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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 1 After our “Haskell in a Nutshell” excursion, let us move on to Search in State Spaces

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 2 Search in State Spaces Many problems in Artificial Intelligence can be mapped onto searches in particular state spaces.Many problems in Artificial Intelligence can be mapped onto searches in particular state spaces. This concept is especially useful if the system (our “world”) can be defined as having a finite number of states, including an initial state and one or more goal states.This concept is especially useful if the system (our “world”) can be defined as having a finite number of states, including an initial state and one or more goal states. Optimally, there are a finite number of actions that we can take, and there are well-defined state transitions that only depend on our current state and current action.Optimally, there are a finite number of actions that we can take, and there are well-defined state transitions that only depend on our current state and current action.

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 3 Search in State Spaces To some extent, it is also possible to account for state changes that the algorithm itself does not initiate.To some extent, it is also possible to account for state changes that the algorithm itself does not initiate. For example, a chess playing program can consider its opponent’s future moves.For example, a chess playing program can consider its opponent’s future moves. However, it is necessary that the set of such actions and their consequences are well-defined.However, it is necessary that the set of such actions and their consequences are well-defined. While computers are able to play chess at a very high level, it is impossible these days to build a robot that, for instance, is capable of reliably carrying out everyday tasks such as going to a supermarket to buy groceries.While computers are able to play chess at a very high level, it is impossible these days to build a robot that, for instance, is capable of reliably carrying out everyday tasks such as going to a supermarket to buy groceries.

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 4 Search in State Spaces Let us consider an easy task in a very simple world with our robot being the only actor in it: The world contains a floor and three toy blocks labeled A, B, and C. The world contains a floor and three toy blocks labeled A, B, and C. The robot can move a block (with no other block on top of it) onto the floor or on top of another block. The robot can move a block (with no other block on top of it) onto the floor or on top of another block. These actions are modeled by instances of a schema, move(x, y). These actions are modeled by instances of a schema, move(x, y). Instances of the schema are called operators. Instances of the schema are called operators.

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 5 Search in State Spaces The robot’s task is to stack the toy blocks so that A is on top of B, B is on top of C, and C is on the floor.The robot’s task is to stack the toy blocks so that A is on top of B, B is on top of C, and C is on the floor. For us it is clear what steps have to be taken to solve the task.For us it is clear what steps have to be taken to solve the task. The robot has to use its world model to find a solution.The robot has to use its world model to find a solution. Let us take a look at the effects that the robot’s actions exert on its world.Let us take a look at the effects that the robot’s actions exert on its world.

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 6 Search in State Spaces Effects of moving a block (illustration and list-structure iconic model notation)

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 7 Search in State Spaces In order to solve the task efficiently, the robot should “look ahead”, that is, simulate possible actions and their outcomes.In order to solve the task efficiently, the robot should “look ahead”, that is, simulate possible actions and their outcomes. Then, the robot can carry out a sequence of actions that, according to the robot’s prediction, solves the problem.Then, the robot can carry out a sequence of actions that, according to the robot’s prediction, solves the problem. A useful structure for such a simulation of alternative sequences of action is a directed graph.A useful structure for such a simulation of alternative sequences of action is a directed graph. Such a graph is called a state-space graph.Such a graph is called a state-space graph.

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 8 State-Space Graphs

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 9 State-Space Graphs To solve a particular problem, the robot has to find a path in the graph from a start node (representing the initial state) to a goal node (representing a goal state).To solve a particular problem, the robot has to find a path in the graph from a start node (representing the initial state) to a goal node (representing a goal state). The resulting path indicates a sequence of actions that solves the problem.The resulting path indicates a sequence of actions that solves the problem. The sequence of operators along a path to a goal is called a plan.The sequence of operators along a path to a goal is called a plan. Searching for such a sequence is called planning.Searching for such a sequence is called planning. Predicting a sequence of world states from a sequence of actions is called projecting.Predicting a sequence of world states from a sequence of actions is called projecting.

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 10 State-Space Graphs There are various methods for searching state spaces.There are various methods for searching state spaces. One possibility is breadth-first search:One possibility is breadth-first search: –Mark the start node with a 0. –Mark all adjacent nodes with 1. –Mark all unmarked nodes adjacent to nodes with a 1 with the number 2, and so on, until you arrive at a goal node. –Finally, trace a path back from the goal to the start along a sequence of decreasing numbers.

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 11 Decision Trees A decision tree is a special case of a state-space graph. It is a rooted tree in which each internal node corresponds to a decision, with a subtree at these nodes for each possible outcome of the decision. Decision trees can be used to model problems in which a series of decisions leads to a solution. The possible solutions of the problem correspond to the paths from the root to the leaves of the decision tree.

