Searching techniques in AI

7/29/2019 Searching techniques in AI

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Simple Hill Climbing

First state that is better than the current state

is selected.


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2

Simple Hill Climbing

Algorithm

1. Evaluate the initial state.

2. Loop until a solution is found or there are no

new operators left to be applied:- Select and apply a new operator

-Evaluate the new state:

goal quit

better than current state new current state


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3

Steepest-Ascent Hill Climbing

(Gradient Search)

Considers all the moves from the current

state.

Selects the best one as the next state.


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Difference between Hill climbing and Best first

search:

In Hill Climbing-one move is selected and all

others are rejected, never to be reconsidered.

In Best first search-one move is selected but

others are kept around so that they can be

revisited later if the selected path becomes less

promising.


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Hill Climbing: Disadvantages

Local maximum

A state that is better than all of its neighbours,

but not

better than some other states far away.


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Plateau

A flat area of the search space in which all

neighbouring

states have the same value.


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Ridge

The orientation of the high region, compared to

the set

of available moves, makes it impossible to climb

up.

However, two moves executed serially may

increasethe height.


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Ways Out

Backtrack to some earlier node and try going

in a different direction.

Make a big jump to try to get in a new section.

Moving in several directions at once.


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9


Hill climbing is a local method:

Decides what to do next by looking only at the

immediate consequences of its choices.

Global information might be encoded in

heuristic functions.


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B

C

D

A

B

C

Start Goal

Blocks World

A D


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B

C

D

A

B

C

Start Goal

Blocks World

A D

Local heuristic:

+1 for each block that is resting on the thing it is supposed to

be resting on.

-1for each block that is resting on a wrong thing.

0 4


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B

C

D

B

C

D

A

A0 2


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B

C

D

A

B

C D

A B

C

DA

0

0

0

B

C

D

A

2


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B

C

D

A

B

C

Start Goal

Blocks World

A D

Global heuristic:

For each block that has the correct support structure: +1to

every block in the support structure.

For each block that has a wrong support structure: -1to

every block in the support structure.

-6 6


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B

C

D

A

B

C D

A B

C

DA

-6

-2

-1

B

C

D

A

-3


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Hill Climbing: Conclusion

Can be very inefficient in a large, rough

problem space.

Global heuristic may have to pay forcomputational complexity.

Often useful when combined with othermethods, getting it started right in the right

general neighbourhood.


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A* Method A* (Aystar) (Hart, 1972) method is a combination of branch &

bound and best search, combined with the dynamic

programming principle.

The heuristic function (or Evaluation function) for a node N is

defined as f(N) = g(N) + h(N)

The function g is a measure of the cost of getting from the Start

node (initial state) to the current node.

It is sum of costs of applying the rules that were applied along the best

path to the current node.

The function h is an estimate of additional cost of getting fromthe current node to the Goal node (final state).

Here knowledge about the problem domain is exploited.

A* algorithm is called OR graph / tree search algorithm.


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Behavior of A* Algorithm

Underestimation

If we can guarantee that h never overestimates actual value from current to goal,

then A* algorithm is guaranteed to find an

optimal path to a goal, if one exists.


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Example Underestimation f=g+hHere h is underestimated

A

Underestimated

(1+3)B (1+4)C (1+5)D

3 moves away from goal

(2+3) E

3 moves away from goal

(3+3) F


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Explanation -Example of Underestimation

Assume the cost of all arcs to be 1. A is expanded to B, C and D. f values for each node is computed.

B is chosen to be expanded to E.

We notice that f(E) = f(C) = 5

Suppose we resolve in favor of E, the path currently we are

expanding. E is expanded to F.

Clearly expansion of a node F is stopped as f(F)=6 and so we willnow expand C.

Thus we see that by underestimating h(B), we have wasted someeffort but eventually discovered that B was farther away than wethought.

Then we go back and try another path, and will find optimal path.


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Example Overestimation

Here h is overestimated

A

Overestimated

(1+3) B (1+4) C (1+5) D

(2+2) E

(3+1)F

(4+0) G


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Explanation Example of Overestimation

A is expanded to B, C and D. Now B is expanded to E, E to F and F to G for a solution path

of length 4.

