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CS621 : Artificial IntelligencePowerPoint Presentation

CS621 : Artificial Intelligence

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S0

G

Tile movement represented as the movement of the blank space.

Operators:

L : Blank moves left

R : Blank moves right

U : Blank moves up

D : Blank moves down

C(L) = C(R) = C(U) = C(D) = 1

8-puzzle: heuristics

Example: 8 puzzle

s

n

g

h*(n) = actual no. of moves to transform n to g

h1(n) = no. of tiles displaced from their destined position.

h2(n) = sum of Manhattan distances of tiles from their destined position.

h1(n) ≤ h*(n) and h1(n) ≤ h*(n)

h*

h2

h1

Comparison

A version A2* of A* that has a “better” heuristic than another version A1* of A* performs at least “as well as” A1*

Meaning of “better”

h2(n) > h1(n) for all n

Meaning of “as well as”

A1* expands at least all the nodes of A2*

h*(n)

h2*(n)

h1*(n)

For all nodes n, except the goal node

Proof by induction on the search tree of A2*.

A* on termination carves out a tree out of G

Induction

on the depth k of the search tree of A2*. A1* before termination expands all the nodes of depth k in the search tree of A2*.

k=0. True since start node S is expanded by both

Suppose A1* terminates without expanding a node n at depth (k+1) of A2* search tree.

Since A1* has seen all the parents of n seen by A2*

g1(n) <= g2(n) (1)

Since A1* has terminated without expanding n,

f1(n) >= f*(S) (2)

Any node whose f value is strictly less than f*(S) has to be expanded.

Since A2* has expanded n

f2(n) <= f*(S) (3)

S

k+1

G

From (1), (2), and (3)

h1(n) >= h2(n) which is a contradiction. Therefore, A1* has to expand all nodes that A2* has expanded.

Exercise

If better means h2(n) > h1(n) for some n and h2(n) = h1(n) for others, then Can you prove the result ?

- Very reasonable assumption
- That is the estimated cost of reaching the goal node for a particular node is no more than the cost of reaching a child and the estimated cost of reaching the goal from the child
- This obviates the need for redirecting parent pointer for a less costly path in the closed list
- This means that for any node taken up for expansion, the optimal path is already found

n1

C(n1,n2)

n2

h(n1)

h(n2)

g

Beam Search

- Run A*, but with a “Beam width K”
- That is, for each iteration keep K most promising nodes and throw away other nodes
- Incomplete and non-optimal, but fast

Genetic Algorithm Based

- Use the operators of mutation and cross over
- Mutation: flip a bit randomly to produce a new state
- Cross over: produce by crossing portion of two states a new state

Blank 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

GA for 8-puzzle4

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Represented as

0100 0011 0110 0010 0001 1000 0111 0000 0101

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Mutation and Crossover

Mutation: flip any bit randomly

A

0100 0011 0110 0010 0001 1000 0111 0000 0101

B

0100 0111 0110 0010 0001 1000 0111 0000 0101

Crossover: cross a set of bit

1000 0111 0000 0101 0001 1000 0111 0000 0101

First 16 bits of

A crossed With

last 16 bits

of B

0100 0111 0110 0010 0001 0100 0011 0110 0010

Care to be exercised

- The bit strings are called chromosomes
- Mutation and crossover produce bit strings which are state descriptions
- But care has to be exercised to see that the produced state is
- Legal
- Reachable

Lab assignment

- Implement A* algorithm for the following problems:
- 8 puzzle

- Specifications:
- Try different heuristics and compare with baseline case, i.e., the breadth first search.
- Violate the condition h ≤ h*. See if the optimal path is still found. Observe the speedup.

- Now implement for the same problem
- Beam Search
- Genetic Algorithm

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