Artificial Intelligence 2007 Spring Midterm Solution

Transcription

1 Artificial Intelligence 2007 Spring Midterm Solution May 31, (10 points) Minesweeper is a famous one-player computer game as shown in Figure 1. The player has to find all the mines in a given minefield without detonating anyone of them. Suppose you are asked to design an intelligent agent to play a minesweeper game, develop the PEAS description of this task.(hint: P: Performance, E: Environment, A: Actuators, S: Sensors) Figure 1: A minesweeper game Performance Environment Actuators Sensors Time spent, Score gained Minefield, Cells Mouse Camera 2. (15 points) Given a Tree G(V, E) as Figure 2. Let the depth for node v V be d(v), and edge cost for edge e E be g(e). (a) (3 points) In what case that Uniform-cost search will be identical to BFS? (b) (5 points) List the order in which nodes will be visited from node 1 to 11 using iterative deepening search. (c) (7 points) Best-first search is an algorithm that a node v is selected for expansion based on an evaluation function f(v). Show that BFS, DFS are special cases of Best-first search. (a) When the costs for each step are the same. (b) 1,1,2,3,1,2,4,5,3,6,7,1,2,4,8,9,5,10,11 (c) When f(v) = d(v), Best-first Search acts the same as BFS. When f(v) = 1/d(v), Best-first search acts the same with DFS. Thus BFS and DFS are all special cases of Best-first search. 3. (20 points) Consider the problem of moving k knights from k starting squares s 1, s 2,..., s k to k goal squares g 1, g 2,..., g k, on an unbounded chessboard, in the smallest number of actions. Each 1

2 Figure 2: G(V, E) Figure 3: Legal moves of a knight on the chess board. action consists of moving from 0 to k knights simultaneously, subject to the rule that no two knights can land on the same square at the same time. The legal moves of a knight are marked in Figure 3. (a) (5 points) Define the maximum branching factor b in this state space as a function of k. (b) (7 points) Suppose h i is an admissible heuristic for the problem of moving knight i to goal g i by itself. We can define a heuristic function for the k-knight problem as: min{h 1, h 2,..., h k } Please prove or disprove whether the heuristic function is admissible for the k-knight problem. (c) (8 points) Define an admissible heuristic function that dominates the heuristic function in (b). Please justify your answer. (a) 9 k (or 9 k 1 if noop is not permitted) (b) It is admissible! Let the real cost be h, since i, h i are admissible, we have j, h j h in any state. So for any integer j {1, 2,..., k}, we have min{h 1, h 2,..., h k } h j h. (c) Define a new heuristic function h = max{h 1, h 2,..., h k }. It is obvious that h is admissible and dominates min{h 1, h 2,..., h k }. 4. (15 points) Here is an instance of a well-known logic puzzle called The Zebra Puzzle: There are five houses, each with a different color, live 5 persons of different nationalities, each of whom prefer a different brand of cigarette, a different drink, and a different pet. Moreover, 2

3 Figure 4: The constraint graph The Englishman lives in the red house. The Spaniard owns the dog. Coffee is drunk in the green house. The Ukrainian drinks tea. The green house is immediately to the right of the ivory house. The Old Gold smoker owns snails. Kools are smoked in the yellow house. Milk is drunk in the middle house. The Norwegian lives in the first house. The man who smokes Chesterfields lives in the house next to the man with the fox. Kools are smoked in the house next to the house where the horse is kept. The Lucky Strike smoker drinks orange juice. The Japanese smokes Parliaments. The Norwegian lives next to the blue house. By the constraints given above, one is asked to find, say, who owns the zebra. Given The Zebra Puzzle as above, formulate it as a Constraint Satisfaction Problem using a constraint graph. You must also define a set of variables with their domain. It s not necessary to solve this puzzle here. The constraint graph is shown as Figure 4. Each node is a variable with domain: {P OSIT IONS = {1, 2, 3, 4, 5}, COLORS = {Red, Blue, Ivory, Green, Y ellow}, N AT ION ALIT IES = {Englishman, Spaniard, U krainian, N orwegian, Japanese}, DRINKS = {Coffee, Milk, Orange juice, T ea, UnknownDrink}, CIGARET T ES = {Old golds, Kools, Chesterf ields, Lucky strikes, P arliaments}, P ET S = {Zebra, Horse, F ox, Dog, Snail}}. 5. (15 points) Given a two-player game tree as Figure 5. Assume that root is a MAX node. (a) (5 points) What is the best reward of the root node in this game tree? (b) (10 points) If we perform a left-to-right alpha-beta pruning to this game tree, which nodes will be pruned? Circle all of these nodes. : (a) 8 3

4 Figure 5: A two-player game tree Figure 6: Applying alpha-beta pruning to the game tree (b) The result after applying alpha-beta pruning to the tree in Figure 5 is shown in Figure 6. All the pruned nodes are circled. 6. (20 points) We are given the following sentences: If a course has a final, no students who take this course will be happy. If a course is easy, some students who take this course will be happy. and the following predicates: hasf inal(x): course x has a final ishappy(x): student x is happy iseasy(x): course x is easy take(x, y): student x takes course y (a) (5 points) Translate these sentences into FOL sentences. (b) (5 points) Convert the preceding FOL sentences to CNF. (c) (10 points) Use resolution to prove that any course without a final is an easy course. : (a) x y hasf inal(x) take(y, x) ishappy(y) x iseasy(x) y take(y, x) ishappy(y) 4

5 (b) (1) hasf inal(x) take(y, x) ishappy(y) (2) iseasy(x) take(f (x), x) (3) iseasy(x) ishappy(f (x)) (c) Negating goal x hasf inal(x) iseasy(x), we will get hasf inal(x) iseasy(x). We can find that it is impossible to use resolution rule to get a empty clause. That is, we can not prove the goal being true or false. 7. (25 points) Short answer questions: (a) (10 points) What is a sound inference algorithm? And what is a complete inference algorithm? (b) (5 points) Inference with Horn clause can be done through the forward chaining and backward chaining. What is Horn clause? Give an example. (c) (5 points) Given n cities on a map. Straight-line heuristic and MST heuristic can be two different admissible heuristic functions for solving TSP problem using A algorithm. Show that MST heuristic dominates straight-line heuristic. (d) (5) Given a informed search strategy. In what case will a heuristic function h be a consistent heuristic function? (Hint: define a cost function c : S A S R +, where S is the set of states, A the set of actions, and R + the real positive numbers). : (a) An sound inference algorithm only derives sentences that can be entailed from a knowledge using this algorithm. A complete inference algorithm can derive any sentence that can be entailed from a knowledge base using this algorithm. (b) A horn clause is a disjunction of literals of which at most one is positive. P Q R is a Horn clause. For example, (c) MST dominates straight-line distance because the shortest distance from any two cities u, v is their straight-line distance. Moreover, let G MST (V, Ē) be the solution of MST problem given the instance G. The cost from u to v for MST will be equal to the straight-line distance only if (u, v) Ē. Thus MST heuristic dominates straight-line distance. (d) Let c(n, a, n ) be the cost from state n to state n that generated by taking action a. A heuristic function h(n) is consistent if for every state n, we have: h(n) c(n, a, n ) + h(n ) 5