CSE 573: Artificial Intelligencecourses.cs.washington.edu/courses/cse573/17wi/slides/18-CSP.pdf ·...

CSE573:ArtificialIntelligenceConstraintSatisfactionProblemsFactored(akaStructured)Search

[WithmanyslidesbyDanKleinandPieterAbbeel (UCBerkeley)availableathttp://ai.berkeley.edu.]

FinalPresentations§ 21groups/40people/110min

§ Minustransfers&tournamentreplay

§ Presentations(withquestions)§ Onepersongroups 2.5min§ Twopersongroups 4.5min§ Threepersongroups 6.5min

§ Everyoneshouldspeak(unlessOOT)§ Rehearse§ AddURLforslidestog-doc

§ https://docs.google.com/spreadsheets/d/1Qt5BW0DkSAg6Q4MOM98jSSwjR2wTZpi5i01XdT0X-fs/edit#gid=02

Finalreport

§ Defaultproject~2pages§ Otherprojects~6pages

§ Experiments§ Lessonslearned§ http://courses.cs.washington.edu/courses/cse573/17wi/reports.html

§ Everyone§ Seenoteonappendices– dynamics&externalcode

3

AITopics

§ Search§ Problemspaces§ BFS,DFS,UCS,A*(treeandgraph),localsearch§ CompletenessandOptimality§ Heuristics:admissibilityandconsistency;patternDBs

§ CSPs§ Constraintgraphs,backtrackingsearch§ Forwardchecking,AC3constraintpropagation,ordering

heuristics§ Games

§ Minimax,Alpha-betapruning,§ Expectimax§ EvaluationFunctions

§ MDPs§ Bellmanequations§ Valueiteration,policyiteration

§ Reinforcement Learning§ Exploration vs Exploitation§ Model-based vs. model-free§ Q-learning§ Linear value function approx.

§ Hidden Markov Models§ Markov chains, DBNs§ Forward algorithm§ Particle Filters

§ POMDPs§ Belief space§ Piecewise linear approximation to value fun

§ Beneficial AI§ Bayesian Networks

§ Basic definition, independence (d-sep)§ Variable elimination§ Sampling (rejection, importance)

§ Learning§ BN parameters with complete data§ Search thru space of BN structures§ Expectation maximization

Whatisintelligence?

§ (bounded)Rationality§ Agenthasaperformancemeasuretooptimize§ Givenitsstateofknowledge§ Chooseoptimalaction§ Withlimitedcomputationalresources

§ Human-likeintelligence/behavior

State-SpaceSearch§ Xasasearchproblem

§ states,actions,transitions,cost,goal-test§ Typesofsearch

§ uninformedsystematic:oftenslow§ DFS,BFS,uniform-cost,iterativedeepening

§ Heuristic-guided:better§ Greedybestfirst,A*§ Relaxationleadstoheuristics

§ Local: fast,fewerguarantees;oftenlocaloptimal§ Hillclimbingandvariations§ SimulatedAnnealing:globaloptimal

§ (Local)BeamSearch

WhichAlgorithm?

§ A*, Manhattan Heuristic:

AdversarialSearch

AdversarialSearch

§ AND/ORsearchspace(max,min)§ minimax objectivefunction§ minimax algorithm(~dfs)

§ alpha-betapruning

§ Utilityfunctionforpartialsearch§ Learningutilityfunctionsbyplayingwithitself

§ Openings/Endgamedatabases

PolicyIteration§ Leti =0§ Initializeπi(s)torandomactions§ Repeat

§ Step1:Policyevaluation:§ Initializek=0;Forall s,V0

π (s)=0§ RepeatuntilVπ converges

§ Foreachstates,

§ Letk+=1§ Step2:Policyimprovement:

§ Foreachstate,s,

§ Ifπi ==πi+1 thenit’soptimal;returnit.§ Elseleti +=1

Example

Initializeπ0to “alwaysgoright”

Performpolicyevaluation

PerformpolicyimprovementIteratethroughstates ?

?

?

Haspolicychanged?

Yes!i +=1

Example

π1says“alwaysgoup”

Performpolicyevaluation

PerformpolicyimprovementIteratethroughstates ?

?

?

Haspolicychanged?

