02 Solving Problems by Searching (Us)

8/9/2019 02 Solving Problems by Searching (Us)

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Artificial Intelligence

Solving problems by searching

Fall 2008

professor: Luigi Ceccaroni


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Problem solving

e !ant:

"o automatically solve a problem

e nee#:

A representation of the problem

Algorithms that use some strategy to solve

the problem #efine# in that representation


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Problem representation

$eneral:

% State space: a problem is #ivi#e# into a set

of resolution steps from the initial state to the

goal state

% Reduction to sub-problems: a problem is

arrange# into a hierarchy of sub&problems

Specific: $ame resolution

Constraints satisfaction


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States

A problem is #efine# by its elements an# theirrelations'

In each instant of the resolution of a problem(

those elements have specific #escriptors )*o!to select them+, an# relations'

A stateis a representation of those elements ina given moment'

"!o special states are #efine#:% Initial state)starting point,

% Final state)goal state,


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State mo#ification:

successor function A successor function is nee#e# to move

bet!een #ifferent states' A successor functionis a #escription of

possible actions( a set of operators' It is atransformation function on a staterepresentation( !hich convert it into anotherstate'

"he successor function #efines a relation of

accessibility among states' -epresentation of the successor function:

Con#itions of applicability "ransformation function


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State space

"he state spaceis the set of all states

reachable from the initial state'

It forms a graph )or map, in !hich the no#es are

states an# the arcs bet!een no#es are actions' A pathin the state space is a se.uence of

states connecte# by a se.uence of actions'

"he solution of the problem is part of the mapforme# by the state space'


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Problem solution

A solutionin the state space is a path from theinitial state to a goal state or( sometimes( /ust agoal state'

Path/solution cost: function that assigns anumeric cost to each path( the cost of applyingthe operators to the states

Solution .uality is measure# by the path costfunction( an# an optimal solutionhas the

lo!est path cost among all solutions' Solutions: any( an optimal one( all' Cost is

important #epen#ing on the problem an# thetype of solution sought'


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Problem #escription

Components: State space )e1plicitly or implicitly #efine#, Initial state $oal state )or the con#itions it has to fulfill, Available actions )operators to change state, -estrictions )e'g'( cost, lements of the #omain !hich are relevant to the

problem )e'g'( incomplete 3no!le#ge of the starting

point, "ype of solution:

Se.uence of operators or goal state

Any( an optimal one )cost #efinition nee#e#,( all


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1ample: 8&pu44le

8

2 3

4

1

6

!


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1ample: 8&pu44le

State space: configuration of the eighttiles on the boar#

Initial state: any configuration

"oal state: tiles in a specific or#er

#perators or actions: 5blan3 moves6 Con#ition: the move is !ithin the boar#

"ransformation: blan3 moves Left( Right( Up(or Down

Solution: optimal se.uence of operators


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1ample: n.ueens

)n 7 ( n 7 8,


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1ample: n.ueens

)n 7 ( n 7 8, State space: configurations from 0 to n .ueens

on the boar# !ith only one .ueen per ro! an#column

Initial state: configuration !ithout .ueens on the

boar# "oal state: configuration !ith n .ueens such that

no .ueen attac3s any other #perators or actions: place a .ueen on the

boar# Con#ition: the ne! .ueen is not attac3e# by any

other alrea#y place# "ransformation: place a ne! .ueen in a particular

s.uare of the boar#

Solution: one solution )cost is not consi#ere#,


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Search as goal satisfaction

Satisfying a goal

Agent 3no!s !hat the goal is

Agent cannot evaluate interme#iate solutions

)uninforme#,

"he environment is:

Static

bservable

9eterministic


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1ample: holi#ay in -omania

n holi#ay in -omania; currently in Ara#

Flight leaves tomorro! from :00 Let?s configure this to be an AI problem


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-omania

hat?s the problem+

Accomplish a goal

-each :00

So this is a goal&base# problem


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-omania

hat?s an e1ample of a non&goal&base#

problem+

Live long an# prosper

@a1imi4e the happiness of your trip to

-omania

9on?t get hurt too much


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-omania

hat .ualifies as a solution+

ou canBcannot reach :00

"he actions one ta3es to travel from Ara# to


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-omania

hat a##itional information #oes one

nee#+

A map


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A state space Which cities could you bein?

