Informed Search
Reading: Chapter 4.5
Agenda
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Introduction of heuristic search
Greedy search
Examples: 8 puzzle, word puzzle
A* search
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Algorithm, admissable heuristics,
optimality
Heuristics for other problems
Homework
2
8-puzzle URLS
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http://www.permadi.com/java/puzzle8
http://www.cs.rmit.edu.au/AISearch/Product
3
4
Breadth first Algorithm
5
6
7
8
9
Depth-first Algorithm
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11
12
13
14
15
16
17
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Heuristics
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Suppose 8-puzzle off by one
Is there a way to choose the best
move next?
Good news: Yes!
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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
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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
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# tiles out of place (h1)
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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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2
3
4
5
6
7
2
5
5
3
1
7
7
6
1
2
3
4
h1=1
h2=1
1
0
0
6
0
Goal State
4
h1=5
h2=1+1+1+2+2=7
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Best first searches
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A class of search functions
Choose the “best” node to expand
next
Use an evaluation function for each
node
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Estimate of desirability
Implementation: sort fringe, open
in order of desirability
Today: greedy search, A* search
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Greedy search
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Evaluation function = heuristic
function
Expand the node that appears to be
closest to the goal
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Greedy Search
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OPEN = start node; CLOSED = empty
While OPEN is not empty do
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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
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http://www.permadi.com/java/puzzle8
http://www.cs.rmit.edu.au/AISearch/Product
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Sodoku
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29
Analysis of Greedy Search
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Like depth-first
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Not Optimal
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Complete if space is finite
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A* Search
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Try to expand node that is on least cost path to
goal
Evaluation function = f(n)
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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
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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
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OPEN = start node; CLOSED = empty
While OPEN is not empty do
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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
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http://www.permadi.com/java/puzzle8
http://www.cs.rmit.edu.au/AISearch/Product
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34
35
36
Admissable heuristics
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A heuristic that never overestimates the
cost to the goal
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h1 and h2 for the 8 puzzle are admissable
heuristics
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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’
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h(n) <=c(n,a,n’)+h(n’)
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Which heuristic is better?
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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
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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
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Shortest path from one city to another
Touring problem: visit every city exactly
once
Challenge: Is there an admissable
heuristic for sodoku?
42
End of Class Questions
43
Homework
44
Homework
45
Fill-In Station
46
Language Models
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Knowledge about what letters in words
are most frequent
Based on a sample of language
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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
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3 letter words only
Medical vocabulary
Newspaper text
47
Formulation
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Left-right, top-down as path
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Choose a letter at a time
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Heuristic: Based on language model
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Cost of choosing letter A first = 1 – probability of A
next
Subtract subsequent probabilities
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Successor function: Choose next best letter, if 3rd
in a row, is it a word?
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Goal test?
48
How does heuristic help us get there?
49