Transcript CS 415 – A.I.

CS 415 – A.I.
Slide Set 6
Chapter 4 – Heuristic Search
 Heuristic – the study of the methods and rules
of discovery and invention
 State Space Heuristics –

Formalized as rules for choosing those branches
in a state space that are most likely to lead to an
acceptable problem solution
 Apply Heuristics When:
1.
2.
A problem is ambiguous and may not have an EXACT solution
The computational cost of finding an exact solution is prohibitive
Simplifying Tic-Tac-Toe
 Brute Force – 9! total states

First use symmetry to reduce the number of
states (see next slide)


7x12 states
Simple Heuristic



Put X where it has the most winning possibilities
See Figure 4.2 and Figure 4.3
This “prunes” the search space


In the first level, pick one of three, ignore the subtrees for
the other two
Approx. 25 total states to consider
Fig 4.1 First three levels of the tic-tac-toe state space reduced by symmetry
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Fig 4.2 The “most wins” heuristic applied to the first children in tic-tac-toe.
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Fig 4.3 Heuristically reduced state space for tic-tac-toe.
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Hill-Climbing and Dynamic Programming



Hill-Climbing
 Expand the current state and evaluate its children
 Pick the best child
Mountain climbing
 Go as steep as you can for as long as you can.
Problem:
 Keeps no backtracking info
 Becomes stuck at “local maxima”
 Example:
 8-square problem
Dynamic Programming
 Dynamic Programming


Don’t solve subproblems multiple times
Instead, keep track of the solutions to these
subproblems

Memoizing – subproblem solution caching
 Two Examples
Finding an optimal global alignment
2. Finding the minimum edit distance between two
strings
1.
Optimal Global Alignment
 Small Example

String #1


BAADDCABDDA
String #2

BBADCBA
 Rules


Cannot change order of respective elements
Can have spaces between elements
 Possible solutions

How do we figure out optimal solution?
Fig 4.5 The initialization stage and first step in completing the
array for character alignment using dynamic programming.
Let’s create a matrix
•The value of each element reflects
the global alignment success to
that point in the matching process
•Our “cost” scheme
•1 – if an element has to be shifted
along the shorter string for better
alignment
•Cost recorded in col
•1 – if a new character is inserted
•Cost recorded in row
•2 – different characters, shift and
insert cost 2
•0 – if elements are identical
•
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The forward stage

Fill the array from the upper left corner
 The value of (x,y) is a function (min cost) of either
 (x-1,y), (x-1,y-1), or (x, y-1)
 If there is a match
 Add 0 to (x-1,y-1)
 If there is no match
 Add 2 to (x-1,y-1)
 If we shift
 Add 1 to the previous column
 If we insert a character
 Add 1 to the previous row
Fig 4.6 The completed array reflecting the maximum alignment information
for the strings.
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The backward stage
 Once the matrix is filled
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From the best alignment count, we produce a
specific alignment of characters


Begin at the lower right-hand corner
Move back through the matrix

Each step, select one of the immediate state’s predecessors
 (previous diagonal, row, or column)
 Choose the minimum
Fig 4.7 A completed backward component of the dynamic programming
example giving one (of several possible) string alignments.
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Example #2 – Min Edit Distance
 Spell Checker


What words from our dictionary best
approximate a word we do not recognize
(misspelled word)
We need to know the “distance” between two
words
 Minimum edit distance

The number of insertions, deletions and
replacements to turn the source word into the
target word
 intention – source word
 execution – target word
 Our “cost”


1 for a character insertion or deletion
2 for a replacement (deletion + insertion)
Fig 4.8 Initialization of minimum edit difference matrix between intention
and execution (adapted from Jurafsky and Martin, 2000).
•
Beginning state
• a sequence of insertions is needed
beginning at null to make either
word look like the other
• (2,2) is cost 2 because a
replacement is required to make an i
be an e
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Complete array of minimum edit difference between intention and execution (adapted
from Jurafsky and Martin, 2000) (of several possible) string alignments.
The value at each (x,y) is the cost of minimum
editing to that point, plus the minimum cost of
either an insertion, deletion, or replacement
•
•
In other words, cost of (x,y) is the minimum of
•Cost of (x-1,y) + insertion
•Cost of (x-1,y-1) + replacement
•Cost of (x, y-1) + deletion
Finally, the backward part will select a list of
optimal changes
•Bold text near the diagonal
•Not strictly required for spell-check case
•
Intention
ntention delete I, cost 1
etention replace n with e, cost 2
exention replace t with x, cost 2
exenution insert u, cost 1
execution replace n with c, cost 2
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Why Dynamic Programming?
 What’s the cost of DP?

n2 where n is the size of the longest string

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What is the cost of exhaustive search?