No!Wehavetheoptimalpolicy

ReinforcementLearning

§ Forall s,a§ InitializeQ(s,a)=0

§ RepeatForeverWhere are you? s.Choose some action aExecute it in real world: transition =(s, a, r, s’)Do update:

ApproximateQ-Learning

§ Interpretationassearch§ Adjustweightsofactivefeatures§ E.g.,ifsomethingunexpectedlybadhappens,blametheactivefeatures

§ Forall s,a§ Initializewi=0

§ RepeatForeverWhere are you? s.Choose some action aExecute it in real world: transition =(s, a, r, s’)Do updates:

ApproximateQ-Learning

§ Q-learningwithlinearQ-functions:

§ Intuitiveinterpretation:§ Adjustweightsofactivefeatures§ E.g.,ifsomethingunexpectedlybadhappens,blamethefeaturesthatwereactive:

disprefer allstateswiththatstate’sfeatures

Old way: Exact Q’s

Now: Approximate Q’s

WhatisSearchFor?

§ Planning:sequencesofactions§ Thepathtothegoal istheimportantthing§ Pathshavevariouscosts,depths§ Assumelittleaboutproblemstructure

§ Identification:assignmentstovariables§ Thegoalitselfisimportant,notthepath§ Allpathsatthesamedepth(forsomeformulations)

ConstraintSatisfactionProblems

CSPs are structured (factored) identification problems


§ Standardsearchproblems:§ Stateisa“blackbox”:arbitrarydatastructure§ Goaltestcanbeanyfunctionoverstates§ Successorfunctioncanalsobeanything

§ Constraintsatisfactionproblems(CSPs):§ Aspecialsubsetofsearchproblems§ StateisdefinedbyvariablesXi withvaluesfroma

domainD (sometimesD dependsoni)§ Goaltestisasetofconstraintsspecifyingallowable

combinationsofvaluesforsubsetsofvariables

§ MakinguseofCSPformulationallowsforoptimizedalgorithms§ Typicalexampleoftradinggeneralityforutility(inthis

case,speed)


§ Constraintsatisfactionproblems(CSPs):§ Aspecialsubsetofsearchproblems§ StateisdefinedbyvariablesXi withvaluesfroma

domainD (sometimesD dependsoni)§ Goaltestisasetofconstraintsspecifyingallowable

combinationsofvaluesforsubsetsofvariables

§ “Factoring”thestatespace

§ Representingthestatespaceinaknowledgerepresentation

CSPExample:N-Queens

§ Formulation1:§ Variables:§ Domains:§ Constraints

CSPExample:N-Queens

§ Formulation2:§ Variables:

§ Domains:

§ Constraints:

Implicit:

Explicit:

CSPExample:Sudoku

§ Variables:§ Each(open)square

§ Domains:§ {1,2,…,9}

§ Constraints:

9-wayalldiff foreachrow9-wayalldiff foreachcolumn

9-wayalldiffforeachregion(orcanhaveabunchofpairwiseinequalityconstraints)

PropositionalLogic

§ Variables:§ Domains:§ Constraints:

propositionalvariables{T,F}logicalformula

CSPExample:MapColoring

§ Variables:

§ Domains:

§ Constraints:adjacentregionsmusthavedifferentcolors

§ Solutionsareassignmentssatisfyingallconstraints,e.g.:

Implicit:

Explicit:

ConstraintGraphs

ConstraintGraphs

§ BinaryCSP:eachconstraintrelates(atmost)twovariables

§ Binaryconstraintgraph:nodesarevariables,arcsshowconstraints

§ General-purposeCSPalgorithmsusethegraphstructuretospeedupsearch.E.g.,Tasmaniaisanindependentsubproblem!

Example:Cryptarithmetic

§ Variables:

§ Domains:

§ Constraints:

29

ChineseConstraintNetwork

Soup

Total Cost< $40

ChickenDish

Vegetable

RiceSeafood

Pork Dish

Appetizer

Must beHot&Sour

No Peanuts

No Peanuts

NotChow Mein

Not BothSpicy

Real-WorldCSPs

§ Assignmentproblems:e.g.,whoteacheswhatclass§ Timetablingproblems:e.g.,whichclassisofferedwhenandwhere?§ Hardwareconfiguration§ Gateassignmentinairports§ SpaceShuttleRepair§ Transportationscheduling§ Factoryscheduling§ …lotsmore!

