Problemas de Inteligencia Artificial Inglés (Curso ) Tercer curso del Grado en Ingeniería Informática, Universidad de Sevilla

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1 Problemas de Inteligencia Artificial Inglés (Curso 0 05) Tercer curso del Grado en Ingeniería Informática, Universidad de Sevilla Search in state spaces. Consider a problem formulated in the states space framework, having constant branching factor b and having a unique solution located at depth d. Calculate, both for the best and the worst cases, the number of nodes that need to be analyzed in order to find the solution when running a breadth-first search. Do the same for depth-first search. Calculate, both for the best and the worst cases, the maximum number of nodes that could be waiting to be analyzed in the OPEN queue when running a breadth-first search. Do the same for iterative deepening search.. Provide an example of a state space problem such that the CLOSED list is superfluous in depth-first and breadth-first search algorithms, explaining why it is so.. Consider a state space problem such that all the solutions, in case they exist, must have a fixed number of actions d. If we need to choose an uninformed searching algorithm, which one is more adequate? The answer should be justified.. Suppose we have two admissible heuristic functions, h and h, one of them being more informed than the other, and we want to apply the A algorithm. Are there any differences from a theoretical point of view between using one or the other? And from a practical point of view? 5. A group of 5 people are going to cross an old and narrow bridge. It is a dark night and the bridge cannot be crossed without a torch. The group has only one torch, whose battery will last only for 5 minutes. a) Person A can cross the bridge in 0 seconds, while B, C, D and E spend 0, 0, 80 and 0 seconds, respectively. b) The bridge can only hold two people at a time, and when two people cross the bridge together, they must move at the slower person s pace. c) The torch cannot be thrown from one end of the bridge to the other, therefore after two people cross, one of them should cross back to bring the torch to the rest, until all of them have crossed. a) Formulate this problem in the states space framework, describing precisely all the elements b) Define two different heuristics, reasoning for each one whether they are admissible. c) Apply the A algorithm with the best one of these heuristics. For the sake of simplicity, start building the search tree using as initial situation having A and C on the left (still have to cross), and B, D and E are on the right having the torch with them. The evolution of the search tree should be explained explicitly: order of analysis of the nodes; their values; justifying why new successors are not included, etc.

2 . We have a toaster that can work on one or two slices of bread at the same time. However, it does not work well and actually only one side of the slices gets toasted. The toaster timer lasts one minute. The goal is to get three slices of bread toasted on their two sides, and spending less than four minutes. Formulate this problem in the state space framework, describing precisely all the elements Indicate which of the studied algorithms is more suitable to solve the problem, explaining your choice. 7. The Missionaries and Cannibals problem can be stated as follows: M missionaries and C cannibals need to cross a river, and the only boat available has a capacity for P people. The problem is to find the way to transport everyone across the river, taking into account that when the boat leaves a bank, the missionaries remaining there cannot be outnumbered by the cannibals. Besides, want to minimize the sum of the total number of river-crossings of each person. Formulate the problem in the states space framework and indicate which informed algorithm (and which heuristic) you should use to find a solution. More info at: problems 8. We want to distribute several computation tasks among several processors. The tasks to be carried out are known a priori, and for each one of them we know exactly the amount of time required by a processor to acomplish it, with no interruptions from the beginning up to the completion of the task (the time is independent of the processor). Any processor can be used for any task, provided that it is not busy doing at this moment some other task. The goal is to minimize the global time for all tasks to be done, considering that processors can work in parallel. a) Formulate this problem in the states space framework, and design an admissible heuristic function allowing us to minimize the global processing time. b) Which search algorithm should be used? Explain your reasoning. 9. Consider the following puzzle: we have a linear board having black pieces (B), white pieces (W) and an empty space (E), in the following initial configuration: B B B W W W E The goal of the puzzle consists on placing all the white pieces to the left of all the black pieces, the position of the empty space is irrelevant. The legal moves are the following: A piece can be moved to an adjacent empty cell, with cost. A piece can go to the empty space by jumping over at most two adjacent pieces, with a cost equal to the length (number of pieces) of the jump. You should: Formulate this problem in the states space framework, describing precisely all the elements Define an admissible heuristic function for this problem, explaining why it is admissible. In order to solve the puzzle, would you use Best-first search algorithm, or A instead? Why?

