On National Teacher Day, meet the 2024-25 Kenan Fellows
Heuristics Search 3249989_slideplayer.ppt
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Ch 4. Heuristic Search
4.0 Introduction(Heuristic)
4.1 An Algorithm for Heuristic Search
4.1.1 Implementing Best-First Search
4.1.2 Implementing Heuristic Evaluation Functions
4.1.3 Heuristic Search and Expert Systems
4.3 Using Heuristics in Games
4.3.1 Minimax Procedure
4.3.2 Minimaxing to Fixed Ply Depth
4.3.3 Alpha-Beta Procedure
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4.0 Heuristic (1)
Definition: the study of the methods and rules of
discovery and invention.
Heuristics are formalized as rules for choosing
those branches in a state space that are most
likely to lead to an acceptable problem solution.
Heuristics are employed in two cases.
A problem may not have an exact solution because of its
inherent ambiguities
e.g. medical diagnosis, vision
A problem may have an exact solution, but the
computational cost of finding it may be prohibitive.
e.g. chess
Heuristics attack this complexity by guiding the search
along the most promising path.
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4.0 Heuristic (2)
Heuristics are fallible
A heuristic is only an informed guess of the next step to be
taken in solving a problem.
Because heuristics use limited information, a heuristic can
lead a search algorithm to a suboptimal solution or fail to find
any solution at all.
Application of AI using Heuristics
Game playing and theorem proving
not feasible to examine every inference that can be made
in a mathematics domain or every possible move that can
be made on a chessboard.
Expert Systems
Expert system designers extract and formalize the
heuristics which human expert uses to solve problems
efficiently.
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4.0 Heuristic (3)
A Heuristic for the Tic-Tac-Toe Game
Exhaustive Search: 9!
Symmetry reduction can decrease the search space a little:
12 * 7! (Fig. 4.1, p125, tp5)
There are nine but really three initial moves: center, corner,
center of the grid.
Symmetry reductions on the second level further reduce the
number of paths.
A simple heuristic (Fig. 4.2, p126, tp6) can eliminate two-
thirds of the search space with the first move. (Fig. 4.3,
p126)
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A Cryptarithmetic Problem
A useful heuristic can
help to select the best
guess to try first.
If there is a letter that
participate in many
constraints, then it is a
good idea to prefer it to a
letter that participates in
a few.
Initial State
M=1, S=8 or 9
O=0 or 1 O=0
N=E or E+1 N=E+1
C2=1, N+R > 8, E<>9
N=3, R=8 or 9
2+D=Y or 2+D=10+Y
2+D=Y
N+R=10+E
R=9, S=8
2+D=10+Y
D=8+Y
D=8 or 9
Y=1
Y=0
Conflict
Conflict
E=2
C1=0 C1=1
SEND
+ MORE
MONEY
D=8 D=9
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Hill Climbing Strategy(1)
Simplest way of implementing heuristic search
Expand the current state and evaluate the children and
select the best child for further expansion. Halt the search
when it reaches a state that is better than any of its children
(i.e. there is not a child who is better than the current state).
Blind mountain climber
go uphill along the steepest possible path until we can go no
farther.
Because it keeps no history, the algorithm cannot recover from
failures.
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Major Problem: tendency to become stuck at
local maxima
If the algorithm reach a local maximum, the algorithm fails to
find a solution.
An example of local maxima in 8-puzzle
In order to move a particular tile to its destination, other
tiles that are already in goal position have to be moved.
This move may temporarily worsen the board state.
If the evaluation function is sufficiently
informative to avoid local maxima and infinite
path, hill climbing can be used effectively.
Hill Climbing Strategy(2)
G
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Blocks World Problem(1)
A
B
C
D
E
F
G
H
initial state
A
B
C
D
E
F
G
H
goal state
A B
C
D
E
F
G
H
state 1
A
B
C
D
E
F
G
H
state 2-(a)
A B
C
D
E
F
G
H
state 2-(b)
A B
C
D
E
F
G
H
state 2-(c)
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Blocks World Problem(2)
Local: Add one point for every block that is resting on the
thing it is supposed to be resting on. Subtract one point for
every block that is sitting on the wrong thing.
initial state
(-1) + 1 + 1 + 1 + 1 + 1 + 1 + (-1) = 4
goal state
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8
state 1
1 + 1 + 1 + 1 + 1 + 1 + (-1) = 5 + 1 6
state 2-(a)
(-1) + 1 + 1 + 1 + 1 + 1 + 1 + (-1) = 4
state 2-(b)
(-1) + 1 = 0
1 + 1 + 1 + 1 + 1 + (-1) = 4 4
state 2-(c)
1, -1, 1 + 1 + 1 + 1 + 1 + (-1) = 4 4
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Blocks World Problem (3)
Global: For each block that has the correct support (i.e. the
complete structure underneath it is exactly as it should be),
add one point for every block in the support structure.
For each block that has an incorrect support structure,
subtract one point for every block in the existing support
structure.
initial state
(-7) + (-6) + (-5) + (-4) + (-3) + (-2) + (-1) = -28
goal state
7 + 6 + 5 + 4 + 3 + 2 + 1 = 28
state 1
(-6) + (-5) + (-4) + (-3) + (-2) + (-1) = -21
state 2-(a)
(-7) + (-6) + (-5) + (-4) + (-3) + (-2) + (-1) = -28
state 2-(b)
(-5) + (-4) + (-3) + (-2) + (-1) = -15
(-1) + 0 = -1 -16
state 2-(c)
(-5) + (-4) + (-3) + (-2) + (-1) = -15
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4.1.1 Best-First Search (1)
Like the depth-first and breadth-first search, best-
first search uses two-lists.
