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Artificial Intelligence In The 21st Century 2nd Edition Stephen Lucci, Danny Kopec - Solutions
4. A fallacy is a type of reasoning that seems to be valid but is not; an example is post hoc reasoning, which literally means “after this.” In this fallacy, one assumes that an event that occurred first is the cause of a later event. For example, this morning your horoscope might have stated
3. How might propositional logic be altered if the Law of excluded middle were not a theorem?
2. What limitations do you believe logic has as an AI knowledge representation language?
1. Comment on the difference in expressive power between the propositional logic and the FOPL.
5. Write a program to play 3D tic-tac-toe (see Exercise 12). Your program should employ the negamax algorithm. Test your program in machine vs. man mode, using a testing procedure similar to programming Exercise 4.
4. Write a program to play tic-tac-toe as discussed in the chapter (minimax is sufficient here).Your program should play in machine vs. machine mode (the computer makes both moves).The first player should follow a procedure that goes three levels into the game tree before heuristic evaluation is
3. Write a program to play Dots and Boxes as described in exercise 10. Employ alpha-beta pruning in your work. Test your program in machine vs. man mode (i.e., let people challenge your program in this game). The machine should move first.
2. Write a program to perform alpha-beta pruning.Test your program on the game trees of exercises 6, 7, and 8.
1. Write a program to perform minimax evaluation. (refer to Exercise 15).Test your program on the game trees of exercises 1, 2, and 3.
17. Problem 16 is reminiscent of the game known as Othello (or Reversi) which is covered in Chapter 16 (Advanced Computer Games). Othello starts with four stones (two White and two Black) placed in the middle of an 8 × 8 board (as in chess and checkers). When a stone is flipped to the opposite
16. The Nine Tails Problem is depicted in Figure 4.29.It is a one-player game where the goal is to flip coins so that they are all showing a Tails face. Once a coin with a Heads face is selected, that coin and the coins perpendicular (not diagonal) to it are flipped. So if the coin on the bottom
15. Give pseudocode for the minimax algorithm.
14. Following are several well-known problems. Discuss whether they are instances of essentially the same problem as the Prisoner’s Dilemma discussed in Section 4.6.If you conclude that a problem is equivalent to the Prisoner’s Dilemma, then design an appropriate payoff matrix. Comment on the
13. Use the negamax algorithm to evaluate the game tree in Figure 4.25.
12. Consider the game of three-dimensional tic-tac-toe. As usual the X and O players alternate and the object of play is to get three in a row on any row, column, or diagonal.a. Compare the complexity for 3-D tic-tac-toe with the more traditional 2-D version of the game described in the chapter.b.
11. The game of Sprouts was developed by M. S. Paterson and J. J. Conway. Draw two or more points on a piece of paper. Two players then alternate making moves according to these rules:• Draw a line (or curve) joining two points, or one point to itself (a self-loop). The line may not pass through
10. Examine the game of Dots and Boxes. Two players take turns drawing lines between adjacent dots on a 3 × 3 grid. Completing the last line on a box makes that box yours. The player that completes the most boxes wins.a. Draw the first several levels in the game tree for this game.b. Compare the
9. Consider the work entailed in evaluating the game trees from exercises 1–3 and 6–8.Minimax evaluation required the same amount of time regardless of the ordering of heuristic evaluations. What impact, if any, did the ordering of heuristic evaluations have on the amount of pruning that
8. Use alpha-beta pruning to evaluate the game trees shown in Figure 4.27. Be sure to indicate all alpha values and all beta values. Specify alpha cutoffs and beta cutoffs, if any.
7. Use alpha-beta pruning to evaluate the game trees shown in Figure 4.26. Be sure to indicate all alpha values and all beta values. Specify alpha cutoffs and beta cutoffs, if any.
6. Use alpha-beta pruning to evaluate the game trees shown in Figure 4.25. Be sure to indicate all alpha values and all beta values. Specify alpha cutoffs and beta cutoffs, if any.
