artificial intelligence chap2-2

7/27/2019 Artificial Intelligence Chap2-2

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Artificial IntelligenceProblem solving by searching

CSC 361

Prof. Mohamed Batouche

Computer Science DepartmentCCIS King Saud University

Riyadh, Saudi Arabia

[email protected]


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Problem Solving by Searching

Search Methods :Uninformed (Blind) search


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3

Search Methods Once we have defined the problem space (state representation, the initial state, the

goal state and operators) is all done?

Lets consider the River Problem:

A farmer wishes to carry a wolf, a duck and corn across a river, from the south to thenorth shore. The farmer is the proud owner of a small rowing boat called Bountywhich he feels is easily up to the job. Unfortunately the boat is only large enough tocarry at most the farmer and one other item. Worse again, if left unattended the wolfwill eat the duck and the duck will eat the corn.

How can the farmer safely transport the wolf, the duck and the corn to the oppositeshore?

Farmer, Wolf,

Duck and Corn

boat

River


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Search Methods The River Problem:

F=Farmer W=Wolf D=Duck C=Corn /=River

How can the farmer safely transport the wolf, the duck andthe corn to the opposite shore?

FWCD/-

-/FWCD


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Search Methods Problem formulation: State representation: location of farmer and items in both sides of river

[items in South shore / items in North shore] : (FWDC/-, FD/WC, C/FWD )

Initial State: farmer, wolf, duck and corn in the south shoreFWDC/-

Goal State: farmer, duck and corn in the north shore-/FWDC

Operators: the farmer takes in the boat at most one item from one side tothe other side

(F-Takes-W, F-Takes-D, F-Takes-C, F-Takes-Self [himself only])

Path cost: the number of crossings


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Search Methods

State space:A problem is solved by moving from the initial state to the goal state by applying validoperators in sequence. Thus the state space is the set of states reachable from a particularinitial state.

F W D C

W D C

F

D C

F W

W C

F D

W D

F C

F W C

D

F W D C

W

F D C

F W C

D

W C

F D

C

F W D

F C

W D

F D C

W

D

F W C

F W D

C

F W

D C

F W C

D

W D

F C

W

F D C

C

F W D

D C

F W

D

F W C

F D C

W

F W D

C

F D

W C

F W D C

D

F W C

Initial state

Goal state

Deadends

Illegal states

repeatedstate intermediate state


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Search Methods Searching for a solution:We start with the initial state andkeep using the operators toexpand the parent nodes till we

find a goal state.

but the search space might belarge

really large

So we need some systematic wayto search.

F W D C

WD C

F

D C

F W

W C

F D

WD

F C

F W C

D

F W D C

W

F D C

F W C

D

W C

F D

C

F WD

F C

W D

F D C

W

D

F W C

F W D

C

F W

D C

F W C

D

WD

F C

W

F D C

C

F W D

D C

F W

D

F W C

F D C

W

F WD

C

F D

W C

F W D C

D

F W C


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Search Methods Problem solution:

A problem solution is simply the setof operators (actions) needed to

reach the goal state from the initialstate:

F-Takes-D, F-Takes-Self, F-Takes-W,

F-Takes-D, F-Takes-C, F-Takes-Self,

F-Takes-D.

F W D C

WD C

F

D C

F W

W C

F D

WD

F C

F W C

D

F W D C

W

F D C

F W C

D

W C

F D

C

F WD

F C

W D

F D C

W

D

F W C

F W D

C

F W

D C

F W C

D

WD

F C

W

F D C

C

F W D

D C

F W

D

F W C

F D C

W

F WD

C

F D

W C

F W D C

D

F W C


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Problem Solving by searching

Basic Search Algorithms


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Basic Search Algorithms uninformed( Blind) search: breadth-first, depth-first,

depth limited, iterative deepening, and bidirectional search

informed (Heuristic) search: search is guided by anevaluation function: Greedy best-first, A*, IDA*, and beamsearch

optimization in which the search is to find an optimal value

of an objective function: hill climbing, simulated annealing,genetic algorithms, Ant Colony Optimization

Game playing, an adversarial search: minimax algorithm,alpha-beta pruning


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What Criteria are used to Compare

different search techniques ?As we are going to consider different techniques to search theproblem space, we need to consider what criteria we will use tocompare them.