Example:TheWaltzAlgorithm

§ TheWaltzalgorithmisforinterpretinglinedrawingsofsolidpolyhedra as3Dobjects

§ AnearlyexampleofanAIcomputationposedasaCSP

?

WaltzonSimpleScenes

§ Assumeallobjects:§ Havenoshadowsorcracks§ Three-facedvertices§ “Generalposition”:nojunctionschangewithsmallmovementsoftheeye.

§ Theneachlineonimageisoneofthefollowing:§ Boundaryline(edgeofanobject)(>)withrighthandofarrowdenoting“solid”andlefthanddenoting“space”

§ Interiorconvexedge(+)§ Interiorconcaveedge(-)

LegalJunctions

§ Onlycertainjunctionsarephysicallypossible§ HowcanweformulateaCSPtolabelanimage?§ Variables:edges§ Domains:>,<,+,-§ Constraints:legaljunctiontypes

SlightProblem:Localvs GlobalConsistency

37

VarietiesofCSPs

VarietiesofCSPVariables

§ DiscreteVariables§ Finitedomains

§ Sized meansO(dn) completeassignments§ E.g.,BooleanCSPs,includingBooleansatisfiability (NP-complete)

§ Infinitedomains(integers,strings,etc.)§ E.g.,jobscheduling,variablesarestart/endtimesforeachjob§ Linearconstraintssolvable,nonlinearundecidable

§ Continuousvariables§ E.g.,start/endtimesforHubbleTelescopeobservations§ Linearconstraintssolvableinpolynomialtimebylinear

programmethods(seeCSE521forabitofLPtheory)

VarietiesofCSPConstraints

§ VarietiesofConstraints§ Unaryconstraintsinvolveasinglevariable(equivalentto

reducingdomains),e.g.:

§ Binaryconstraintsinvolvepairsofvariables,e.g.:

§ Higher-orderconstraintsinvolve3ormorevariables:e.g.,cryptarithmetic columnconstraints

§ Preferences(softconstraints):§ E.g.,redisbetterthangreen§ Oftenrepresentable byacostforeachvariableassignment§ Givesconstrainedoptimizationproblems§ (We’llignoretheseuntilwegettoBayes’nets)

SolvingCSPs

CSPasSearch

§ States§ Operators§ InitialState§ GoalState

StandardDepthFirstSearch

StandardSearchFormulation

§ StandardsearchformulationofCSPs

§ Statesdefinedbythevaluesassignedsofar(partialassignments)§ Initialstate:theemptyassignment,{}§ Successorfunction:assignavaluetoanunassignedvariable

§ Goaltest:thecurrentassignmentiscompleteandsatisfiesallconstraints

§ We’llstartwiththestraightforward,naïveapproach,thenimproveit

BacktrackingSearch

BacktrackingSearch

§ BacktrackingsearchisthebasicuninformedalgorithmforsolvingCSPs

§ Idea1:Onevariableatatime§ Variableassignmentsarecommutative,sofixordering§ I.e.,[WA=redthenNT=green]sameas[NT=greenthenWA=red]§ Onlyneedtoconsiderassignmentstoasinglevariableateachstep

§ Idea2:Checkconstraintsasyougo§ I.e.consideronlyvalueswhichdonotconflictpreviousassignments§ Mighthavetodosomecomputationtochecktheconstraints§ “Incrementalgoaltest”

§ Depth-firstsearchwiththesetwoimprovementsiscalledbacktrackingsearch

§ Cansolven-queensforn» 25

BacktrackingExample

BacktrackingSearch

§ Whatarethechoicepoints?

[Demo:coloring-- backtracking]

BacktrackingSearch

§ Kindofdepthfirstsearch§ Isitcomplete?

ImprovingBacktracking

§ General-purposeideasgivehugegainsinspeed

§ Ordering:§ Whichvariableshouldbeassignednext?§ Inwhatordershoulditsvaluesbetried?

§ Filtering:Canwedetectinevitablefailureearly?

§ Structure:Canweexploittheproblemstructure?