5. A group < G, o > is a set G together with a binary operation o such that:• the operation o has closure—for all x, y in G, x o y is in G• associativity (x o y) o z = x o (y o z)—for all x, y, z in G• identity exists ∃ e ∈ G such that∀ x ∈ G x o e = e o = x• inverses
4. One variation on the game of nim is described in Luger. 7 The game begins with a single pile of stones. The move by a player consists of dividing a pile into two piles that contain an unequal number of stones. For example, if one pile contains six stones, it could be subdivided into piles of
3. Perform a minimax evaluation for the game tree in Figure 4.27.
2. Perform a minimax evaluation for the game tree in Figure 4.26.4
1. Perform a minimax evaluation for the game tree in Figure 4.25.
11. Does a Nash equilibrium always correspond to an optimal strategy? Explain.
10. Define the following terms.a. Nash equilibriumb. Pareto Optimal
9. What is the Prisoner’s Dilemma? Why has it received so much attention?
8. What is the expectiminimax algorithm? For what types of games is it useful?
7. What is progressive deepening? When is it useful?
6. What is the principle behind alpha-beta pruning? Why is it helpful in minimax evaluation?
5. What do we mean by symmetry in a game move? How does it help in game tree evaluation?
4. Briefly explain the principle behind minimax evaluation.
3. What is heuristic evaluation and why is it helpful in games with large game trees?
2. What is combinational explosion?
1. How does a game tree help to evaluate moves in a game?
14. Develop a solution to the Donkey Sliding Block Puzzle (Figure 3.26 in this Chapter). It requires 81 moves at minimum. Consider a subgoal that may apply to developing this solutions.++
13. In the 64 move solution to the Knight’s Tour Problem at some point it becomes necessary to abandon the heuristic you were asked to find in Problem 10.b. Try to identify that point. ++
12. In Chapter 2, we presented the n-Queens Problem. Write a program to solve the Eight-Queens Problem by applying constraints that remove any row or column from consideration once a queen has been placed.
11. Write a program to apply the primary heuristic search algorithms, described in this chapter, to Figure 3.17. That is, hill climbing, beam search, best first search, branch and bound with and without underestimates, and the A* algorithm.
10. Recall that in Chapter 2 we posed the Knight’s Tour problem, in which a chess Knight is to visit each of the n × n squares on a chessboard. The challenge is to start on a given source square on a full 8 × 8 chess board (say (1,1)), and to find a sequence of moves that visits every square on
9.a. Suggest an admissible heuristic for the Water Jug Problem.b. Employ your heuristic to conduct an A* search to solve the instance of this problem presented in Chapter 1.
8.a. Develop an admissible heuristic to solve the Maze Problem from Chapter 2 (Exercise 13).b. Employ your heuristic to conduct an A* search to solve this problem.
7. Use both the “plain vanilla” branch and bound, and branch and bound with dynamic programming, to get from the start node (S)to the goal node (G) in parts (a) and (b) of Figure 3.31. When all else is equal, explore nodes in alphabetical order.
6. Consider the following variation of the n-Queens Problem:If some of the squares that would be attacked by the placement are obstructed by the placement of pawns on an n x n chessboard, can more than n-Queens be placed on the partial board that remains? For example, if five pawns are added to a 3
5.a. Provide a heuristic that is appropriate for graph coloring.b. Employ your heuristic to find the chromatic number of the graph in Figure 2.41 in Chapter 2.
4.a. Suggest an admissible heuristic for the Missionaries and Cannibals Problem that is robust enough to avoid unsafe states.b. Is your heuristic informed enough to appreciably reduce the search-base explored by an A* search?
3. Suggest an admissible heuristic, not mentioned in the text, for solving the 3-puzzle.a. Employ your heuristic to conduct an A* search to solve the instance of the puzzle presented in this chapter.b. Is your heuristic more or less informed than the two heuristics that are presented?
2. Explain why hill climbing is called a “greedy algorithm.”a. Describe some other algorithms that you know that are “greedy.”b. Explain how steepest-ascent hill climbing is an improvement over simple hill climbing.c. How does the best first search improve over hill climbing?
1. Give three examples of heuristics and explain how they play a significant role ina. your day-to-day life, andb. the problem-solving process for some challenge that faces you.
14. Describe how the bidirectional search works.a. How is it different from the other techniques discussed in the chapter?b. Describe the frontiers problem and the Missiles Metaphor.c. What are wave-shaping algorithms?
13. Explain how AND/OR trees can be used to divide a search problem.
12. Explain the ideas behind the constraint satisfaction search and how it might apply to the Donkey Puzzle.
11. Why should the A* algorithm be better than branch and bound with underestimates or branch and bound with dynamic programming?
10.a. What is the notion of dynamic programming?b. Describe the Principle of Optimality.
9. Explain why underestimates are likely to result in better solutions.
8. What is the idea behind the branch and bound search?
7. What does it mean for a heuristic to be more informed than another heuristic?
6. What does it mean for a heuristic to be admissible?a. How does admissibility relate to monotonicity?b. Can there be monotonicity without admissibility? Explain.
