W umpus W orld PEAS description

Logical a gents

Chapter7

Chapter71

Outline

♦Knowledge-basedagents

♦Wumpusworld

♦Logicingeneral—modelsandentailment

♦Propositional(Boolean)logic

♦Equivalence,validity,satisfiability

♦Inferencerulesandtheoremproving–forwardchaining–backwardchaining–resolution

Chapter72

Kno wledge bases

Inference engine

Knowledge basedomain−specific content domain−independent algorithms

Knowledgebase=setofsentencesinaformallanguage

Declarativeapproachtobuildinganagent(orothersystem):Tellitwhatitneedstoknow

ThenitcanAskitselfwhattodo—answersshouldfollowfromtheKB

Agentscanbeviewedattheknowledgeleveli.e.,whattheyknow,regardlessofhowimplemented

Orattheimplementationleveli.e.,datastructuresinKBandalgorithmsthatmanipulatethem

Chapter73

A simple kno wledge-based agen t

functionKB-Agent(percept)returnsanactionstatic:KB,aknowledgebaset,acounter,initially0,indicatingtime

Tell(KB,Make-Percept-Sentence(percept,t))action←Ask(KB,Make-Action-Query(t))Tell(KB,Make-Action-Sentence(action,t))t←t+1returnaction

Theagentmustbeableto:Representstates,actions,etc.IncorporatenewperceptsUpdateinternalrepresentationsoftheworldDeducehiddenpropertiesoftheworldDeduceappropriateactions

Chapter7

W umpus W orld PEAS description

Performancemeasuregold+1000,death-1000-1perstep,-10forusingthearrowEnvironmentSquaresadjacenttowumpusaresmellySquaresadjacenttopitarebreezyGlitteriffgoldisinthesamesquareShootingkillswumpusifyouarefacingitShootingusesuptheonlyarrowGrabbingpicksupgoldifinsamesquareReleasingdropsthegoldinsamesquare BreezeBreeze Breeze BreezeBreeze

Stench Stench BreezePIT

PIT

PIT

123 1 2 3 4

START Gold Stench

ActuatorsLeftturn,Rightturn,Forward,Grab,Release,Shoot

SensorsBreeze,Glitter,Smell

Chapter7

W umpus w orld c haracterization

Observable??

Chapter7

W umpus w orld c haracterization

Observable??No—onlylocalperception

Deterministic??

Chapter77

W umpus w orld c haracterization

Observable??No—onlylocalperception

Deterministic??Yes—outcomesexactlyspecified

Episodic??

Chapter78

W umpus w orld c haracterization

Observable??No—onlylocalperception

Deterministic??Yes—outcomesexactlyspecified

Episodic??No—sequentialatthelevelofactions

Static??

Chapter79

W umpus w orld c haracterization

Observable??No—onlylocalperception

Deterministic??Yes—outcomesexactlyspecified

Episodic??No—sequentialatthelevelofactions

Static??Yes—WumpusandPitsdonotmove

Discrete??

Chapter710

W umpus w orld c haracterization

Observable??No—onlylocalperception

Deterministic??Yes—outcomesexactlyspecified

Episodic??No—sequentialatthelevelofactions

Static??Yes—WumpusandPitsdonotmove

Discrete??Yes

Single-agent??

Chapter711

W umpus w orld c haracterization

Observable??No—onlylocalperception

Deterministic??Yes—outcomesexactlyspecified

Episodic??No—sequentialatthelevelofactions

Static??Yes—WumpusandPitsdonotmove

Discrete??Yes

Single-agent??Yes—Wumpusisessentiallyanaturalfeature

Chapter712

Exploring a wumpus w orld

A OK

OKOK

Chapter713

Exploring a wumpus w orld

OK

OKOK A

A B

Chapter714

Exploring a wumpus w orld

OK

OKOK A

A B P?

P?

Chapter715

Exploring a wumpus w orld

OK

OKOK A

A B P?