Filtering

§ Filtering:Keeptrackofdomainsforunassignedvariablesandcrossoffbadoptions§ Forwardchecking:Crossoffvaluesthatviolateaconstraintwhenaddedtotheexisting

assignment

Filtering:ForwardChecking

WASANT Q

NSWV

[Demo:coloring-- forwardchecking]

Filtering:ConstraintPropagation

§ Forwardcheckingonlypropagatesinformationfromassignedtounassigned§ Itdoesn'tcatchwhentwounassignedvariableshavenoconsistentassignment:

§ NTandSAcannotbothbeblue!§ Whydidn’twedetectthisyet?§ Constraintpropagation:reasonfromconstrainttoconstraint

WA SA

NT Q

NSW

V

ConsistencyofaSingleArc

§ AnarcX® Yisconsistent iff foreveryxinthetailthereissomeyintheheadwhichcouldbeassignedwithoutviolatingaconstraint

§ Forwardchecking:Enforcingconsistencyofarcspointingtoeachnewassignment

Deletefromthetail!

WA SA

NT Q

NSW

V

ArcConsistencyofanEntireCSP§ Asimpleformofpropagationmakessureallarcsareconsistent:

§ Important:IfXlosesavalue,neighborsofXneedtoberechecked!§ Arcconsistencydetectsfailureearlier thanforwardchecking§ Canberunasapreprocessoror aftereachassignment§ What’sthedownside ofenforcingarcconsistency?

WA SANT Q

NSW

V

AC-3algorithmforArcConsistency

§ Runtime:O(n2d3),canbereducedtoO(n2d2)§ …butdetectingall possiblefutureproblemsisNP-hard– why?

[Demo:CSPapplet(madeavailablebyaispace.org)-- n-queens]

LimitationsofArcConsistency

§ Afterenforcingarcconsistency:§ Canhaveonesolutionleft§ Canhavemultiplesolutionsleft§ Canhavenosolutionsleft

(andnotknowit)

§ EvenwithArcConsistencyyoustillneedbacktrackingsearch!§ Couldrunatevenstepofthatsearch§ Usuallybettertorunitonce,beforesearch

Whatwentwronghere?

VideoofDemoColoring– BacktrackingwithForwardChecking–ComplexGraph

VideoofDemoColoring– BacktrackingwithArcConsistency–ComplexGraph

K-Consistency

K-Consistency§ Increasingdegreesofconsistency

§ 1-Consistency(NodeConsistency):Eachsinglevariable’sdomainhasavaluewhichmeetsthatvariablesunaryconstraints

§ 2-Consistency(ArcConsistency):Foreachpairofvariables,anyconsistentassignmenttoonecanbeextendedtotheother

§ 3-Consistency(PathConsistency):Foreverysetof3vars,anyconsistentassignmentto2ofthevariablescanbeextendedtothethirdvar

§ K-Consistency:Foreachknodes,anyconsistentassignmenttok-1canbeextendedtothekth node.

§ Higherkmoreexpensivetocompute

§ (Youneedtoknowthealgorithmfork=2case:arcconsistency)

StrongK-Consistency

§ Strongk-consistency:alsok-1,k-2,…1consistent

§ Claim:strongn-consistencymeanswecansolvewithoutbacktracking!

§ Why?§ Chooseanyassignmenttoanyvariable§ Chooseanewvariable§ By2-consistency,thereisachoiceconsistentwiththefirst§ Chooseanewvariable§ By3-consistency,thereisachoiceconsistentwiththefirst2§ …

Ordering

BacktrackingSearch

Ordering:MinimumRemainingValues

§ VariableOrdering:Minimumremainingvalues(MRV):§ Choosethevariablewiththefewestlegalleftvaluesinitsdomain

§ Whyminratherthanmax?§ Alsocalled“mostconstrainedvariable”§ “Fail-fast”ordering

§ Tie-breakeramongMRVvariables§ Whatistheveryfirststatetocolor?(Allhave3valuesremaining.)

§ Maximumdegreeheuristic:§ Choosethevariableparticipatinginthemostconstraintsonremainingvariables

§ Whymostratherthanfewestconstraints?

Ordering:MaximumDegree

Ordering:LeastConstrainingValue

§ ValueOrdering:LeastConstrainingValue§ Givenachoiceofvariable,choosetheleastconstrainingvalue

§ I.e.,theonethatrulesoutthefewestvaluesintheremainingvariables

§ Notethatitmaytakesomecomputationtodeterminethis!(E.g.,rerunningfiltering)

§ Whyleastratherthanmost?

