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Informed Search. Reading: Chapter 4.5. Agenda. Introduction of heuristic search Greedy search Examples: 8 puzzle, word puzzle A* search Algorithm, admissable heuristics, optimality Heuristics for other problems Homework. 8-puzzle URLS. http://www.permadi.com/java/puzzle8

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informed search

Informed Search

Reading: Chapter 4.5

agenda
Agenda
  • Introduction of heuristic search
  • Greedy search
  • Examples: 8 puzzle, word puzzle
  • A* search
    • Algorithm, admissable heuristics, optimality
  • Heuristics for other problems
  • Homework
8 puzzle urls
8-puzzle URLS
  • http://www.permadi.com/java/puzzle8
  • http://www.cs.rmit.edu.au/AI-Search/Product
heuristics
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
nature of heuristics
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
heuristic for the 8 puzzle
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
slide23

Goal State

h1=5

h2=1+1+1+2+2=7

h1=1

h2=1

best first searches
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
greedy search
Greedy search
  • Evaluation function = heuristic function
  • Expand the node that appears to be closest to the goal
greedy search26
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
8 puzzle urls27
8-puzzle URLS
  • http://www.permadi.com/java/puzzle8
  • http://www.cs.rmit.edu.au/AI-Search/Product
analysis of greedy search
Analysis of Greedy Search
  • Like depth-first
  • Not Optimal
  • Complete if space is finite
a search
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
a search32
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
8 puzzle urls33
8-puzzle URLS
  • http://www.permadi.com/java/puzzle8
  • http://www.cs.rmit.edu.au/AI-Search/Product
admissable heuristics
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’)
which heuristic is better
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?
relaxed problems
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.
heuristics for other problems
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?
language models
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
formulation
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?