P?A S

Chapter7

Exploring a wumpus w orld

OK

OKOK A

A B P?

P?A S OK

P

W

Chapter7

Exploring a wumpus w orld

OK

OKOK A

A B P?

P?A S OK

P

W

Chapter7

Exploring a wumpus w orld

OK

OKOK A

A B P?

P?A S OK

P

W

Chapter719

Exploring a wumpus w orld

OK

OKOK A

A B P?

P?A S OK

P

W

A BGS

Chapter720

Other tigh t sp ots

A BOK

OKOK

A B

A P?

P?P?

P? Breezein(1,2)and(2,1)⇒nosafeactions

Assumingpitsuniformlydistributed,(2,2)haspitw/prob0.86,vs.0.31

A S Smellin(1,1)⇒cannotmoveCanuseastrategyofcoercion:shootstraightaheadwumpuswasthere⇒dead⇒safewumpuswasn’tthere⇒safe

Chapter721

Logic in general

Logicsareformallanguagesforrepresentinginformationsuchthatconclusionscanbedrawn

Syntaxdefinesthesentencesinthelanguage

Semanticsdefinethe“meaning”ofsentences;i.e.,definetruthofasentenceinaworld

E.g.,thelanguageofarithmetic

x+2≥yisasentence;x2+y>isnotasentence

x+2≥yistrueiffthenumberx+2isnolessthanthenumbery

x+2≥yistrueinaworldwherex=7,y=1x+2≥yisfalseinaworldwherex=0,y=6

Chapter722

En tailmen t

Entailmentmeansthatonethingfollowsfromanother:

KB|=α

KnowledgebaseKBentailssentenceαifandonlyifαistrueinallworldswhereKBistrue

E.g.,theKBcontaining“theGiantswon”and“theRedswon”entails“EithertheGiantswonortheRedswon”

E.g.,x+y=4entails4=x+y

Entailmentisarelationshipbetweensentences(i.e.,syntax)thatisbasedonsemantics

Note:brainsprocesssyntax(ofsomesort)

Chapter723

Mo dels

Logicianstypicallythinkintermsofmodels,whichareformallystructuredworldswithrespecttowhichtruthcanbeevaluated

Wesaymisamodelofasentenceαifαistrueinm

M(α)isthesetofallmodelsofα

ThenKB|=αifandonlyifM(KB)⊆M(α)

E.g.KB=GiantswonandRedswonα=GiantswonM( )

M(KB) x

x

x x x

x xx x

xxxxx x xxx

xxxxx xx xx xxx x x x xxx

x

xx

x x

xx

x xx

x

Chapter724

En tailmen t in the wumpus w orld

Situationafterdetectingnothingin[1,1],movingright,breezein[2,1]

Considerpossiblemodelsfor?sassumingonlypits AA B

? ? ?