Completeness: Is the technique guaranteed to find an answer(if there is one).

Optimality/Admissibility : does it always find a least-costsolution?- an admissible algorithm will find a solution with minimum cost

Time Complexity: How long does it take to find a solution.

Space Complexity: How much memory does it take to find asolution.


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Time and Space Complexity ?Time and space complexity are measured in terms of:

The average number of new nodes we create when expanding

a new node is the (effective) branching factor b.

The (maximum) branching factor b is defined as the maximumnodes created when a new node is expanded.

The length of a path to a goal is the depth d.

The maximum length of any path in the state space m.


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Branching factors for some

problems The eight puzzle has a (effective)

branching factor of 2.13, so a search treeat depth 20 has about 3.7 million nodes:O(bd)

Rubiks cube has a (effective) branchingfactor of 13.34. There are901,083,404,981,813,616 different states.The average depth of a solution is about18.

Chess has a branching factor of about 35,there are about 10120 states (there areabout 1079 electrons in the universe).

2 1 34 7 65 8


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The state Space


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Generic Search Algorithms

Basic Idea: Off-line exploration of state space bygenerating successors of already-explored states (alsoknown as expanding states).

Function GENERAL-SEARCH (problem, strategy)

returns a solution or failure

Initialize the search tree using the initial state of problem

loop do

ifthere are no candidates for expansion, then return failureChoose a leaf node for expansion according tostrategy

if node contains goal state then returnsolution

else expand node and add resulting nodes to search tree.

end


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Representing Search

Arad

Zerind Sibiu Timisoara

Oradea Fagaras Rimnicu VilceaArad

Sibiu Bucharest


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Solution


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Implementation of GenericSearch Algorithm

function general-search(problem, QUEUEING-FUNCTION)

nodes = MAKE-QUEUE(MAKE-NODE(problem.INITIAL-STATE))

loop do

ifEMPTY(nodes) then return "failure"

node = REMOVE-FRONT(nodes)

ifproblem.GOAL-TEST(node.STATE) succeeds then return solution(node)

nodes = QUEUEING-FUNCTION(nodes, EXPAND(node, problem.OPERATORS))

end

A nice fact about this search algorithm is that we can use a single algorithm to do

many kinds of search. The only difference is in how the nodes are placed in the

queue.The choice of queuing function is the main feature.


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Key Issues of State-Spacesearch algorithm

Search process constructs a search tree root is the start node leaf nodes are:

unexpanded nodes (in the nodes list)

dead ends (nodes that arent goals and have no successors becauseno operators were applicable)

Loops in a graph may cause a search tree to be infinite even if thestate space is small

changing definition of how nodes are added to list leads to a differentsearch strategy

Solution desired may be: just the goal state a path from start to goal state (e.g., 8-puzzle)


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Uninformed search strategies(Blind search)

Uninformed (blind) strategies use only the information available inthe problem definition. These strategies order nodes withoutusing any domain specific information

Contrary to Informed search techniques which might haveadditional information (e.g. a compass).

Breadth-first search Uniform-cost search

Depth-first search Depth-limited search Iterative deepening search Bidirectional search


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Basic Search AlgorithmsUninformed Search

Breadth First Search (BFS)


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Complete? Yes.

Optimal? Yes, if path cost is nondecreasing function of depth

Time Complexity: O(bd)

Space Complexity: O(bd

), note that every node in the fringe is kept in the queue.

Main idea: Expand all nodes at depth (i) before expanding nodes at depth (i + 1)

Level-order Traversal.