Consider a scenario when there a direct path from D to G witha solution giving a path of length 2.

We will never find it because of overestimating h(D).

Thus, we may find some other worse solution without everexpanding D.

So by overestimating h, we can not be guaranteed to

find the cheaper path solution.


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Admissibility of A*

A search algorithm is admissible, if

for any graph, it always terminates in an optimalpath from initial state to goal state, if path exists.

If heuristic function h is underestimate ofactual value from current state to goal state,then the it is called admissible function.

Alternatively we can say that A* alwaysterminates with the optimal path in case

h(x) is an admissible heuristic function.


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Start state Goal state

3 7 6 5 3 6

5 1 2 7 2

4 8 4 1 8

Example: Solve Eight puzzle problem using A* algorithm

Evaluation function f (X) = g (X) + h(X)

h (X) = the number of tiles not in their goal

position in a given state X

g(X) = depth of node X in the search tree Initial node has f(initial_node) = 4

Apply A* algorithm to solve it.

The choice of evaluation function critically determines search

results.


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Example: Eight puzzle problem (EPP)

Start state Goal state

3 7 6 5 3 6

5 1 2 7 2

4 8 4 1 8


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Evaluation function - f for EPP

The choice of evaluation function criticallydetermines search results.

Consider Evaluation function

f (X) = g (X) + h(X) h (X) = the number of tiles not in their goal

position in a given state X

g(X) = depth of node X in the search tree

For Initial node f(initial_node) = 4

Apply A* algorithm to solve it.


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Start State

Search Tree f = 0+4

3 7 65 1 2

4 8

up

(1+3)

left

(1+5)

right

(1+5)

3 7 6

5 2

4 1 8

3 7 6

5 1 2

4 8

3 7 6

5 1 2

4 8

up

(2+3)

left

(2+3)

right

(2+4)

3 65 7 2

4 1 8

left

(3+2)

3 7 65 2

4 1 8

right

(3+4)

3 7 65 2

4 1 8

3 6

5 7 2

4 1 8down

(4+1)

3 6

5 7 2

4 1 8

5 3 6

7 24 1 8

right 5 3 6

7 24 1 8

Goal

State


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Harder Problem

Harder problems (8 puzzle) cant be solved by

heuristic function defined earlier.

Initial State Goal State

2 1 6 1 2 34 8 8 4

7 5 3 7 6 5

A better estimate function is to be thought.

h (X) = the sum of the distances of the tiles

from their goal position in a given state X

Initial node has h(initial_node) = 1+1+2+2+1+3+0+2=12


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Problem Reduction

Goal: Acquire TV set

AND-OR Graphs

Goal: Steal TV set Goal: Earn some money Goal: Buy TV set

Algorithm AO* (Martelli & Montanari 1973, Nilsson 1980)


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Problem Reduction: AO*

A

DCB

43 5

A

5

6

FE

44

A

DCB

43

10

9

9

9

FE

44

A

DCB

4

6 10

11

12

HG

75


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Problem Reduction: AO*

A

G

CB 10

5

11

13

ED 65 F 3

A

G

CB 15

10

14

13

ED 65 F 3

H 9Necessary backward propagation


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Constraint Satisfaction

Many AI problems can be viewed as problems

ofconstraint satisfaction.

Cryptarithmetic puzzle:

SEND

MORE

MONEY


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As compared with a straightforard search

procedure, viewing a problem as one of

constraint satisfaction can reduce substantially

the amount of search.


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Operates in a space of constraint sets.

Initial state contains the original constraints

given in the problem.

A goal state is any state that has been

constrained enough.


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Two-step process:

1. Constraints are discovered and propagated as

far as possible.2. If there is still not a solution, then search

begins, adding new constraints.


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M = 1

S = 8 or 9

O = 0N = 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 = YN + R = 10 + E

R = 9

S =8

2 + D = 10 + YD = 8 + Y

D = 8 or 9

Y = 0 Y = 1

E = 2

C1 = 0 C1 = 1

D = 8 D = 9

Initial state:

No two letters have

the same value.

The sum of the digits

must be as shown.

SEND

MORE

MONEY


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Two kinds of rules:

1. Rules that define valid constraint

propagation.2. Rules that suggest guesses when necessary.

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Searching techniques in AI