3Booleanchoices⇒8possiblemodels

Chapter725

W umpus mo dels

123 1 2

Breeze PIT 123 1 2

Breeze PIT

123 1 2

Breeze PITPIT

PIT 123 1 2

Breeze PIT

PIT 123 1 2

BreezePIT

123 1 2

Breeze PIT

PIT

123 1 2

Breeze PITPIT 123 1 2

Breeze

Chapter726

W umpus mo dels

123 1 2

Breeze PIT 123 1 2

Breeze PIT

123 1 2

Breeze PITPIT

PIT 123 1 2

Breeze PIT

PIT 123 1 2

BreezePIT

123 1 2

Breeze PIT

PIT

123 1 2

Breeze PITPIT 123 1 2

Breeze KB

KB=wumpus-worldrules+observations

Chapter727

W umpus mo dels

123 1 2

Breeze PIT 123 1 2

Breeze PIT

123 1 2

Breeze PITPIT

PIT 123 1 2

Breeze PIT

PIT 123 1 2

BreezePIT

123 1 2

Breeze PIT

PIT

123 1 2

Breeze PITPIT 123 1 2

Breeze KB1

KB=wumpus-worldrules+observations

α1=“[1,2]issafe”,KB|=α1,provedbymodelchecking

Chapter7

W umpus mo dels

123 1 2

Breeze PIT 123 1 2

Breeze PIT

123 1 2

Breeze PITPIT

PIT 123 1 2

Breeze PIT

PIT 123 1 2

BreezePIT

123 1 2

Breeze PIT

PIT

123 1 2

Breeze PITPIT 123 1 2

Breeze KB

KB=wumpus-worldrules+observations

Chapter7

W umpus mo dels

123 1 2

Breeze PIT 123 1 2

Breeze PIT

123 1 2

Breeze PITPIT

PIT 123 1 2

Breeze PIT

PIT 123 1 2

BreezePIT

123 1 2

Breeze PIT

PIT

123 1 2

Breeze PITPIT 123 1 2

Breeze KB 2

KB=wumpus-worldrules+observations

α2=“[2,2]issafe”,KB6|=α2

Chapter7

Inference

KB`iα=sentenceαcanbederivedfromKBbyprocedurei

ConsequencesofKBareahaystack;αisaneedle.Entailment=needleinhaystack;inference=findingit

Soundness:iissoundifwheneverKB`iα,itisalsotruethatKB|=α Completeness:iiscompleteifwheneverKB|=α,itisalsotruethatKB`iα

Preview:wewilldefinealogic(first-orderlogic)whichisexpressiveenoughtosayalmostanythingofinterest,andforwhichthereexistsasoundandcompleteinferenceprocedure.

Thatis,theprocedurewillansweranyquestionwhoseanswerfollowsfromwhatisknownbytheKB.

Chapter731

Prop ositional logic: Syn tax

Propositionallogicisthesimplestlogic—illustratesbasicideas

ThepropositionsymbolsP1,P2etcaresentences

IfSisasentence,¬Sisasentence(negation)

IfS1andS2aresentences,S1∧S2isasentence(conjunction) IfS1andS2aresentences,S1∨S2isasentence(disjunction) IfS1andS2aresentences,S1⇒S2isasentence(implication) IfS1andS2aresentences,S1⇔S2isasentence(biconditional)

Chapter732

Prop ositional logic: Seman tics

Eachmodelspecifiestrue/falseforeachpropositionsymbol

E.g.P1,2P2,2P3,1truetruefalse

(Withthesesymbols,8possiblemodels,canbeenumeratedautomatically.)

Rulesforevaluatingtruthwithrespecttoamodelm:

¬SistrueiffSisfalseS1∧S2istrueiffS1istrueandS2istrueS1∨S2istrueiffS1istrueorS2istrueS1⇒S2istrueiffS1isfalseorS2istruei.e.,isfalseiffS1istrueandS2isfalseS1⇔S2istrueiffS1⇒S2istrueandS2⇒S1istrue Simplerecursiveprocessevaluatesanarbitrarysentence,e.g.,¬P1,2∧(P2,2∨P3,1)=true∧(false∨true)=true∧true=true

Chapter733

T ruth tables for connectiv es

PQ¬PP∧QP∨QP⇒QP⇔Qfalsefalsetruefalsefalsetruetruefalsetruetruefalsetruetruefalsetruefalsefalsefalsetruefalsefalsetruetruefalsetruetruetruetrue

Chapter734

W umpus w orld sen tences

LetPi,jbetrueifthereisapitin[i,j].LetBi,jbetrueifthereisabreezein[i,j].

¬P1,1¬B1,1B2,1

“Pitscausebreezesinadjacentsquares”

Chapter735

W umpus w orld sen tences

LetPi,jbetrueifthereisapitin[i,j].LetBi,jbetrueifthereisabreezein[i,j].