Implementation: Use of a First-In-First-Out queue (FIFO). Nodes visited first

are expanded first. Enqueue nodes in FIFO (first-in, first-out) order.


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Breadth First Search

QUEUING-FN:- successors added to endof queue Arad


Oradea Fagaras Rimnicu VilceaAradArad Oradea Arad Lugoj

Shallow nodes are expanded before deeper nodes.


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Uniform Cost Search (UCS)


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25

1 7

4 5

[5] [2]

[9][3]

[7] [8]

1 4

[9][6]

[x] = g(n)

path cost of node n

Goal state


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25

[5] [2]


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25

1 7

[5] [2]

[9][3]


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25

1 7

4 5

[5] [2]

[9][3]

[7] [8]


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25

1 7

4 5

[5] [2]

[9][3]

[7] [8]

1 4

[9][6]


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25

1 7

4 5

[5] [2]

[9][3]

[7] [8]

1 4

[9]

Goal state

path cost

g(n)=[6]


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In case of equal step costs, Breadth First search findsthe optimal solution.

For any step-cost function, Uniform Cost searchexpands the node n with the lowest path cost.

UCS takes into account the total cost: g(n).

UCS is guided by path costs rather than depths.Nodes are ordered according to their path cost.


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Main idea: Expand the cheapest node. Where the cost is the path cost g(n).

Implementation:

Enqueue nodes in order of cost g(n).QUEUING-FN:- insert in order of increasing path cost.

Enqueue new node at the appropriate position in the queue so that wedequeue the cheapest node.

Complete? Yes. Optimal? Yes, if path cost is nondecreasing function of depth

Time Complexity: O(bd)

Space Complexity: O(bd), note that every node in the fringe keep in the queue.


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Depth First Search (DFS)


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Depth First Search (DFS)


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Depth First Search


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Depth-First Search (DFS)

QUEUING-FN:- insert successors atfront of queue

Arad


Arad Oradea



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Depth-Limited Search (DLS)


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Depth Bound = 3


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It is simply DFS with a depth bound.

Searching is not permitted beyond the depth bound.

Works well if we know what the depth of the solution is.

Termination is guaranteed.

If the solution is beneath the depth bound, the search

cannot find the goal (hence this search algorithm isincomplete).

Otherwise use Iterative deepening search (IDS).


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Main idea: Expand node at the deepest level, but limit depth to L.

Implementation:Enqueue nodes in LIFO (last-in, first-out) order. But limit depth to L

Complete? Yes if there is a goal state at a depth less than L

Optimal? No

Time Complexity: O(bL), where L is the cutoff.

Space Complexity: O(bL), where L is the cutoff.


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Iterative Deepening Search (IDS)


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function ITERATIVE-DEEPENING-SEARCH():

for depth = 0 to infinity do

ifDEPTH-LIMITED-SEARCH(depth) succeeds

then return its result

end

return failure


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Key idea: Iterative deepening search (IDS) applies DLS repeatedlywith increasing depth. It terminates when a solution is found or nosolutions exists.

IDS combines the benefits of BFS and DFS: Like DFS the memoryrequirements are very modest (O(bd)). Like BFS, it is completewhen the branching factor is finite.

The total number of generated nodes is :

N(IDS)=(d)b +(d-1) b2++(1)bd

In general, iterative deepening is the preferred uninformed searchmethod when there is a large search space and the depth of the

solution is not known.


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L = 0

L = 1

L = 2

L = 3


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Bi-Directional Search (BDS)


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Bi-directional Search (BDS)

Main idea: Start searching fromboth the initial state and the goalstate, meet in the middle.

Complete? Yes

Optimal? Yes

Time Complexity: O(bd/2), whered is the depth of the solution.

Space Complexity: O(bd/2), whered is the depth of the solution.


