Artificial Intelligence Search Problem. Search is a problem-solving technique to explores successive...

Artificial Intelligence

Search Problem

Search Problem

Search is a problem-solving technique to explores

successive stages in problem-solving process.

A simple example: traveling on a graph

A

BC

F

D E3 4 4

3

92

2

start stategoal state

Search treestate = A ,

cost = 0

state = B ,cost = 3

state = D ,cost = 3

state = C ,cost = 5

state = F ,cost = 12

state = A ,cost = 7

goal state!

search tree nodes and states are not the same thing!

Full search treestate = A ,

cost = 0

state = B ,cost = 3

state = D ,cost = 3

state = C ,cost = 5


state = A ,cost = 7

goal state!

state = E ,cost = 7


goal state!

state = B ,cost = 10

state = D ,cost = 10...

...

Problem types

• Deterministic, fully observable single-state problem– Solution is a sequence of states

• Non-observable sensorless problem – Problem-solving may have no idea where it is;

solution is a sequence

• Nondeterministic and/or partially observable

Unknown state space

Algorithm types

• There are two kinds of search algorithm –Complete• guaranteed to find solution or prove there is

none

– Incomplete• may not find a solution even when it exists• often more efficient (or there would be no

point)

Example: the 8-puzzle.Given: a board situation for the 8-puzzle:

11 33 88

22 77

55 44 66

11 22 33

5566774488

Problem: find a sequence of moves that transform this board situation in a desired goal situation:

State Space representation

In the space representation of a problem, the

nodes of a graph correspond to partial

problem solution states and arcs correspond

to steps (action) in a problem-solving process

Key concepts in search

• Set of states that we can be in– Including an initial state…– … and goal states (equivalently, a goal test)

• For every state, a set of actions that we can take– Each action results in a new state– Given a state, produces all states that can be reached from it

• Cost function that determines the cost of each action (or path = sequence of actions)

• Solution: path from initial state to a goal state– Optimal solution: solution with minimal cost

( NewYork )( NewYork )

( NewYork,( NewYork, Boston )Boston )

( NewYork,( NewYork, Miami )Miami )

( NewYork,( NewYork, Dallas )Dallas )

( NewYork,( NewYork, Frisco )Frisco )

( NewYork,( NewYork, Boston,Boston, Miami )Miami )

( NewYork,( NewYork, Frisco,Frisco, Miami )Miami )

250250 12001200 1500150029002900

00

250250 12001200 15001500 29002900

14501450 33003300

17001700 62006200

Keep track of Keep track of accumulated costsaccumulated costs in each state if you want in each state if you wantto be sure to get the best path.to be sure to get the best path.

Example: Route Finding

• Initial state– City journey starts in

• Operators– Driving from city to city

• Goal test– Is current location the

destination city?

Liverpool

London

Nottingham

Leeds

Birmingham

Manchester

State space representation (salesman)• State:– the list of cities that are already visited

• Ex.: ( NewYork, Boston )• Initial state:

• Ex.: ( NewYork )• Rules:– add 1 city to the list that is not yet a member– add the first city if you already have 5 members

• Goal criterion:– first and last city are equal

Example: The 8-puzzle

• states? locations of tiles • actions? move blank left, right, up, down • goal? = goal state (given)• path cost? 1 per move

Example: robotic assembly

• states?: real-valued coordinates of robot joint angles parts of the object to be assembled

• actions?: continuous motions of robot joints• goal test?: complete assembly• path cost?: time to execute••••

123

84

765

13

746

582

143

76

582

143

786

52

143

76

582

13

746

582

13

746

582

43

176

582

143

576

82

Example: Chess• Problem: develop a program that plays chess

1122334455667788

A B C D E F G A B C D E F G HH

1. A way to represent board situations Ex.:

List:List:(( king_black, 8, C),(( king_black, 8, C), ( knight_black, 7, B),( knight_black, 7, B), ( pawn_black, 7, G),( pawn_black, 7, G), ( pawn_black, 5, F),( pawn_black, 5, F), ( pawn_white, 2, H),( pawn_white, 2, H), ( king_white, 1, E))( king_white, 1, E))

Chess

Move 1Move 1

Move 2Move 2

Move 3Move 3

search treesearch tree~15~15

~ (15)~ (15)22

~ (15)~ (15)33

Need Need very efficient searchvery efficient search techniques to find good techniques to find goodpaths in such combinatorial trees.paths in such combinatorial trees.

independence of states:Ex.: Blocks world problem.• Initially: C is on A and B is on the table.• Rules: to move any free block to another or to the table• Goal: A is on B and B is on C.

