Module Bank-1-SEC-B 1. Consider a water jug problem. You are given 3 jugs: a 10-gallon 5-gallon and a 6-gallon jugs. Neither has any measuring mark in it. There is a pump that can be used to fill the jugs with water. How can you get exactly 8-gallon of water into a 10-gallon jug? State the production rules for the water jug problem. Formulate this problem as state space search problem and answer the following questions according to your formulation. a. How do you represent a state? b. What is your representation of the start state? c. Give your representation of the set of goal states. d. Draw the state space graph of this problem e. Show any one solution to the problem by specifying the sequence of operations. 2. Consider the problem of finding the shortest path between two points on a plane that has convex polygonal obstacles as shown in. This is an idealization of the problem that a robot has to solve to navigate in a crowded environment. a. Suppose the state space consists of all positions (x,y)(x,y) in the plane. How many states are there? How many paths are there to the goal? b. Explain briefly why the shortest path from one polygon vertex to any other in the scene must consist of straight-line segments joining some of the vertices of the polygons. Define a good state space now. How large is this state space? c. Define the necessary functions to implement the search problem, including a function that takes a vertex as input and returns a set of vectors, each of which maps the current vertex to one of the vertices that can be reached in a straight line. (Do not forget the neighbors on the same polygon.) Use the straight- line distance for the heuristic function. d. Apply one or more of the algorithms in this chapter to solve a range of problems in the domain, and comment on their performance. 3. Consider the Z-to-A example below (a reversed mangle of the class example). Trace the sequence of transformations for the "expanding all ways from a chosen state" approach, to get from Z to A, for a a. Consecutively bounded DFS to find any solution, depth increment 1. b. BFS, using the indicated estimated costs to the goal. 4. Consider the problem of finding the optimal route between two cities on a map, represented as a graph. The objective is to minimize the distance traveled. a. Explain the hill climbing algorithm's application to this problem, considering each city as a state and the edges between cities as possible moves. b. Describe the evaluation function that assigns a value (distance) to each state (city). c. Perform a step-by-step illustration of the hill climbing algorithm on a small example, starting from an initial city and selecting the neighbor with the lowest distance at each step. d. Identify the challenges faced by the hill climbing algorithm in finding the optimal route and potential issues related to local optima. e. Discuss how the hill climbing algorithm could be enhanced by incorporating random restarts or simulated annealing to overcome local optima and improve the quality of the solution. 5. The following question pertains to adversarial search. Consider the game tree in Figure in which static scores are all from the first players point of view. (Triangle is maximizer. Upside-down triangle is minimizer.) MAX MIN 3 12 82 4 6 2 5 14 Figure: A Min- max Game Tree 6. The missionaries and cannibals problem is usually stated as follows. Three missionaries and three cannibals are on one side of a river, along with a boat that can hold one or two people. Find a way to get everyone to the other side without ever leaving a group of missionaries in one place outnumbered by the cannibals in that place. This problem is famous in AI because it was the subject of the first paper that approached problem formulation from an analytical viewpoint (Amarel, 1968). A. Formulate the problem precisely, making only those distinctions necessary to ensure a valid solution. Draw a diagram of the complete state space. B. Implement and solve the problem optimally using an appropriate search algorithm. Is it a good idea to check for repeated states? C. Why do you think people have a hard time solving this puzzle, given that the state space is so simple? 