3 0. Explain, without getting into implementation details, how to formulate the following problem in the states space framework. Provide the representation for the states (specifying initial state and final states) and the actions (applicability conditions, how the actions are applied over a state, and cost of such application). Define also an admissible heuristic function, explaining why it is admissible. The problem can be stated as follows: given four natural numbers n, m, r and T, find the minimal sequence of basic arithmetic operations (addition, subtraction, multiplication and division) starting from 0 and getting T as final result, in such a way that using only numbers n and m are used in the operations, having the additional constraint that neither n or m can be used more than r times. For example, let n =, m =, r = and T = 8, a possible solution (not necessarily minimal) is ((((((0 + ) ) ) ) ) ), since the total result is 8 and neither nor have been used more than three times each.. Consider the problem of the -puzzle, a reduced version of the 8-puzzle, where there are three tiles (tagged with,, ) in a board (hence there is a blank space). Initial and final states are, respectively: Initial state Final state The actions to be considered are: moving the space up, moving the space down, moving the space left, and moving the space right (exactly in this order), and the cost of application for all of them is. Represent graphically the search trees corresponding to: a) Depth-first search. b) A search using heuristic h = Manhattan distance from the current position of the space to its position in the final state. c) Best-first search using heuristic h = number of tiles which are not in the same position as in the final state. For each of them, indicate by each node the order of analysis, and if it the case, its heuristic value (or cost+heuristic). In case of nodes with the same values, select the node which has been longer in the open queue. Is h admissible? Is h admissible? Which one is more informed?

4 . Let us consider a robot that can navigate through the following net, where each edge is meter long. The position of the robot is determined by the coordinates (i, j) of the node where it is currently, together with the orientation info: North, South, East, West. For example, the location of the robot in the previous picture is (,), orientation West. At each instant, the robot can: move forward according to its current orientation until the next node, or rotate 90 degrees clockwise or counterclockwise, staying at the same coordinates. Besides, we know that it: spends s. in completing a 90 degrees rotation. runs at 0.5 m/s speed along the thick edges. runs at m/s speed along the thin edges. (a) Formulate the problem of sending the robot from a location to a different one spending the minimum possible time in the framework of states space search. (b) For the particular case of sending the robot from location L ((, ), orientation West), to location L ((, ), orientation North): Define an admissible heuristic function, as informed as possible, and justify why it is admissible. Apply the A* algorithm, using the heuristic function defined in the previous item, and draw the search tree indicating the order of generation and order of analysis of the nodes. Describe the solution found and its cost. Is the solution found the one having minimum path length? Explain your answer. Note:In case of nodes with the same values, sort them according to the list (E, S, W, N).. Formulate the following problem in the states space framework, describing precisely all the elements required for the representation, and indicating also which algorithm should be applied to solve it (just name the algorithm, do not run it). In case of an heuristic been needed, define one and indicate whether it is admissible or not (if this makes sense in the context of the chosen representation). Two bike travellers start their trip simultaeously, one of them from La Coruña, and the other one from Almería. At each stage of their journey, a traveller can move from his/her current city to any other capital among the neighbouring provinces. We want to find a route for the travellers such that they finally meet at some city. Besides, the goal is to minimize the time until they meet, taking into account that they always synchronize their stages (i.e. they start a stage from a city to a neighbouring one simultaneously, and when they complete a stage they make sure by exchanging messages that the other traveller also completed his/her stage before starting the next). Assume that we know for every city the duration of the trip to each of its neighbours.