AACC

BBGoalGoal:: A A onon B B andand B B onon C C

AACC

BBGoalGoal:: A A onon B B

AACC

BBGoalGoal:: B B onon C C

AND-OR-tree?AND-OR-tree?

ANDAND

Search in State Spaces

•Effects of moving a block (illustration and list-structure iconic model notation)

Avoiding Repeated States

• In increasing order of effectiveness in reducing size of state space and with increasing computational costs:1. Do not return to the state you just came from. 2. Do not create paths with cycles in them. 3. Do not generate any state that was ever created

before. • Net effect depends on frequency of “loops” in state

space.

Forward versus backward reasoning:Initial statesInitial states

Goal statesGoal states

Forward reasoning (or Data-driven): from initial states to goal states.

Forward versus backward reasoning:Initial statesInitial states

Goal statesGoal states

Backward reasoning (or backward chaining / goal-driven ): from goal states to initial states.

Data-Driven search• It is called forward chaining

• The problem solver begins with the given facts

and a set of legal moves or rules for changing

state to arrive to the goal.

Goal-Driven Search• Take the goal that we want to solve and see

what rules or legal moves could be used to

generate this goal.

• So we move backward.

Search Implementation

• In both types of moving search, we must find

the path from start state to a goal.

• We use goal-driven search if

– The goal is given in the problem

– There exist a large number of rules

– Problem data are not given

Search Implementation

• The data-driven search is used if

– All or most data are given

– There are a large number of potential goals

– It is difficult to form a goal

Criteria:

• Sometimes: no way to start from the goal states– because there are too many (Ex.: chess)– because you can’t (easily) formulate the rules

in 2 directions.

11 33 88

22 77

55 44 66

11 22 33

5566774488

In this case: even the same rules !!In this case: even the same rules !!

Sometimes equivalent:

Classical Search Strategies

• Breadth-first search• Depth-first search• Bidirectional search• Depth-bounded depth first search– like depth first but set limit on depth of search in

tree• Iterative Deepening search– use depth-bounded search but iteratively increase

limit

Breadth-first search:

• Move downwards, level by level, until goal is reached.

SS

AA DD

BB DD AA EE

CC EE EE BB BB FF

DD FF BB FF CC EE AA CC GG

GG

GGGG FFCC

It explores the space in a level-by-level fashion.

Breadth-first search• BFS is complete: if a solution exists, one will

be found• Expand shallowest unexpanded node– fringe is a FIFO queue, i.e., new successors go at

end

–

• Implementation:

Breadth-first search

• Expand shallowest unexpanded node– fringe is a FIFO queue, i.e., new successors go at

end

–

•Implementation:

Depth-First Order

• When a state is examined, all of its children and

their descendants are examined before any of its

siblings.

• Not complete (might cycle through nongoal states)

• Depth- First order goes deeper whenever this is

possible.

Depth-first search= Chronological backtracking

• Select a child – convention: left-to-right

• Repeatedly go to next child, as long as possible.

• Return to left-over alternatives (higher-up) only when needed.

BB

CC EE

DD FF

GG

SS

AA

Depth-first search

• Expand deepest unexpanded node– fringe = LIFO queue, i.e., put successors at front

–

•Implementation:

A

C D E FB

G H I J K L M N O P

Q R S T U V W X Y Z

The example node set

Initial state

Goal state

A

L

Press space to see a BFS of the example node set

A

C D E FB

G H I J K L

Q R S T U

A

B C D

We begin with our initial state: the node labeled A. Press space

to continue

This node is then expanded to reveal further (unexpanded)

nodes. Press space

Node A is removed from the queue. Each revealed node is

added to the END of the queue. Press space to continue the

search.