§ Combiningtheseorderingideasmakes1000queensfeasible

RationaleforMRV,MD,LCV

§ Wewanttoenterthemostpromisingbranch,butwealsowanttodetectfailurequickly

§ MRV+MD:§ Choosethevariablethatismostlikelytocausefailure§ Itmustbeassignedatsomepoint,soifitisdoomedtofail,bettertofindoutsoon

§ LCV:§ Wehopeourearlyvaluechoicesdonotdoomustofailure§ Choosethevaluethatismostlikelytosucceed

Trapped

§ Pacman istrapped!Heissurroundedbymysteriouscorridors,eachofwhichleadstoeitherapit(P),aghost(G),oranexit(E).Inordertoescape,heneedstofigureoutwhichcorridors,ifany,leadtoanexitandfreedom,ratherthanthecertaindoomofapitoraghost.

§ Theonesignofwhatliesbehindthecorridorsisthewind:apitproducesastrongbreeze(S)andanexitproducesaweakbreeze(W),whileaghostdoesn’tproduceanybreezeatall.Unfortunately,Pacman cannotmeasurethestrengthofthebreezeataspecificcorridor.Instead,hecanstandbetweentwoadjacentcorridorsandfeelthemaxofthetwobreezes.Forexample,ifhestandsbetweenapitandanexithewillsenseastrong(S)breeze,whileifhestandsbetweenanexitandaghost,hewillsenseaweak(W)breeze.Themeasurementsforallintersectionsareshowninthefigurebelow.

§ Also,whilethetotalnumberofexitsmightbezero,one,ormore,Pacmanknowsthattwoneighboringsquareswillnotbothbeexits.

75

2 CSPs: Trapped Pacman

Pacman is trapped! He is surrounded by mysterious corridors, each of which leads to either a pit (P), a ghost(G), or an exit (E). In order to escape, he needs to figure out which corridors, if any, lead to an exit and freedom,rather than the certain doom of a pit or a ghost.

The one sign of what lies behind the corridors is the wind: a pit produces a strong breeze (S) and an exitproduces a weak breeze (W), while a ghost doesn’t produce any breeze at all. Unfortunately, Pacman cannotmeasure the strength of the breeze at a specific corridor. Instead, he can stand between two adjacent corridorsand feel the max of the two breezes. For example, if he stands between a pit and an exit he will sense a strong(S) breeze, while if he stands between an exit and a ghost, he will sense a weak (W) breeze. The measurementsfor all intersections are shown in the figure below.

Also, while the total number of exits might be zero, one, or more, Pacman knows that two neighboring squareswill not both be exits.

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Pacman models this problem using variables Xi for each corridor i and domains P, G, and E.

1. State the binary and/or unary constraints for this CSP (either implicitly or explicitly).

2. Cross out the values from the domains of the variables that will be deleted in enforcing arc consistency.

X1 P G E

X2 P G E

X3 P G E

X4 P G E

X5 P G E

X6 P G E

2

Variables?

Trapped

76





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X1 P G E

X2 P G E

X3 P G E

X4 P G E

X5 P G E

X6 P G E

2

Variables? X1, … X6Domains {P, G, E}

§ Pacman istrapped!Heissurroundedbymysteriouscorridors,eachofwhichleadstoeitherapit(P),aghost(G),oranexit(E).Inordertoescape,heneedstofigureoutwhichcorridors,ifany,leadtoanexitandfreedom,ratherthanthecertaindoomofapitoraghost.

§ Theonesignofwhatliesbehindthecorridorsisthewind:apitproducesastrongbreeze(S)andanexitproducesaweakbreeze(W),whileaghostdoesn’tproduceanybreezeatall.Unfortunately,Pacman cannotmeasurethestrengthofthebreezeataspecificcorridor.Instead,hecanstandbetweentwoadjacentcorridorsandfeelthemaxofthetwobreezes.Forexample,ifhestandsbetweenapitandanexithewillsenseastrong(S)breeze,whileifhestandsbetweenanexitandaghost,hewillsenseaweak(W)breeze.Themeasurementsforallintersectionsareshowninthefigurebelow.

§ Also,whilethetotalnumberofexitsmightbezero,one,ormore,Pacmanknowsthattwoneighboringsquareswillnotbothbeexits.