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Comparison of search algorithms

C i f h


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Comparison of searchalgorithms

b: Branching factord: Depth of solutionm: Maximum depth

l : Depth Limit


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Blind Search Algorithms

Tree Search:

BFS, DFS, DLS, IDS


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Breadth First Search

BFS


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Breadth First Search Application1:

Given the following state space (tree search), give the sequenceof visited nodes when using BFS (assume that the nodeOis thegoal state):

A

B C ED

F G H I J

K L

O

M N


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Breadth First Search A,

A

B C ED


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B,C,D

A

B C ED

F G H I J


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B,C,D,E

A

B C ED

F G H I J


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B,C,D,E,

F,

A

B C ED

F G H I J


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B,C,D,E,

F,G

A

B C ED

F G H I J

K L


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B,C,D,E,

F,G,H

A

B C ED

F G H I J

K L


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B,C,D,E,

F,G,H,I

A

B C ED

F G H I J

K L M


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B,C,D,E,

F,G,H,I,J,

A

B C ED

F G H I J

K L M N


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B,C,D,E,

F,G,H,I,J,

K, A

B C ED

F G H I J

K L M N


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B,C,D,E,

F,G,H,I,J,

K,L A

B C ED

F G H I J

K L

O

M N


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B,C,D,E,

F,G,H,I,J,

K,L, M, A

B C ED

F G H I J

K L

O

M N


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B,C,D,E,

F,G,H,I,J,

K,L, M,N, A

B C ED

F G H I J

K L

O

M N


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B,C,D,E,

F,G,H,I,J,

K,L, M,N,

Goal state: O

A

B C ED

F G H I J

K L

O

M N


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Breadth First Search The returned solution is the sequence of operators in the path:

A, B, G, L, O

A

B C ED

F G H I J

K L

O

M N


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Depth First Search

DFS


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Depth First Search A,

A

B C ED


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Depth First Search A,B,

A

B C ED

F G


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Depth First Search A,B,F,

A

B C ED

F G


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G,

A

B C ED

F G

K L


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G,K,

A

B C ED

F G

K L


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G,K,

L,

A

B C ED

F G

K L

O


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G,K,

L, O: Goal State

A

B C ED

F G

K L

O


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Depth First SearchThe returned solution is the sequence of operators in the path:

A, B, G, L, O

A

B C ED

F G

K L

O


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Depth-Limited Search

DLS


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Depth-Limited Search (DLS) Application3:

Given the following state space (tree search), give the sequenceof visited nodes when using DLS (Limit = 2):

A

B C ED

F G H I J

K L

O

M N

Limit = 0

Limit = 1

Limit = 2


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Depth-Limited Search (DLS) A,

A

B C ED

Limit = 2


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Depth-Limited Search (DLS) A,B,

A

B C ED

F GLimit = 2


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Depth-Limited Search (DLS) A,B,F,

A

B C ED

F GLimit = 2


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G,

A

B C ED

F GLimit = 2


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G,

C,

A

B C ED

F G HLimit = 2


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G,

C,H,

A

B C ED

F G HLimit = 2


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G,

C,H,

D, A

B C ED

F G H I JLimit = 2


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G,

C,H,

D,I A

B C ED

F G H I JLimit = 2


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G,

C,H,

D,I

J,

A

B C ED

F G H I JLimit = 2


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G,

C,H,

D,I

J, E

A

B C ED

F G H I JLimit = 2


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G,

C,H,

D,I

J, E, Failure

A

B C ED

F G H I JLimit = 2


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Depth-Limited Search (DLS) DLS algorithm returns Failure (no solution)

The reason is that the goal is beyond the limit (Limit =2): thegoal depth is (d=4)

A

B C ED

F G H I J

K L

O

M N

Limit = 2


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Iterative Deepening Search

IDS


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Iterative Deepening Search (IDS) Application4:

Given the following state space (tree search), give the sequenceof visited nodes when using IDS:

A

B C ED

F G H I J

K L

O

M N

Limit = 0

Limit = 1

Limit = 2

Limit = 3

Limit = 4


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Iterative Deepening Search(IDS)

DLS with bound = 0


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Iterative Deepening Search (IDS) A, Failure