¬P1,1¬B1,1B2,1

“Pitscausebreezesinadjacentsquares”

B1,1⇔(P1,2∨P2,1)B2,1⇔(P1,1∨P2,2∨P3,1)

“Asquareisbreezyifandonlyifthereisanadjacentpit”

Chapter736

T ruth tables for inference

B1,1B2,1P1,1P1,2P2,1P2,2P3,1R1R2R3R4R5KBfalsefalsefalsefalsefalsefalsefalsetruetruetruetruefalsefalsefalsefalsefalsefalsefalsefalsetruetruetruefalsetruefalsefalse.......................................

falsetruefalsefalsefalsefalsefalsetruetruefalsetruetruefalsefalsetruefalsefalsefalsefalsetruetruetruetruetruetruetruefalsetruefalsefalsefalsetruefalsetruetruetruetruetruetruefalsetruefalsefalsefalsetruetruetruetruetruetruetruetruefalsetruefalsefalsetruefalsefalsetruefalsefalsetruetruefalse.......................................

truetruetruetruetruetruetruefalsetruetruefalsetruefalse

Enumeraterows(differentassignmentstosymbols),ifKBistrueinrow,checkthatαistoo

Chapter737

Inference b y en umeration

Depth-firstenumerationofallmodelsissoundandcomplete

functionTT-Entails?(KB,α)returnstrueorfalseinputs:KB,theknowledgebase,asentenceinpropositionallogicα,thequery,asentenceinpropositionallogic

symbols←alistofthepropositionsymbolsinKBandαreturnTT-Check-All(KB,α,symbols,[])

functionTT-Check-All(KB,α,symbols,model)returnstrueorfalseifEmpty?(symbols)thenifPL-True?(KB,model)thenreturnPL-True?(α,model)elsereturntrueelsedoP←First(symbols);rest←Rest(symbols)returnTT-Check-All(KB,α,rest,Extend(P,true,model))andTT-Check-All(KB,α,rest,Extend(P,false,model))

O(2n)fornsymbols;problemisco-NP-complete

Chapter738

Logical equiv alence

Twosentencesarelogicallyequivalentifftrueinsamemodels:α≡βifandonlyifα|=βandβ|=α

(α∧β)≡(β∧α)commutativityof∧(α∨β)≡(β∨α)commutativityof∨((α∧β)∧γ)≡(α∧(β∧γ))associativityof∧((α∨β)∨γ)≡(α∨(β∨γ))associativityof∨¬(¬α)≡αdouble-negationelimination(α⇒β)≡(¬β⇒¬α)contraposition(α⇒β)≡(¬α∨β)implicationelimination(α⇔β)≡((α⇒β)∧(β⇒α))biconditionalelimination¬(α∧β)≡(¬α∨¬β)DeMorgan¬(α∨β)≡(¬α∧¬β)DeMorgan(α∧(β∨γ))≡((α∧β)∨(α∧γ))distributivityof∧over∨(α∨(β∧γ))≡((α∨β)∧(α∨γ))distributivityof∨over∧

Chapter739

V alidit y and satisfiabilit y

Asentenceisvalidifitistrueinallmodels,e.g.,True,A∨¬A,A⇒A,(A∧(A⇒B))⇒B

ValidityisconnectedtoinferenceviatheDeductionTheorem:KB|=αifandonlyif(KB⇒α)isvalid

Asentenceissatisfiableifitistrueinsomemodele.g.,A∨B,C

Asentenceisunsatisfiableifitistrueinnomodelse.g.,A∧¬A

Satisfiabilityisconnectedtoinferenceviathefollowing:KB|=αifandonlyif(KB∧¬α)isunsatisfiablei.e.,proveαbyreductioadabsurdum

Chapter7

Pro of metho ds

Proofmethodsdivideinto(roughly)twokinds:

Applicationofinferencerules–Legitimate(sound)generationofnewsentencesfromold–Proof=asequenceofinferenceruleapplicationsCanuseinferencerulesasoperatorsinastandardsearchalg.–Typicallyrequiretranslationofsentencesintoanormalform

Modelcheckingtruthtableenumeration(alwaysexponentialinn)improvedbacktracking,e.g.,Davis–Putnam–Logemann–Lovelandheuristicsearchinmodelspace(soundbutincomplete)e.g.,min-conflicts-likehill-climbingalgorithms