OPEN: to keep track of the current fringe of the search.
CLOSED: to record states already visited.
Order the states on OPEN according to some
heuristic estimate of their closeness to a goal.
Each iteration of the loop considers the most promising state
on the OPEN list.
Procedure best_first_search (p. 128, tp15~16)
The states on OPEN is sorted according to their heuristic
values.
If a child state is already on OPEN or CLOSED, the
algorithm checks if the child is reached by a shorter path.
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4.1.1 Best-First Search(4)
A hypothetical state space with heuristic evaluation
(Fig. 4.4 p129)
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4.1.1 Best-First Search(4)
A trace of the execution of best_first_search
1. OPEN=[A5]; CLOSED=[ ]
2. evaluate A5; OPEN=[B4,C4,D6]; CLOSED=[A5]
3. evaluate B4; OPEN=[C4,E5,F5,D6]; CLOSED=[B4,A5]
4. evaluate C4; OPEN=[H3,G4,E5,F5,D6]; CLOSED=[C4,B4,A5]
5. Evaluate H3; OPEN=[O2,P3,G4,E5,F5,D6];
CLOSED=[H3,C4,B4,A5] …………
In the event a heuristic leads the search down a path that
proves incorrect, the algorithm shifts its focus to another part of
the space. A B … (E, F) C
Shift the focus from B to C, but the children of B (E and F) are
kept on OPEN in case the algorithm returns to them later.
The goal of best-first search is to find the goal state
by looking at as few states as possible. (Fig. 4.5,
p130, tp19)
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4.1.2. Heuristic Evaluation
Functions (1)
Several heuristics for solving the 8-puzzle (Fig. 4.8,
p132, tp21)
count the number of tiles out of place compared with the goal.
sum all the distances by which the tiles are out of place.
multiplies a small number times each direct tile reversal.(Fig.4.7)
adds the sum of the distances out of place and 2 times the
number of direct reversals.
The design of good heuristics is an empirical
problem: the measure of a heuristic must be its
actual performance on problem instances.
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4.1.2 Heuristic Evaluation
Functions (3)
With the same heuristic evaluations, it is
preferable to examine the state that is nearest to
the root.
The distance from the starting state to its descendants can
be measured by maintaining a depth count for each state.
f(n) = g(n) + h(n) where g(n) measures the actual
length of the path from the state n to the start
state, h(n) is a heuristic estimate of the distance
from state n to a goal.
Example of heuristic f in 8-puzzle (Fig. 4.9, p133, tp23)
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4.1.2 Heuristic Evaluation
Function (5)
State space generated in heuristic search (Fig 4.10, p135, tp25)
Each state is labeled with a letter and its heuristic weight, f(n) =
g(n) + h(n) where g(n)=actual distance from n to the start state,
and h(n)=number of tiles out of place.
The successive stages of OPEN and CLOSED are:
1. OPEN=[a4]; CLOSED=[ ];
2. OPEN=[c4,b6,d6]; CLOSED=[a4];
3. OPEN=[e5,f5,g6,b6,d6]; CLOSED=[a4, c4];
4. OPEN=[f5,h6,g6,b6,d6,i7]; CLOSED=[a4,c4,e5]; ………….
Although the state h, the immediate child of e, has the same
number of tiles out of place as f, it is one level deeper in the state
space. The depth measure, g(n), causes the algorithm to select f
for evaluation.
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4.1.2 Heuristic Evaluation
Function (7)
1. Operations on states generate children of the state currently under
examination.
2. Each new state is checked to see whether it has occurred before,
thereby preventing loops
3. Each state n is given an f value equal to the sum of its depth in the
search space g(n) and a heuristic estimate of its distance to a goal
h(n). The h value guides search toward heuristically promising
states while the g value prevents search from persisting
indefinitely on a fruitless path.
4. States on OPEN are sorted by their f values. By keeping all states
on OPEN until they are examined or a goal is found, the algorithm
can go back from fruitless paths. At any one time, OPEN may
contain states at different levels of the state space graph
(Fig. 4.11), allowing full flexibility in changing the focus of the
search.
5. The efficiency of the algorithm can be improved by careful
maintenance of the OPEN and CLOSED lists.
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Example of Best First Search(1)
a
b d
e
c
f g h
i j k
l m n
o p
q
initial node: a goal node: q
3
2
3
3
3
2
2
1
2
2
2 2
3 5 4
1
3 1
1
3
2
3
2
2
G value: cost of getting from initial
node to the current node(1, 2…)
H value: estimated cost to goal
node(, ...)
F = G + H
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Example of Best First Search(2)
OPEN PATH
1. ((A 2)) Node A
2. ((B 4) (C 5) (D 8)) Node B (AB)
3. ((C 5) (D 8) (F 10) (E 11)) Node C (AC)
4. ((G 6) (D 8) (F 8) (E 11)) Node G (ACG)
5. ((D 8) (F 8) (E 11) (J 11) (K 12)) Node D (AD)
6. ((F 8) (H 10) (E 11) (J 11) (K 12)) Node F (ACF) /* (G 6) */
7. ((H 10) (I 10) (J 10) (E 11) (K 12)) Node H (ADH)
8. ((I 10) (J 10) (E 11) (K 11)) Node I (ACFI)
9. ((J 10) (E 11) (K 11) (L 13)) Node J (ACFJ)
10. ((M 10) (E 11) (K 11) (L 13)) Node M (ACFJM)
11. ((E 11) (K 11) (O 11) (L 13)) Node E (ABE)
12. ((K 11) (O 11) (L 13)) Node K (ADHK)
13. ((O 11) (L 13) (N 15)) Node O(ACFJM O)
14. ((Q 11) (L 13) (N 15)) (ACFJMOQ)