7. You are tasked with creating a restaurant ordering agent that can assist customers in placing their orders based on their preferences. The agent will use various search algorit hms to navigate through the menu and find suitable recommendations for the customers. To simplify the problem, assume that the restaurant has a fixed menu and specific rules for recommending dishes to customers. The agent will gather information from customers, search through the menu using different searching algorithms, and provide appropriate suggestions for their orders. Your task is to implement a searching algorithm in Python that can assist the agent in finding suitable recommendations based on predefined rules and customer preferences. Here are the details of the scenario: 1. Menu: The restaurant has a fixed menu of dishes, each identified by a unique name and possibly accompanied by additional information such as ingredients or dietary restrictio ns. 2. Customer Preferences: Customers will provide their preferences, such as dietary restrictions (e.g., vegetarian, gluten- free), preferred cuisine types, and any specific ingredients they dislike. 3. Rules and Recommendations: Define a set of rules that map customer preferences to appropriate recommendations. Each rule consists of a condition (e.g., vegetarian and Italian cuisine) and a corresponding recommendation (e.g., "We recommend the Margherita pizza or the Vegetable pasta"). To complete this task, you will need to implement the following components: 1. Menu Representation: Define the restaurant's menu as a data structure in Python, such as a list or a dictionary, where each item represents a dish with its relevant details. 2. Searching Algorithm: Implement a searching algorithm, such as linear search, binary search, or even a more complex algorithm like A* search, to search through the menu and find suitable recommendations based on customer preferences. 3. Recommendation Generation: Based on the search results, generate appropriate recommendations for the customer by applying the predefined rules and matching them with the customer's preferences. Once you have implemented the searching algorithm in Python, you can test it by simulating customer interactions, providing their preferences, and evaluating the agent's ability to find suitable recommendations based on the predefined rules and the menu. 8. Imagine you are planning a road trip across the United States, starting from New York City and ending in San Francisco. You want to find the optimal route to reach your destination, considering the shortest distance traveled. To find the optimal route, you decide to use a state space search algorithm. The state space search involves exploring different states (cities) and the possible transitions (roads) between them. 1. Start State: New York City 2. Goal State: San Francisco State Space: - New York City - Boston - Philadelphia - Washington D.C. - Pittsburgh - Chicago - Denver - Las Vegas - Los Angeles - San Francisco Transitions (Roads) and Distances: 1. New York City to Boston: 225 miles 2. New York City to Philadelphia: 95 miles 3. New York City to Washington D.C.: 225 miles 4. Boston to Pittsburgh: 520 miles 5. Boston to Chicago: 1,000 miles 6. Philadelphia to Pittsburgh: 305 miles 7. Philadelphia to Washington D.C.: 140 miles 8. Pittsburgh to Chicago: 470 miles 9. Pittsburgh to Denver: 1,460 miles 10. Chicago to Denver: 1,020 miles 11. Chicago to Las Vegas: 1,730 miles 12. Denver to Las Vegas: 750 miles 13. Denver to Los Angeles: 1,020 miles 14. Las Vegas to Los Angeles: 270 miles 15. Los Angeles to San Francisco: 380 miles Using a state space search algorithm, such as breadth-first search or A*, you can explore the different cities and the distances between them to find the optimal route from New York City to San Francisco. The algorithm will expand nodes (cities) based on the available transitions (roads) and the distance from the start state to the current state. Note: The distances mentioned here are for illustrative purposes only and do not represent actual distances between cities. Draw a State space diagram for each move. 9. Consider the game tree below, which contains maximizer nodes and chance nodes. A. Assume that for the chance nodes the probability of each outcome is equally likely. Is pruning possible if the nodes are evaluated in left-to-right order? If NO, provide a brief justification. if YES, cross out the branches that can be pruned. B. What is the propagated value at the root node under the condition as in (A)? C. What is the minimum value of the root node assuming a fully rational opponent but considering the worst-case outcome at the chance nodes? 10. Consider the following game: there are three piles of sticks, with 1,1 and 2 sticks in each pile, respectively. There are THREE players A, B, C who play in turn in that order. Each player can take one or more sticks from one and only one pile. The player who takes last stick wins one treat from each of the other two players (i.e., two treats total). i. How would you represent the game value of the leaf nodes in the game tree now that there are three players? ii. Draw the complete game tree (hint: the piles (1,2,1) are equivalent to (1,1,2). You may want to always sort the numbers from small to large). iii. What should A’s initial move be?
