Informed Search

Reading: Chapter 4.5

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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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8-puzzle URLS

http://www.permadi.com/java/puzzle8

http://www.cs.rmit.edu.au/AI-Search/Product

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Breadth first Algorithm

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7

8

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Depth-first Algorithm

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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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0 1 2

3 4 5

6 7

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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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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

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0 1 2

3 4 5

6 7

0 1 2

3 4 5

6 7

0 2 5

3 1 7

6 4

Goal State

h1=1h2=1

h1=5h2=1+1+1+2+2=7

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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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Greedy search

Evaluation function = heuristic function

Expand the node that appears to be closest to the goal

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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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8-puzzle URLS

http://www.permadi.com/java/puzzle8

http://www.cs.rmit.edu.au/AI-Search/Product

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Sodoku

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Analysis of Greedy Search

Like depth-first

Not Optimal

Complete if space is finite

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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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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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8-puzzle URLS

http://www.permadi.com/java/puzzle8

http://www.cs.rmit.edu.au/AI-Search/Product

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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’)

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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?

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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.

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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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End of Class Questions

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Homework

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Homework

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Fill-In Station

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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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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?

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How does heuristic help us get there?