Trapped

§ Apitproducesastrongbreeze(S)andanexitproducesaweakbreeze(W),whileaghostdoesn’tproduceanybreezeatall.

§ Pacmanfeelsthemaxofthetwobreezes.§ thetotalnumberofexitsmightbezero,one,ormore,§ twoneighboringsquareswillnotbothbeexits.

77





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X1 P G E

X2 P G E

X3 P G E

X4 P G E

X5 P G E

X6 P G E

2

Variables? X1, … X6Domains {P, G, E}

Constraints?

Trapped


§ Pacman feelsthemaxofthetwobreezes.§ thetotalnumberofexitsmightbezero,one,ormore,§ twoneighboringsquareswillnotbothbeexits.

78





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X1 P G E

X2 P G E

X3 P G E

X4 P G E

X5 P G E

X6 P G E

2

Constraints?








X1 P G E

X2 P G E

X3 P G E

X4 P G E

X5 P G E

X6 P G E

2

X1 = P or X2= P

Xi = E nand Xi+1|7 = E

X3 = E or X4= EX5 = P or X6= PX2 = E or X3= EX4 = P or X5= P

X6 = P or X1= P

Also! X2 =/= PX3 =/= PX4 =/= P

Trapped



79





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X1 P G E

X2 P G E

X3 P G E

X4 P G E

X5 P G E

X6 P G E

2

Arc consistent?

Constraints?








X1 P G E

X2 P G E

X3 P G E

X4 P G E

X5 P G E

X6 P G E

2

X1 = P or X2= P



X6 = P or X1= P

Also! X2 =/= PX3 =/= PX4 =/= P

Trapped



80





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X1 P G E

X2 P G E

X3 P G E

X4 P G E

X5 P G E

X6 P G E

2

Arc consistent?

Constraints?








X1 P G E

X2 P G E

X3 P G E

X4 P G E

X5 P G E

X6 P G E

2

MRV heuristic?X1 = P or X2= P



X6 = P or X1= P

Also! X2 =/= PX3 =/= PX4 =/= P

Structure

ProblemStructure

§ Extremecase:independentsubproblems§ Example:Tasmaniaandmainlanddonotinteract

§ Independentsubproblems areidentifiableasconnectedcomponentsofconstraintgraph

§ Supposeagraphofnvariablescanbebrokenintosubproblems ofonlycvariables:§ Worst-casesolutioncostisO((n/c)(dc)),linearinn§ E.g.,n=80,d=2,c=20§ 280 =4billionyearsat10millionnodes/sec§ (4)(220)=0.4secondsat10millionnodes/sec

Tree-StructuredCSPs

§ Theorem:iftheconstraintgraphhasnoloops,theCSPcanbesolvedinO(nd2)time§ ComparetogeneralCSPs,whereworst-casetimeisO(dn)

§ Thispropertyalsoappliestoprobabilisticreasoning(later):anexampleoftherelationbetweensyntacticrestrictionsandthecomplexityofreasoning

Tree-StructuredCSPs§ Algorithmfortree-structuredCSPs:

§ Order:Choosearootvariable,ordervariablessothatparentsprecedechildren

§ Removebackward:Fori =n:2,applyRemoveInconsistent(Parent(Xi),Xi)§ Assignforward:Fori =1:n,assignXi consistentlywithParent(Xi)

§ Runtime:O(nd2)(why?)

Tree-StructuredCSPs

§ Claim1:Afterbackwardpass,allroot-to-leafarcsareconsistent§ Proof:EachX®YwasmadeconsistentatonepointandY’sdomaincouldnothave

beenreducedthereafter(becauseY’schildrenwereprocessedbeforeY)

§ Claim2:Ifroot-to-leafarcsareconsistent,forwardassignmentwillnotbacktrack§ Proof:Inductiononposition

§ Whydoesn’tthisalgorithmworkwithcyclesintheconstraintgraph?

