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Informed Search Reading: Chapter 4.5

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2 Agenda Introduction of heuristic search Greedy search Examples: 8 puzzle, word puzzle A* search Algorithm, admissable heuristics, optimality Heuristics for other problems Homework

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3 8-puzzle URLS http://www.permadi.com/java/puzzle8 http://www.cs.rmit.edu.au/AI- Search/Product http://www.cs.rmit.edu.au/AI- Search/Product

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5 Breadth first Algorithm

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10 Depth-first Algorithm

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19 Heuristics Suppose 8-puzzle off by one Is there a way to choose the best move next? Good news: Yes! We can use domain knowledge or heuristic to choose the best move

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20 012 345 67

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21 Nature of heuristics Domain knowledge: some knowledge about the game, the problem to choose Heuristic: a guess about which is best, not exact Heuristic function, h(n): estimate the distance from current node to goal

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22 Heuristic for the 8-puzzle # tiles out of place (h 1 ) Manhattan distance (h 2 ) Sum of the distance of each tile from its goal position Tiles can only move up or down city blocks

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23 012 345 67 012 345 67 025 317 64 Goal State h 1 =1 h 2 =1 h 1 =5 h 2 =1+1+1+2+2=7

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24 Best first searches A class of search functions Choose the best node to expand next Use an evaluation function for each node Estimate of desirability Implementation: sort fringe, open in order of desirability Today: greedy search, A* search

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25 Greedy search Evaluation function = heuristic function Expand the node that appears to be closest to the goal

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26 Greedy Search OPEN = start node; CLOSED = empty While OPEN is not empty do Remove leftmost state from OPEN, call it X If X = goal state, return success Put X on CLOSED SUCCESSORS = Successor function (X) Remove any successors on OPEN or CLOSED Compute f=heuristic function for each node Put remaining successors on either end of OPEN Sort nodes on OPEN by value of heuristic function End while

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27 8-puzzle URLS http://www.permadi.com/java/puzzle8 http://www.cs.rmit.edu.au/AI- Search/Product http://www.cs.rmit.edu.au/AI- Search/Product

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28 Sodoku

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30 Analysis of Greedy Search Like depth-first Not Optimal Complete if space is finite

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31 A* Search Try to expand node that is on least cost path to goal Evaluation function = f(n) f(n)=g(n)+h(n) h(n) is heuristic function: cost from node to goal g(n) is cost from initial state to node f(n) is the estimated cost of cheapest solution that passes through n If h(n) is an underestimate of true cost to goal A* is complete A* is optimal A* is optimally efficient: no other algorithm using h(n) is guaranteed to expand fewer states

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32 A* Search OPEN = start node; CLOSED = empty While OPEN is not empty do Remove leftmost state from OPEN, call it X If X = goal state, return success Put X on CLOSED SUCCESSORS = Successor function (X) Remove any successors on OPEN or CLOSED Compute f(n)= g(n) + h(n) Put remaining successors on either end of OPEN Sort nodes on OPEN by value of heuristic function End while

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33 8-puzzle URLS http://www.permadi.com/java/puzzle8 http://www.cs.rmit.edu.au/AI- Search/Product http://www.cs.rmit.edu.au/AI- Search/Product

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37 Admissable heuristics A heuristic that never overestimates the cost to the goal h 1 and h 2 for the 8 puzzle are admissable heuristics Consistency: the estimated cost of reaching the goal from n is no greater than the step cost of getting to n plus estimated cost to goal from n h(n) <=c(n,a,n)+h(n)

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38 Which heuristic is better? Better means that fewer nodes will be expanded in searches h 2 dominates h 1 if h 2 >= h 1 for every node n Intuitively, if both are underestimates, then h 2 is more accurate Using a more informed heuristic is guaranteed to expand fewer nodes of the search space Which heuristic is better for the 8-puzzle?

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41 Relaxed Problems Admissable heuristics can be derived from the exact solution cost of a relaxed version of the problem If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h 1 gives the shortest solution If the rules are relaxed so that a tile can move to any adjacent square, then h 2 gives the shortest solution Key: the optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem.

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42 Heuristics for other problems Problems Shortest path from one city to another Touring problem: visit every city exactly once Challenge: Is there an admissable heuristic for sodoku?

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43 End of Class Questions

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44 Homework

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45 Homework

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46 Fill-In Station

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47 Language Models Knowledge about what letters in words are most frequent Based on a sample of language What is the probability that A is the first letter Having seen A, what do we predict is the next letter? Tune our language model to a situation 3 letter words only Medical vocabulary Newspaper text

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48 Formulation Left-right, top-down as path Choose a letter at a time Heuristic: Based on language model Cost of choosing letter A first = 1 – probability of A next Subtract subsequent probabilities Successor function: Choose next best letter, if 3 rd in a row, is it a word? Goal test?

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49 How does heuristic help us get there?

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