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Informed Search

Informed Search. Reading: Chapter 4.5 HW #1 out today, due Sept 26th. Pending Questions. The state space for the word problem: Isn’t it all permutations of the letters, not the power set? Yes. Agenda. Introduction of heuristic search Greedy search Examples: 8 puzzle, word puzzle

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Informed Search

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  1. Informed Search Reading: Chapter 4.5 HW #1 out today, due Sept 26th

  2. Pending Questions • The state space for the word problem: Isn’t it all permutations of the letters, not the power set? • Yes

  3. Agenda • Introduction of heuristic search • Greedy search • Examples: 8 puzzle, word puzzle • A* search • Algorithm, admissable heuristics, optimality • Heuristics for other problems • Beginning of online search if time

  4. 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

  5. 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

  6. Heuristic for the 8-puzzle • # tiles out of place (h1) • Manhattan distance (h2) • Sum of the distance of each tile from its goal position • Tiles can only move up or down  city blocks

  7. Goal State h1=1 h2=1 h1=5 h2=1+1+1+2+2=7

  8. 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

  9. Greedy search • Evaluation function = heuristic function • Expand the node that appears to be closest to the goal

  10. 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

  11. Fill-In Station

  12. 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

  13. 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 3rd in a row, is it a word? • Goal test?

  14. Solution to 2nd puzzle

  15. How does heuristic help us get there?

  16. Analysis of Greedy Search • Like depth-first • Not Optimal • Complete if space is finite

  17. 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

  18. 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

  19. Admissable heuristics • A heuristic that never overestimates the cost to the goal • h1 and h2 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’)

  20. Which heuristic is better? • Better means that fewer nodes will be expanded in searches • h2 dominates h1 if h2 >= h1 for every node n • Intuitively, if both are underestimates, then h2 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?

  21. 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 h1 gives the shortest solution • If the rules are relaxed so that a tile can move to any adjacent square, then h2 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.

  22. Heuristics for other problems • Shortest path from one city to another • Straight line distance • Touring problem: visit every city exactly once • Challenge: What is an admissable heuristic for the word problem?

  23. End of Class Questions

  24. Online Search • Agent operates by interleaving computation and action • No time for thinking • The agent only knows • Actions (s) • The step-cost function c(s,a,s’) • Goal-text (s) • Cannot access the successors of a state without trying all actions

  25. Assumptions • Agent recognizes a state it has seen before • Actions are deterministic • Admissable heuristics • Competitive ratio: Compare cost that agent actually travels with cost of the actual shortest path

  26. What properties of search are desirable? • Will A* work? • Expand nodes in a local order • Depth first • Variant of greedy search • Difference from offline search: • Agent must physically backtrack • Record states to which agent can backtrack and has not yet explored

  27. Depth-first • 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 • Put remaining successors on left end of OPEN • End while

  28. Online DFS - setup • Inputs: s’, a percept that identifies the current state Static: • result, a table indexed by action and state, initially empty • unexplored: a table that lists, for each visited state, the actions not yet tried • unbacktracked: a table that lists, for each visited state, the backtracks not yet tried • s,a: the previous state and action, initially null

  29. Online DFS – the algorithm • If Goal-test(s’) then return stop • If s’ is a new state then unexplored[s’] actions(s’) • If s is not null then do • result[a,s]s’ • Add s to the front of unbacktracked[s’] • If unexplored[s’] is empty • Then return stop • Else a action b such that result[b,s’]=pop(unbacktracked[s’]) • Else apop(unexplored[s’]) • s s’ • Return a

  30. Homework

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