5 . Given a set of integer numbers I = {i, i, i,..., i N }, find a non-empty subset S I such that its elements add up exactly zero. a) Formulate this problem in the states space framework, describing precisely all the elements b) If we wish to find a solution minimizing the sum of the square values of the elements in S, which algorithm guarantees to find it? (Extend the representation by providing the additional information required by the algorithm, if needed) 5. Problem of the towers of Hanoi: The legend says that there is a secret Hindu temple where there are three spikes of platinum. On one of them, disks of gold (all of them of different sizes) are piled with the largest on the bottom and the smallest on top. The monks of this temple have the duty to transfer all disks to the third spike, using the second one as an auxiliary intermediate location, and under the following conditions: Only one disk can be moved at a time A larger disk may never be placed on top of a smaller one, in none of the spikes a) Formulate this problem in the states space framework, describing precisely all the elements b) Define an admissible heuristic that allows to find an optimal solution (considering that the cost of applying movements is always ). c) Which search algorithm should be used? The answer should be justified.. The following graph represents a states space for a problem. Nodes of the graph are the states of the problem, the edges connect the states to their successors, and the number on each edge stands for the cost of going from a state to its successor. The initial state is I, and the only final state is F. We consider the heuristic functions H and H given by the following table: I E A B 5 C 8 5 G 87 F D State H H I 0 A 8 8 B 0 C D E 9 9 F 0 0 G a) Build the search trees generated by the depth-first search and by the optimum search. Indicate in each case the order in which nodes are analyzed, the solutions found and their cost. b) Analogously for the A search algorithm, using the two heuristics H and H previously defined. Are they admissible? The answer should be justified. Note: When calculating successors of a node, consider them in alphabetical order. 5

6 7. The following graph represents a states space for a problem. Nodes of the graph are the states of the problem, the edges connect the states to their successors, and the number on each edge stands for the cost of going from a state to its successor. The initial state of the problem is A, and the final states are H and I. We consider the heuristic function h given by the following table: A B D H 9 C G I 0 5 E F Node Heuristic A 7 B 7 C D 7 E F 9 G H 0 I 0 (a) Build the search trees generated by the best-first search and by the optimum search. (b) Indicate in each case the number of nodes analyzed, the solutions found and their cost. Is any of the solutions found optimal? Which one? Why the other one is non-optimal? (c) Analogously for the A search algorithm. Taking into account the solution found, is h admissible? Make a small modification on the definition of the heuristic such that A finds an optimal solution. Note: When calculating successors of a node, consider them in alphabetical order. In case of nodes with the same value on the OPEN queue, also sort them alphabetically. 8. The following graph represents a states space for a problem. Nodes of the graph are the states of the problem, the edges connect the states to their successors, and the number on each edge stands for the cost of going from a state to its successor. The initial state is A, and the only final state is L. A 5 B C D 7 E F G I J K H 5 L

7 a) Find the shortest path (in number of actions) to go from A to L, by using either Depth-first search or Iterative deepening depth-first search algorithm. Explain your choice. b) Is it possible to find the same shortest solution by using any other searching algorithm? Which one(s)? c) Run the optimal search algorithm. c) Run the A search using the heuristic of the following table: A B C D E F G H I J K L h Does the obtained solution have the minimal cost? How many nodes less have been analyzed (wrt optimal search)? Once the result obtained by A is known, can we claim that the heuristic h is admissible? Note: The evolution of the search trees should be indicated explicitly: order of analysis of the nodes; their evaluation; justifying if some successors are discarded; etc. In case of having nodes with the same value, sort them alphabetically. Analogously with the successors of a node. 9. The following graph represents a states space for a problem. Nodes of the graph are the states of the problem, the edges connect the states to their successors (in both directions), and the number on each edge stands for the cost of going from a state to its successor. The initial state is A, and the only final state is I. E H A B C 5 5 F G I 5 D a) Find the shortest path (in number of actions) to go from A to I, by using some uninformed search algorithm that guarantees to find it. b) Run the A search using the heuristic of the following table: A B C D E F G H I h Note: The evolution of the search trees should be indicated explicitly: order of analysis of the nodes; their evaluation; justifying if some successors are discarded; etc. In case of having nodes with the same value, sort them alphabetically. Analogously with the successors of a node. 7