Chapter7

F orw ard and bac kw ard c haining

HornForm(restricted)KB=conjunctionofHornclausesHornclause=♦propositionsymbol;or♦(conjunctionofsymbols)⇒symbolE.g.,C∧(B⇒A)∧(C∧D⇒B)

ModusPonens(forHornForm):completeforHornKBs

α1,...,αn,α1∧···∧αn⇒ββ

Canbeusedwithforwardchainingorbackwardchaining.Thesealgorithmsareverynaturalandruninlineartime

Chapter7

F orw ard c haining

Idea:fireanyrulewhosepremisesaresatisfiedintheKB,additsconclusiontotheKB,untilqueryisfound

P⇒Q

L∧M⇒P

B∧L⇒MA∧P⇒L

A∧B⇒L

A

B QP

M

L

BA

Chapter743

F orw ard c haining algorithm

functionPL-FC-Entails?(KB,q)returnstrueorfalseinputs:KB,theknowledgebase,asetofpropositionalHornclausesq,thequery,apropositionsymbollocalvariables:count,atable,indexedbyclause,initiallythenumberofpremisesinferred,atable,indexedbysymbol,eachentryinitiallyfalseagenda,alistofsymbols,initiallythesymbolsknowninKB

whileagendaisnotemptydop←Pop(agenda)unlessinferred[p]doinferred[p]←trueforeachHornclausecinwhosepremisepappearsdodecrementcount[c]ifcount[c]=0thendoifHead[c]=qthenreturntruePush(Head[c],agenda)returnfalse

Chapter744

F orw ard c haining example

QP

M

L

BA 22 2 2 1

Chapter745

F orw ard c haining example

QP

M

L

B 2 1

A 11 2

Chapter746

F orw ard c haining example

QP

M 2 1

A 1

B 0 1L

Chapter747

F orw ard c haining example

QP

M 1

A 1

B 0 L 0 1

Chapter748

F orw ard c haining example

Q1

A 1

B 0 L 0 M 0 P

Chapter749

F orw ard c haining example

QAB 0 L 0 M 0 P 0

0

Chapter750

F orw ard c haining example

QAB 0 L 0 M 0 P 0

0

Chapter751

F orw ard c haining example

AB 0 L 0 M 0 P 0

0 Q

Chapter7

Pro of of completeness

FCderiveseveryatomicsentencethatisentailedbyKB

1.FCreachesafixedpointwherenonewatomicsentencesarederived

2.Considerthefinalstateasamodelm,assigningtrue/falsetosymbols

3.EveryclauseintheoriginalKBistrueinmProof:Supposeaclausea1∧...∧ak⇒bisfalseinmThena1∧...∧akistrueinmandbisfalseinmThereforethealgorithmhasnotreachedafixedpoint!

4.HencemisamodelofKB

5.IfKB|=q,qistrueineverymodelofKB,includingm

Generalidea:constructanymodelofKBbysoundinference,checkα

Chapter7

Bac kw ard c haining

Idea:workbackwardsfromthequeryq:toproveqbyBC,checkifqisknownalready,orprovebyBCallpremisesofsomeruleconcludingq

Avoidloops:checkifnewsubgoalisalreadyonthegoalstack

Avoidrepeatedwork:checkifnewsubgoal1)hasalreadybeenprovedtrue,or2)hasalreadyfailed

Chapter7

Bac kw ard c haining example

QP

M

L

AB

Chapter755

Bac kw ard c haining example

P

M

L

A Q

B

Chapter756

Bac kw ard c haining example

M

L

A QP

B

Chapter757

Bac kw ard c haining example

M

A QP

L

B

Chapter758

Bac kw ard c haining example

M

L

A QP

B

Chapter759

Bac kw ard c haining example

M

A QP

L

B

Chapter760

Bac kw ard c haining example

M

A QP

L

B

Chapter761

Bac kw ard c haining example

A QP

L

B M

Chapter762

Bac kw ard c haining example

A QP

L

B M

Chapter763

Bac kw ard c haining example

A QP

L

B M

Chapter7

Bac kw ard c haining example

A QP

L

B M

Chapter7

F orw ard vs. bac kw ard c haining

FCisdata-driven,cf.automatic,unconsciousprocessing,e.g.,objectrecognition,routinedecisions

