. So, the state space is {S, A, B, C, D, E, G}. The initial state is S and the goal state is G. Each arc represents a possible transition from one state to another and the number besides each arch is the cost of the transition. We would like to use the A* algorithm to generate a minimum-cost path from S to G, where the cost of a path is the sum of the costs of the arcs forming this path. The h values given for states are the values of the admissible monotonic heuristic function h used by A*.

2 Figure 1 Fill in the following table with the contents of the fringe priority queue after each iteration of the A* algorithm (the best lowest scoring entry in the queue is written to the left, the worst to the right). After the iteration where A* decides to exit, write Finished. Note that the same state may be achieved several times. When this happens update the fringe priority queue appropriately. When there is a tie in the queue, break this tie by choosing an arbitrary order. (The table is longer than needed, that is, you are not expected to fill in all lines of the table. A* will terminate before the 10 th line. We ve started the table for you.) Answer: After iteration The fringe priority queue contains Initialization (S, 3) 1 (B, 3), (A, 5), (C, 7)

3 2 (E,3), (A,5), (C,6) 3 (A,5), (C,6), (G,7) 4 (C,4), (D,5), (G,7) 5 (G,4), (D,5), (E,5) 6 FINISHED 3) If a (finite) data set representing a Boolean function f of three real variables is linearly separable, is it true that the projection of f on XY plane, namely f x (y, z) = f(x, y, 0) linearly separable? If the answer is YES, prove it. If the answer is NO, give a counter-example. Answer: No. Consider f defined as follows: f(0, 0, 0) = 0, f(1, 0, 0) = 0, f(0, 1, 0) = 0, f(0.5, 0.5, 0.5) = 1. Clearly, f is linearly separable with weights (0, 0, 0.5) and threshold = The projection of the points on XY plane will give a function that is NOT linearly separable. 4) What are the memory requirements of each of the following the following algorithms to search a tree of depth D and branching factor B? (a) BFS (b) DFS (c) iterative deepening search Solution: (a) O(B^D) (b) O(B*D) (c) O(B*D) 5) Suppose the search tree for a problem is as shown below. The cost of each edge is 1. The goal nodes are 5 and 12. A heuristic function h for this problem is shown in the table: h a. List the order in which A* will expand the nodes of this tree. b. Is h admissible? Explain your answer. c. Is h monotonic? Explain your answer. d. Suggest an admissible heuristic h that dominates h.

4 Solution: (a) Display the successive nodes expanded by A* 1, 2, 5 (b) Is h^ admissible? Yes. Some of you said it is not by claiming that actual distance from 6 to goal is 2 while h(6) = 5 > 2. But actual distance from 6 to goal is infinity since the distance is measured along a path from root to leaf, not by a path going up and down the tree. (c) Suggest an admissible heuristic function h that dominates h^. One such function is h where h (x) = h(x) for all x except at 8, and h (8) = 3. 6) Consider a 1-dimensional sliding block puzzle with the following initial configuration: B B B W W W E There are three black tiles (B), three white tiles (W) and an empty cell (E). The puzzle has the following moves: A tile may move to an adjacent empty cell with unit cost.

5 A tile may hop over at most two other tiles into an empty cell with a cost equal to the number of tiles hopped over. The goal of the puzzle is to have all of the white tiles to the left of all the black tiles (without regard to the position of the blank cell). (a) Does the (state space) search graph have cycles (of length 3 or more)? If so, give an example. No. Here is a cycle of length 3. BBBWWWE BBBWWEW BBBWEWW BBBWWWE. (b) Exhibit the first 4 nodes that will be expanded by DFS and BFS. (Assume arbitrary order of children for each node.) BFS will expand the nodes BBBWWWE, BBBWWEW, BBBWEWW, BBBEWWW. One possible solution for DFS is: BBBWWWE, BBBWWEW, BBBWEWW, BBBEWWW (Assuming that the sliding move is first tried). (c) For each black tile (B), assign a value 1 if there is at least 1 white tile (W) to its right, 0 otherwise. For each white tile (W), assign a value 1 if there is at least 1 black tile (B) to its left, 0 otherwise. Define a function h(n) as the sum over the assigned values for all tiles. What is the value of h(root)? What is the value of h(goal) for all the goal nodes? Is h admissible? Is h monotonic? h(root) = 6 since 6 tiles are out of place. h(goal) = 0 obviously. h is admissible, as well as monotonic. 7) Shown below is the MIN-MAX algorithm. Modify it as follows: instead of searching the entire tree, we will stop as soon as an inactive node is reached. Assume that there is a

6 Boolean function inactive(x) that takes as input a node x, and returns true if x is inactive. Solution: Only one change is needed. The condition if (leaf(u)) return Eval(u) should be replaced by if (leaf(u) or inactive(u)) return Eval(u) 8) Consider the game tree below, in which the values assigned by the evaluation function are shown at the leaves. The root is a MAX player. Suppose MAX uses alpha-beta pruning. Write down the final α and β values assigned to each node. What move will MAX choose? What nodes will be pruned by the alpha-beta algorithm? Short answers: 1) Which of the following are true of BFS? DFS? Iterative deepening? (a) It uses a stack to implement the fringe. (b) It always finds the shortest path in the unweighted cost model. (c) It will always find a solution in finite number of steps.

7 (d) It is not complete DFS: true, false, false, false BFS: false, true, true, true ID: true, true, true, false 2) Under what conditions does A* reduce to BFS? Choosing h = 0 makes A* reduce to BFS. 3) If a heuristic function h is admissible, does it imply it is monotonic? Explain. No. We gave an example in class of a h function that is admissible but not monotonic. 4) Converse of (3). YES. We proved this in class. 5) Define a threshold gate or perceptron. More precisely, suppose x1, x2,, xn are inputs to a perceptron with weights w1, w2,, wn, and threshold value is C. State the output of this gate. (i.e., when is it 1 and when is it 0)? 6) What is horizon effect? How is it handled in a game search algorithm? 7) Define the alpha (beta) value of a MAX (MIN) node. (can be found in the text) 8) Which of the following are true of alpha-beta pruning algorithm? (i) it uses horizon effect to prune some nodes in the tree NO (ii) the final decision on which move to make at the root node is the same as the MINMAX algorithm applied to the same tree. YES (iii) The alpha (beta) values computed at all the unpruned nodes are the same as the max (min) values computed by the min-max algorithm. NO (iv) The success of alpha-beta pruning depends on the order in which the children of a node are expanded. YES 9) What is the main difference between supervised and unsupervised learning? 10) What are the main challenges of reinforcement learning? 11) What are the main techniques used by the following famous AI algorithms? (i) Slagle s SAINT (AND/OR graph searching) (ii) Sejnowski s NET Talk (neural network) (iii) Samuel s Checkers program (reinforcement and self-play)