§ Note:we’llseethisbasicideaagainwithBayes’nets

ConnectiontoBayesNets

BayesNetExample:AlarmNetwork

Burglary Earthqk

Alarm

Johncalls

Marycalls

B P(B)

+b 0.001

-b 0.999

E P(E)

+e 0.002

-e 0.998

B E A P(A|B,E)

+b +e +a 0.95+b +e -a 0.05+b -e +a 0.94+b -e -a 0.06-b +e +a 0.29-b +e -a 0.71-b -e +a 0.001-b -e -a 0.999

A J P(J|A)

+a +j 0.9+a -j 0.1-a +j 0.05-a -j 0.95

A M P(M|A)

+a +m 0.7+a -m 0.3-a +m 0.01-a -m 0.99

MoreComplexBayes’ Net:AutoDiagnosis

HiddenMarkovModel(TreeStructured)

§ AnHMMisdefinedby:§ Initialdistribution:§ Transitions:§ Emissions:

P(R1 )0.6

Rt-1 tf

P(Rt | Rt-1 )0.70.1

Rttf

P(Ut | Rt )0.90.2

ForwardAlgorithm

Umbr1=T Umbr2=T

Rain0 Rain1 Rain2

B(x0=r)=0.5

P(R1 )0.5

Rt-1 tf

P(Rt | Rt-1 )0.80.6

Rttf

P(Ut | Rt )0.90.3

B

0(Xt+1) =

X

xt

P (X 0|xt

)B(xt

)

B’(x1=r) = 0.7

B(x1=r)=0.875

B’(x2=r) = P(x2=r | x1=r)*0.875 + P(x2=r | x1=s)*0.125= 0.8*0.875 + 0.6*0.125= 0.775

B(x1=r)∝ 0.9*0.775=0.6975B(x1=s)∝ 0.3*0.225=0.0675

Divideby0.765tonormalizeB(x1=r)=0.912

B(Xt+1) /Xt+1 P (et+1|Xt+1)B0(Xt+1)

B(Xt+1) /Xt+1 P (et+1|Xt+1)B0(Xt+1)

MoreComplexHMMInference

§ ForwardBackward§ Computesmarginalprobabilitiesofall hiddenstatesgivensequenceofobservations

P(xt = value)

MoreComplexHMMInference

§ ForwardBackward§ Computesmarginalprobabilities ofallhiddenstatesgivensequenceofobservations

§ Viterbi§ Computesmostlikelysequenceofstates

valuevaluevaluevalue value

ImprovingStructure

NearlyTree-StructuredCSPs

§ Conditioning:instantiateavariable,pruneitsneighbors'domains

§ Cutset conditioning:instantiate(inallways)asetofvariablessuchthattheremainingconstraintgraphisatree

§ Cutset sizecgivesruntimeO((dc)(n-c)d2),veryfastforsmallc

Cutset Conditioning

SA

SA SA SA

Instantiatethecutset(allpossibleways)

ComputeresidualCSPforeachassignment

SolvetheresidualCSPs(treestructured)

Chooseacutset

Cutset Quiz

§ Findthesmallestcutset forthegraphbelow.

LocalSearchforCSPs

IterativeAlgorithmsforCSPs

§ Localsearchmethodstypicallyworkwith“complete”states,i.e.,allvariablesassigned

§ ToapplytoCSPs:§ Takeanassignmentwithunsatisfiedconstraints§ Operatorsreassignvariablevalues§ Nofringe!Liveontheedge.

§ Algorithm:Whilenotsolved,§ Variableselection:randomlyselectanyconflictedvariable§ Valueselection:min-conflictsheuristic:

§ Chooseavaluethatviolatesthefewestconstraints§ I.e.,hillclimbwithh(n)=totalnumberofviolatedconstraints

Example:4-Queens

§ States:4queensin4columns(44 =256states)§ Operators:movequeenincolumn§ Goaltest:noattacks§ Evaluation:c(n)=numberofattacks

[Demo:n-queens– iterativeimprovement(L5D1)][Demo:coloring– iterativeimprovement]

PerformanceofMin-Conflicts

§ Givenrandominitialstate,cansolven-queensinalmostconstanttimeforarbitrarynwithhighprobability(e.g.,n=10,000,000)!

§ Thesameappearstobetrueforanyrandomly-generatedCSPexcept inanarrowrangeoftheratio

Summary:CSPs

§ CSPsareaspecialkindofsearchproblem:§ Statesarepartialassignments§ Goaltestdefinedbyconstraints

§ Basicsolution:backtrackingsearch

§ Speed-ups:§ Ordering§ Filtering§ Structure(cutset conditioning)

§ Iterativemin-conflictsisofteneffectiveinpractice

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