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n3 in the worst case if we need to consider other
row/col values to determine the current state
Between 2n and 3n
Useful heuristics for DP
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
Usually the solution lies near the horizontal
Prune the considerations if the cost passes a
predetermined threshold
Best-First Search Algorithm


Need to have a good evaluation function
 Avoid
 Local maxima
 Dead-ends
 Anomalies in search space
Return to general form of state space search algorithm
 Lists
 open – states we haven’t explored
 closed – states we have explored
 Now, order states in open according to some heuristic

Thus, each loop considers the most “promising” state first
Keeps track of ancestor
information to ensure
that we can check to see
if state already reached
and so we can return the
chosen path at the end
Can make use of a
“priority queue”
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Fig 4.10 Heuristic search of a hypothetical state space.
Notice how states are
selected for next
examination.
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A trace of the execution of best_first_search for Figure 4.4
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Implementing Heuristic Evaluation
Functions
 Evaluate the performance of heuristics for
solving 8-puzzle.
 Ideas?
Count number of tiles out of place
1.

Doesn’t consider how far out of place they are
Get a total distance out of place
2.

But what if tiles are right next to each other but out of
order

Very difficult to switch tiles
Examining these Heuristics
Fig 4.12 The start state, first moves, and goal state for an example-8 puzzle.
Which seems closest?
What the Heuristics Give Us
Fig 4.14 Three heuristics applied to states in the 8-puzzle.
What if two states have (nearly) the same
heuristic evaluation?
The heuristic
f applied to
states in the
8-puzzle.
How Good is a Heuristic?
 Evaluated along several dimensions
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Goals?
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Guaranteed Solution?
Shortest Path?
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Better heuristics
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Shortest path heuristics are said to be admissible
Concept of informedness
When a state is found is there a guarantee that a
better state won’t be found later in the search?

Concept of monotonicity
Admissibility Measures
 Admissible – guaranteed to find a minimal
path to a solution if it exists

Example: BFS
 Use the function f(n) = g(n) + h(n)



f(n) = evaluation function
g(n) = actual length from any state n to the start
state
h(n) = heuristic estimate from any state n to the
goal
 Using f(n) to discuss admissibility
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f(n) estimates the total cost of the path from the
start state through n to the goal state
 Define f*(n) = g*(n) + h*(n)


All the same, but now only dealing with the
shortest path from start to n and from n to goal
f*(n) is often called an oracle


Often do not exist for most real problems
Still we want f to be a close estimate to f*
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Examples
 BFS is an A* algorithm in which

f(n) = g(n) + 0

The decision for considering a state is based solely on
the distance from the start state
 Several Heuristics from 8-Puzzle are A*

Can’t always compute value of h*(n), but we can
bound a heuristic cost above by the actual
shortest-path cost
A* Examples for 8-puzzle

Counting the number of tiles out of place is always less
than or equal to the number of moves to move them to
their goal

Thus: heuristic is admissible
 The sum of the direct distances of tiles out of
place is also less than or equal to the
minimum actual path

Thus: heuristic is admissible
Monotonicity
 The definition of A* does not require that g(n)
= g*(n)

So, A* algorithms may initially reach non-goal
states along sub-optimal paths before eventually
finding an optimal path
 Monotonicity – property that guarantees that
the heuristic is “locally optimal”

Consistently follows only the optimal path
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 Monotonicity is possible when:
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Search space is everywhere locally consistent
with the heuristic employed
The difference between a heuristic measure of a
state and any of its successors is bound by the
actual cost of moving from predecessor to
successor states
 As the path is followed when using a
monotone heuristic

value of f is monotonically nondecreasing (hence
the name)
Monotonicity and Admissibility
 All monotone heuristics are admissible

In fact, monotonicity is a refinement of
admissibility
 Not all admissible heuristics are monotone
Relatively Better Heuristics
 Informedness
Example
BFS is equivalent to A* with heuristic h1 such that
h1(x)=0 for all states x
 This is always less than h*(x)
 Lets call the number of tiles out of place, h2
 This is also less than h*
 But, we have h1<=h2<=h*
 Thus, h2 is “more informed” than h1
 Additionally, we can argue that calculation of direct
distance of out of place tiles is more informed than
h2