The search then moves to the first node in the queue. Press

space to continue.

Node B is expanded then removed from the queue. The

revealed nodes are added to the END of the queue. Press space.

Size of Queue: 0

Nodes expanded: 0

Current Action: Current level: n/a

Queue: EmptyQueue: ASize of Queue: 1

Nodes expanded: 1

Queue: B, C, D, E, F

Press space to begin the search

Size of Queue: 5

Current level: 0Current Action: Expanding

Queue: C, D, E, F, G, HSize of Queue: 6

Nodes expanded: 2

Current level: 1

We then backtrack to expand node C, and the process

continues. Press space

Current Action: Backtracking

Current level: 0Current level: 1

Queue: D, E, F, G, H, I, JSize of Queue: 7

Nodes expanded: 3

Current Action: ExpandingCurrent Action: Backtracking

Current level: 0Current level: 1

Queue: E, F, G, H, I, J, K, LSize of Queue: 8

Nodes expanded: 4

Current Action: ExpandingCurrent Action: Backtracking

Current level: 0Current level: 1Current Action: Expanding

NM

Queue: F, G, H, I, J, K, L, M, NSize of Queue: 9

Nodes expanded: 5

E


Current level: 0Current Action: Expanding Current level: 1

O P

Queue: G, H, I, J, K, L, M, N, O, PSize of Queue: 10Nodes

expanded: 6

F


Current level: 0Current level: 1Current level: 2Current Action: Expanding

Queue: H, I, J, K, L, M, N, O, P, Q

Nodes expanded: 7

G


Current level: 1Current Action: Expanding

Queue: I, J, K, L, M, N, O, P, Q, R

Nodes expanded: 8

H


Current level: 2Current level: 1Current level: 0Current level: 1Current level: 2Current Action: Expanding

Queue: J, K, L, M, N, O, P, Q, R, S

Nodes expanded: 9

I


Current level: 1Current level: 2Current Action: Expanding

Queue: K, L, M, N, O, P, Q, R, S, T

Nodes expanded: 10

J


Current level: 1Current level: 0Current level: 1Current level: 2Current Action: Expanding

Queue: L, M, N, O, P, Q, R, S, T, U

Nodes expanded: 11

K


Current level: 1

LLLL

Node L is located and the search returns a solution.

Press space to end.

FINISHED SEARCH

Queue: EmptySize of Queue: 0

Current level: 2

BREADTH-FIRST SEARCH PATTERN

L

Press space to continue the searchPress space to continue the searchPress space to continue the searchPress space to continue the searchPress space to continue the searchPress space to continue the searchPress space to continue the searchPress space to continue the searchPress space to continue the search

Aside: Internet Search

• Typically human search will be “incomplete”, • E.g. finding information on the internet before

google, etc– look at a few web pages,– if no success then give up

Example

• Determine whether data-driven or goal-driven and depth-first or breadth-first would be preferable for solving each of the following– Diagnosing mechanical problems in an automobile– You have met a person who claims to be your

distant cousin, with a common ancestor named John. You like to verify her claim

– A theorem prover for plane geometry

• A program for examining sonar readings and interpreting them

• An expert system that will help a human

classify plants by species, genus,etc.

Example

Any path, versus shortest path, versus best path:

Ex.: Traveling salesperson problem:

• Find a sequence of cities ABCDEA such that the total distance is MINIMAL.

BostonBoston

MiamiMiami

NewYorkNewYork

SanFranciscoSanFrancisco

DallasDallas

30003000

250250

14501450

12001200 17001700

3300330029002900

15001500

16001600

17001700

55

Bi-directional search• IF you are able to EXPLICITLY describe the GOAL state, AND you have

BOTH rules for FORWARD reasoning AND BACKWARD reasoning• Compute the tree both from the start-node and from a goal node,

until these meet.

GoalGoalStartStart

Example Search Problem• A genetics professor– Wants to name her new baby boy– Using only the letters D,N & A

• Search through possible strings (states)– D,DN,DNNA,NA,AND,DNAN, etc.– 3 operators: add D, N or A onto end of string– Initial state is an empty string

• Goal test– Look up state in a book of boys’ names, e.g. DAN

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