5. Explain how beam search works.
4. Why is the best-first search more effective than hill climbing?
3. Explain how steepest-ascent hill climbing can also provide an optimal solution.
2. Explain why hill climbing would be classified as a greedy algorithm.
1. What distinguishes heuristic search methods from those discussed in Chapter 2?a. Give three definitions of heuristic search.b. Give three ways heuristic information can be added to a search.
6. When you color a graph, you assign colors to the nodes of a graph so that no two adjacent nodes have the same color. For example, in Figure 2.39, if node V1 is colored red, then none of the vertices V2, V3, nor V4 could be colored red. Vertex V5, however, might be colored with red as V1 and V5
8. Down to left one A Knight’s tour of an n × n chess board is a sequence of n2 – 1 moves so that a Knight visits each square on the board only once when started from an arbitrary square.Write a program to perform a Knight’s tour when n = 3, 4, and 5. Employ a random number generator to
7. Down two, right one
6. Down one, left two
5. Down one right two
4. Up two, left one
3. Up two, right one
2. Up one, left two
1. Up one square, right two squares
5. In Chess, a Knight may move in eight different ways:
4. Write a program to solve the 8-Queens problem that employs the generator suggested in Exercise 5.
3. Write a program that employs backtracking to solve the 8–Queens problem and then answer the following questions:a. How many solutions are there?b. How many of these solutions are distinct? (You can look ahead to Exercise 5 in Chapter 4 for hints.)c. Which generator did your program employ?
2. Write a program that employs the greedy algorithm to find the minimum spanning tree for a graph. A spanning tree T for a graph G is a tree whose vertex set is the same as the vertex set for the graph. Consider the graph in 2.38(a). A spanning tree is provided in Figure 2.38(b).Observe that the
1. Write a program to solve an instance of the 15-puzzle that first checks if a goal state is reachable. Your program should employ:a. Depth first searchb. Breadth first searchc. Depth first search with iterative deepening.
15. We discussed The Missionaries and Cannibals Problem. Develop a solution to this problem given that the “moves” or “transitions” are forced.Identify “subgoal states” to the problem solution state which must be achieved in order to solve the problem. ++
14. Having determined that the Twelve Coins Problem requires us to weigh three combinations of coins to identify the false coin: How many weighings would be needed to determine a false coin amongst 15 coins? What about 20 coins? Can you develop an algorithm to prove your conclusions?Hint: Consider
13. Use bfs and then dfs to get from the start to the goal, for the maze in Figure 2.37.
12. Label the maze in Figure 2.37.
11. Use bfs and then dfs to get from start node (S) to goal node (G) in parts (a) and (b) of Figure 2.36. At each step, explore nodes in alphabetical order.
10. A farmer with a wolf, a goat, and a container of cabbage are on the west bank of the river. On the river is a boat in which the farmer and one of the other three (wolf, goat, or cabbage) can fit. If the wolf is left alone with the goat, the wolf will eat the goat. If the goat is left alone with
9. Solve the Missionaries and Cannibals Problem using breadth first search.
8. Create a representation for a puzzle such as the 15-puzzle that is appropriate for checking repeated states.
7. Use Dijkstra’s Algorithm to find the shortest path from the source vertex Vo to all other vertices in Figure 2.35.
6. A number is said to be perfect if it equals the sum of its divisors (excluding itself). For example, 6 is perfect because 6 = 1 + 2 + 3, where each of the integers 1, 2, and 3 are divisors of 6. Give the most informed generator that you can think of to find all perfect numbers between 1 and 100,
5. Consider the following generator for the 4-Queens problem: for i = 1 to 4, randomly assign Queen i to a row.Is this generator complete? Is it nonredundant? Explain your answer.
4. Another generator for the n-Queens Problem is: Place a Queen in row 1.Do not place the second Queen in any square that is attacked by the first Queen. In state i, place a Queen in column i in a square that is not under attack from any of the previous i-1 Queens. See Figure 2.34.a. Solve the
3. A blind search not discussed in this chapter is a nondeterministic search. It is a form of blind search in which the children of nodes that have just been expanded are placed on the open list in random order. Is a nondeterministic search complete? Optimal?
2. Solve the Mini False Coin Problem weighing only twice, or prove that this is not possible.
1. Solve the False Coin Problem for 12 coins. Only three combinations of coins are permitted to be weighed. Recall that a balance scale returns one of three results: equal, left side is lighter, or left side is heavier.
13. In what sense is dfs-id a compromise between bfs and dfs?
12. When is bfs preferable to dfs?
11. When is dfs preferable to bfs?
10. Compare the three blind search algorithms described in this chapter with respect to:completeness, optimality, and time-and-space complexity.
9. In what sense are blind search algorithms blind?
8. Name three blind search algorithms.
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