An initial state Which city do you start

from?A goal state

Which city do you aim to

reach?

A function #efining state

transitions

When in city foo, the

following cities can be

reached

A function #efining the

5cost6 of a state

se.uence

How long does it take to

travel through a city

sequence?

@ore concrete problem

#efinition


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@ore concrete problem

#efinition

A state space Choose a representation

An initial stateChoose an element from the

representation

A goal stateCreate goal_function(state)such

that TR! is returned upon reaching

goal

A function #efining state

transitions

successor_function(statei)"

#$actiona, statea%, $actionb, stateb%,

&'

A function #efining the 5cost6 of a

state se.uencecost (sequence) " number


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Important notes about this

e1ample Static environment )available states(

successor function( an# cost functions #on?t

change,

bservable )the agent 3no!s !here it is, 9iscrete )the actions are #iscrete,

9eterministic )successor function is al!ays

the same,


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"ree search algorithms


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Implementation: general search

algorithm,l*orithm$eneral Search

penDstates'insert )InitialDstate,

Current7 penDstates'first),

$hile notisDfinal+)Current, and notpenDstates'empty+), do

penDstates'#eleteDfirst),

Close#Dstates'insert)Current,

Successors7 generateDsuccessors)Current,

Successors7 processDrepeate#)Successors( Close#Dstates(

penDstates,

penDstates'insert)Successors,

Current7 penDstates'first),

ehile

e,l*orithm


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,rad

1ample: Ara#


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,rad

.erind %!& imisoara %118& Sibiu %140&

1ample: Ara#


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,rad


1ample: Ara#


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,rad


radea %1!1& Fara*as %& Rimnicu ilcea %80&

1ample: Ara#


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Implementation: states vs'

no#es State

)-epresentation of, a physical configuration

Eo#e

9ata structure constituting part of a search

tree

Inclu#esparent, children, depth, path costg)1,

States #o not have parents( chil#ren(#epth( or path cost


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Search strategies

A strategy is #efine# by pic3ing the or#er of no#e

e1pansion

Strategies are evaluate# along the follo!ing

#imensions: Completeness % #oes it al!ays fin# a solution if

one e1ists+

"ime comple1ity % number of no#es

generate#Be1pan#e#

Space comple1ity % ma1imum no#es in memory

ptimality % #oes it al!ays fin# a least&cost

solution+


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Search strategies

"ime an# space comple1ity are measure# in

terms of:

b % ma1imum branching factor of the search tree

)may be infinite, # % #epth of the least&cost solution

m % ma1imum #epth of the state space )may be

infinite,


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Gninforme# Search Strategies

Gninforme# strategies use only the

information available in the problem

#efinition


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>H

Eo#es

pen no#es:

$enerate#( but not yet e1plore#

1plore#( but not yet e1pan#e#

Close# no#es:

1plore# an# e1pan#e#

>H


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Space cost of


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Properties of brea#th&first

search Complete+

es )if b )ma1 branch factor,is finite,

"ime+

= b b2

J b#

b)b#

&=, 7 )b#=

,( i'e'( e1ponential in # Space+

)b#=, )3eeps every no#e in memory,

ptimal+

nly if cost 7 = per step( other!ise not optimal in general Space is the big problem; it can easily generate no#es at

=0 @


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=

9epth&limite# search

It is #epth&first search !ith an impose#

limit on the #epth of e1ploration( to

guarantee that the algorithm en#s'

=


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2

"reatment of repeate# states


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Iterative #eepening search


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H


"he algorithm consists of iterative( #epth&first

searches( !ith a ma1imum #epth that increases at

each iteration' @a1imum #epth at the beginning is ='


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Summary

All uninforme# searching techni.ues are more ali3e

than #ifferent'


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8

Gninforme# vs' informe#

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02 Solving Problems by Searching (Us)