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 12 Decision Trees Example: The n-queens problem How can we place n queens on an n n chessboard so that no two queens can capture each other? A queen can move any number of squares horizontally, vertically, and diagonally. Here, the possible target squares of the queen Q are marked with an x.

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 13 Decision Trees Let us consider the 4-queens problem. Question: How many possible configurations of 4 4 chessboards containing 4 queens are there? Answer: There are 16!/(12! 4!) = (13 14 15 16)/(2 3 4) = 13 7 5 4 = 1820 possible configurations. Shall we simply try them out one by one until we encounter a solution? No, it is generally useful to think about a search problem more carefully and discover constraints on the problem’s solutions. Such constraints can dramatically reduce the size of the relevant state space.

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 14 Decision Trees Obviously, in any solution of the n-queens problem, there must be exactly one queen in each column of the board. Otherwise, the two queens in the same column could capture each other. Therefore, we can describe the solution of this problem as a sequence of n decisions: Decision 1: Place a queen in the first column. Decision 2: Place a queen in the second column. … Decision n: Place a queen in the n-th column. This way, “only” 256 configurations need to be tested.

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 15 Backtracking in Decision Trees There are problems that require us to perform an exhaustive search of all possible sequences of decisions in order to find the solution.There are problems that require us to perform an exhaustive search of all possible sequences of decisions in order to find the solution. We can solve such problems by constructing the complete decision tree and then find a path from its root to a leave that corresponds to a solution of the problem (breadth-first search often requires the construction of an almost complete decision tree).We can solve such problems by constructing the complete decision tree and then find a path from its root to a leave that corresponds to a solution of the problem (breadth-first search often requires the construction of an almost complete decision tree). In many cases, the efficiency of this procedure can be dramatically increased by a technique called backtracking (depth-first search with “sanity checks”).In many cases, the efficiency of this procedure can be dramatically increased by a technique called backtracking (depth-first search with “sanity checks”).

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 16 Backtracking in Decision Trees Idea: Start at the root of the decision tree and move downwards, that is, make a sequence of decisions, until you either reach a solution or you enter a situation from where no solution can be reached by any further sequence of decisions. In the latter case, backtrack to the parent of the current node and take a different path downwards from there. If all paths from this node have already been explored, backtrack to its parent. Continue this procedure until you find a solution or establish that no solution exists (there are no more paths to try out).

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 17 Backtracking in Decision Trees Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q place 1 st queen place 2 nd queen place 3 rd queen place 4 th queen empty board

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 18 Breadth-First vs. Depth-First Uninformed breadth-first search: Requires the construction and storage of almost the complete search tree.Requires the construction and storage of almost the complete search tree. Space complexity for search depth n is O(e n ). Is guaranteed to find the shortest path to a solution.Is guaranteed to find the shortest path to a solution. Uninformed depth-first search: Requires the storage of only the current path and the branches from this path that were already visited.Requires the storage of only the current path and the branches from this path that were already visited. Space complexity for search depth n is O(n). May search unnecessarily deep for a shallow goal.May search unnecessarily deep for a shallow goal.

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 19 Iterative Deepening Iterative deepening is an interesting combination of breadth-first and depth-first strategies: Space complexity for search depth n is O(n). Space complexity for search depth n is O(n). Is guaranteed to find the shortest path to a solution without searching unnecessarily deep. Is guaranteed to find the shortest path to a solution without searching unnecessarily deep. How does it work? The idea is to successively apply depth-first searches with increasing depth bounds (maximum search depth).

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 20 Iterative Deepening maximum search depth = 0 (only root is tested)

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 21 Iterative Deepening maximum search depth = 1

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 22 Iterative Deepening maximum search depth = 2

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 23 Iterative Deepening maximum search depth = 3

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 24 Iterative Deepening maximum search depth = 4

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 25 Iterative Deepening But it seems that the time complexity of iterative deepening is much higher than that of breadth-first search! Well, if we have a branching factor b and the shallowest goal at depth d, then the worst-case number of nodes to be expanded by breadth-first is:

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 26 Iterative Deepening In order to determine the number of nodes expanded by iterative deepening, we have to look at depth-first search. What is the worst-case number of nodes expanded by depth-first search for a branching factor b and a maximum search level j ?

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 27 Iterative Deepening Therefore, the worst-case number of nodes expanded by iterative deepening from depth 0 to depth d is:

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September 26, 2012Introduction to Artificial Intelligence Lecture 7: Search in State Spaces I 28 Iterative Deepening Let us now compare the numbers for breadth-first search and iterative deepening: For large d, you see that N id /N bf approaches b/(b – 1), which in turn approaches 1 for large b. So for big trees (large b and d), iterative deepening does not expand many more nodes than does breadth- first search (about 11% for b = 10 and large d).

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