ALimit = 0


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DLS with bound = 1


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Iterative Deepening Search (IDS) A,

A

B C EDLimit = 1


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Iterative Deepening Search (IDS) A,B,

A

B C EDLimit = 1


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C,

A

B C EDLimit = 1


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C,

D,

A

B C EDLimit = 1


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Iterative Deepening Search (IDS) A,B

C,

D,

E, A

B C EDLimit = 1


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C,

D,

E, Failure A

B C EDLimit = 1


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A

B C ED

Limit = 2


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A

B C ED

F GLimit = 2


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Iterative Deepening Search (IDS) A,B,F,

A

B C ED

F GLimit = 2


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G,

A

B C ED

F GLimit = 2


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G,

C,

A

B C ED

F G HLimit = 2


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G,

C,H,

A

B C ED

F G HLimit = 2


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G,

C,H,

D, A

B C ED

F G H I JLimit = 2


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G,

C,H,

D,I A

B C ED

F G H I JLimit = 2


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G,

C,H,

D,I

J,

A

B C ED

F G H I JLimit = 2


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G,

C,H,

D,I

J, E

A

B C ED

F G H I JLimit = 2


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G,

C,H,

D,I

J, E, Failure

A

B C ED

F G H I J

K L

O

M N

Limit = 2


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DLS with bound = 3


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A

B C ED

Limit = 3


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A

B C ED

F G

Limit = 3


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A

B C ED

F G

Limit = 3


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G,

A

B C ED

F G

K LLimit = 3


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G,K,

A

B C ED

F G

K LLimit = 3


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G,K,

L,

A

B C ED

F G

K LLimit = 3


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G,K,

L,

C,H, A

B C ED

F G H

K LLimit = 3


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G,K,

L,

C,H,

D,

A

B C ED

F G H I J

K LLimit = 3


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G,K,

L,

C,H,

D,I,

A

B C ED

F G H I J

K L MLimit = 3


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G,K,

L,

C,H,

D,I,M,

A

B C ED

F G H I J

K L MLimit = 3


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G,K,

L,

C,H,

D,I,M, J,

A

B C ED

F G H I J

K L M NLimit = 3

h ( )


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G,K,

L,

C,H,

D,I,M, J,N,

A

B C ED

F G H I J

K L M NLimit = 3

i i S h ( S)


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G,K,

L,

C,H,

D,I,M, J,N,

E,

A

B C ED

F G H I J

K L M NLimit = 3


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DLS with bound = 4

It ti D i S h (IDS)


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A

B C ED

Limit = 4

It ti D i S h (IDS)


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A

B C ED

F G

Limit = 4

It ti D i S h (IDS)


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A

B C ED

F G

Limit = 4

It ti D i S h (IDS)


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G,

A

B C ED

F G

K L

Limit = 4

It ti D i S h (IDS)


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G,K,

A

B C ED

F G

K L

Limit = 4

It ti D i S h (IDS)


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G,K,

L,

A

B C ED

F G

K L

OLimit = 4

It ti D i S h (IDS)


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G,K,

L, O:Goal State

A

B C ED

F G

K L

OLimit = 4

Ite ti e Deepening Se h (IDS)


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Iterative Deepening Search (IDS)The returned solution is the sequence of operators in the path:

A, B, G, L, O

A

B C ED

F G

K L

O

Summary


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Summary

Search: process of constructing sequences of actions that achieve a goal given aproblem.

The studied methods assume that the environment is observable, deterministic,static and completely known.

Goal formulation is the first step in solving problems by searching. It facilitatesproblem formulation.

Formulating a problem requires specifying four components: Initial states,operators, goal test and path cost function. Environment is represented as astate space.

A solution is a path from the initial state to a goal state.

Search algorithms are judged on the basis of completeness, optimality, timecomplexity and space complexity.

Several search strategies: BFS, DFS, DLS, IDS,

artificial intelligence chap2-2

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