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Mahgul Gulzai
Moomal Umer
Rabail Hafeez
 Introduction
 Definition of heuristic search
 Algorithms of heuristic search
 Best first search
 Hill climbing strategy
 Implementing heuristic evaluation functions
 Admissibility, Monotonicity and Informedness
 Using heuristic in games
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 Human generally consider number of alternative
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strategies on their way to solving a problem .
To obtain the best possible strategy, humans use
search.
For example : A chess player consider number of
possible moves, A Doctor examine several possible
diagnoses .
Human Problem solving seems to be based on
judgmental rules that guide our search to those
portion of state Space that seems some how promising.
These rules are known as “Heuristics”
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 George Polya defines Heuristics as, “The study of
methods and rules of discovery and invention”.
 A heuristic is a method that
 might not always find the best solution .
 but is guaranteed to find a good solution in
reasonable time.
 By sacrificing completeness it increases efficiency.
 Useful in solving tough problems which
 could not be solved any other way.
 solutions take an infinite time or very long time to
compute.
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Heuristic Search is used in AI in two situations:
1. When a problem doesn’t have an exact solution.
2. There is an exact solution but the computational cost
of finding it exceeds the limit.
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 Consider the game of tic-tac-toe.
 Even if we use symmetry to reduce the search space of
redundant moves, the number of possible paths
through the search space is something like 12 x 7! =
60480.
 That is a measure of the amount of work that would
have to be done by a brute-force search.
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First three levels of the tic-tac-toe state space
reduced by symmetry
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The “most wins” heuristic applied to the first
children in tic-tac-toe
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Heuristically reduced state space for tic-tac-toe
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 Using this rule, we can see that a corner square has
heuristic value of 3, a side square has a heuristic value of 2,
but the centre square has a heuristic value of 4.
 So we can prune the left and right branches of the search
tree.
 This removes 2/3 of the search space on the first move.
 If we apply the heuristic at each level of the search, we will
remove most of the states from consideration thereby
greatly improving the efficiency of the search.
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 Heuristic search is implemented in two parts:
 The heuristic measure.
 The search algorithm.
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 Use heuristic to move only to states that are better than
the current state.
 Always move to better state when possible.
 The process ends when all operators have been applied
and none of the resulting states are better than the
current state.
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• Will terminate when at local optimum.
• The order of application of operators can make a big
difference.
• Can’t see past a single move in the state space
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The local maximum problem for hill-climbing
with 3-level look ahead
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 Also heuristic search – use heuristic (evaluation)
function to select the best state to explore
 Can be implemented with a priority queue
 Breadth-first implemented with a queue
 Depth-first implemented with a stack
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The bestfirst search
algorithm
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Heuristic search of a hypothetical state
space
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A trace of the execution of best-firstsearch
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Heuristic search of a hypothetical state space
with open and closed states highlighted
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 Heuristics can be evaluated in different ways
 8-puzzle problem
 Heuristic 1: count the tiles out of places compared with
the goal state
 Heuristic 2: sum all the distances by which the tiles are
out of pace, one for each square a tile must be moved to
reach its position in the goal state
 Heuristic 3: multiply a small number (say, 2) times each
direct tile reversal (where two adjacent tiles must be
exchanged to be in the order of the goal)
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The start state, first moves, and goal state for
an example-8 puzzle
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Three heuristics applied to states in the 8puzzle
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 Use the limited information available in a single state to make
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intelligent choices
Empirical, judgment, and intuition
Must be its actual performance on problem instances
The solution path consists of two parts: from the starting state
to the current state, and from the current state to the goal state
The first part can be evaluated using the known information
The second part must be estimated using unknown
information
The total evaluation can be
f(n) = g(n) + h(n)
g(n) – from the starting state to the current state n
h(n) – from the current state n to the goal state
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The heuristic f applied to states in the 8-puzzle
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State space
generated in
heuristic
search of
the 8-puzzle
graph
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The successive stages of open and closed that generate
the graph are:
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Open and closed as they appear after the 3rd iteration of
heuristic search
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 evaluation function for the states in a search space, you
are interested in two things:
g(n): How far is state n from the start state?
h(n): How far is state n from a goal state?
 Evaluation function. This gives us the following
evaluation function: f(n) = g(n) + h(n)
where g(n) measures the actual length of the path
from the start state to the state n, and h(n) is a
heuristic estimate of the distance from a state n to a
goal state.
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 Expert System employ confidence measures to select
the conclusions with the highest likelihood of the
success through heuristics implementation.
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 A best-first search algorithm guarantee to find a best
path, if exists, if the algorithm is admissible.
 A best-first search algorithm is admissible if its
heuristic function h is monotone.
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Admissibility and Algorithm A*
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Monotonicity and Informedness
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Comparison of
state space
searched using
heuristic search
with space
searched by
breadth-first
search. The
proportion of the
graph searched
heuristically is
shaded. The
optimal search
selection is in
bold. Heuristic
used is f(n) = g(n)
+ h(n) where h(n)
is tiles out of
place.
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 Games
 Two players attempting to win
 Two opponents are referred to as MAX and MIN
 A variant of game nim
 A number of tokens on a table between the 2 opponents
 Each player divides a pile of tokens into two nonempty
piles of different sizes
 The player who cannot make division losses
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Exhaustive Search
State space for a
variant of nim.
Each state
partitions the
seven matches
into one or more
piles
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 Principles
 MAX tries to win by maximizing her score, moves to a state that is
best for MAX
 MIN, the opponent, tries to minimize the MAX’s score, moves to a
state that is worst for MAX
 Both share the same information
 MIN moves first
 The terminating state that MAX wins is scored 1, otherwise 0
 Other states are valued by propagating the value of terminating
states
 Value propagating rules
 If the parent state is a MAX node, it is given the maximum value
among its children
 If the parent state is a MIN state, it is given the minimum value of
its children
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Exhaustive
minimax for
the game of
nim. Bold
lines indicate
forced win
for MAX.
Each node is
marked with
its derived
value (0 or 1)
under
minimax.
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 If cannot expand the state space to terminating
(leaf) nodes (explosive), can use the fixed ply
depth
 Search to a predefined number, n, of levels from
the starting state, n-ply look-ahead
 The problem is how to value the nodes at the
predefined level – heuristics
 Propagating values is similar
 Maximum children for MAX nodes
 Minimum children for MIN nodes
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Minimax to a hypothetical state space. Leaf
states show heuristic values; internal states
show backed-up values.
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Heuristic measuring conflict applied to states of
tic-tac-toe
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Two-ply minimax applied to the opening move of
tic-tac-toe
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Two ply minimax, and one of two possible MAX
second moves
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Two-ply minimax applied to X’s move near the end of
the game
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 Alpha-beta pruning to improve search efficiency
 Proceeds in a depth-first fashion and creates two values alpha
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and beta during the search
Alpha associated with MAX nodes, and never decreases
Beta associated with MIN nodes, never increases
To begin, descend to full ply depth in a depth-first search, and
apply heuristic evaluation to a state and all its siblings. The
value propagation is the same as minimax procedure
Next, descend to other grandchildren and terminate
exploration if any of their values is >= this beta value
Terminating criteria
 Below any MIN node having beta <= alpha of any of its MAX ancestors
 Below any MAX node having alpha >= beta of any of its MIN ancestors
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