Maydolotsofworkthatisirrelevanttothegoal

BCisgoal-driven,appropriateforproblem-solving,e.g.,Wherearemykeys?HowdoIgetintoaPhDprogram?

ComplexityofBCcanbemuchlessthanlinearinsizeofKB

Chapter7

Resolution

ConjunctiveNormalForm(CNF—universal)conjunctionofdisjunctionsofliterals|{z}clausesE.g.,(A∨¬B)∧(B∨¬C∨¬D)

Resolutioninferencerule(forCNF):completeforpropositionallogic

`1∨···∨`k,m1∨···∨mn`1∨···∨`i−1∨`i+1∨···∨`k∨m1∨···∨mj−1∨mj+1∨···∨mn

where`iandmjarecomplementaryliterals.E.g.,

OK

OKOK A

A B P?

P?

A S OK P

W A P1,3∨P2,2,¬P2,2P1,3

Resolutionissoundandcompleteforpropositionallogic

Chapter767

Con v ersion to CNF

B1,1⇔(P1,2∨P2,1)

1.Eliminate⇔,replacingα⇔βwith(α⇒β)∧(β⇒α).

(B1,1⇒(P1,2∨P2,1))∧((P1,2∨P2,1)⇒B1,1)

2.Eliminate⇒,replacingα⇒βwith¬α∨β.

(¬B1,1∨P1,2∨P2,1)∧(¬(P1,2∨P2,1)∨B1,1)

3.Move¬inwardsusingdeMorgan’srulesanddouble-negation:

(¬B1,1∨P1,2∨P2,1)∧((¬P1,2∧¬P2,1)∨B1,1)

4.Applydistributivitylaw(∨over∧)andflatten:

(¬B1,1∨P1,2∨P2,1)∧(¬P1,2∨B1,1)∧(¬P2,1∨B1,1)

Chapter768

Resolution algorithm

Proofbycontradiction,i.e.,showKB∧¬αunsatisfiable

functionPL-Resolution(KB,α)returnstrueorfalseinputs:KB,theknowledgebase,asentenceinpropositionallogicα,thequery,asentenceinpropositionallogic

clauses←thesetofclausesintheCNFrepresentationofKB∧¬αnew←{}loopdoforeachCi,Cjinclausesdoresolvents←PL-Resolve(Ci,Cj)ifresolventscontainstheemptyclausethenreturntruenew←new∪resolventsifnew⊆clausesthenreturnfalseclauses←clauses∪new

Chapter769

Resolution example

KB=(B1,1⇔(P1,2∨P2,1))∧¬B1,1α=¬P1,2

P1,2 P1,2

P2,1 P1,2B1,1

B1,1P2,1B1,1P1,2P2,1P2,1 P1,2B1,1B1,1 P1,2B1,1P2,1B1,1P2,1B1,1

P1,2P2,1P1,2

Chapter770

Summary

Logicalagentsapplyinferencetoaknowledgebasetoderivenewinformationandmakedecisions

Basicconceptsoflogic:–syntax:formalstructureofsentences–semantics:truthofsentenceswrtmodels–entailment:necessarytruthofonesentencegivenanother–inference:derivingsentencesfromothersentences–soundess:derivationsproduceonlyentailedsentences–completeness:derivationscanproduceallentailedsentences

Wumpusworldrequirestheabilitytorepresentpartialandnegatedinforma-tion,reasonbycases,etc.

Forward,backwardchainingarelinear-time,completeforHornclausesResolutioniscompleteforpropositionallogic

Propositionallogiclacksexpressivepower

Chapter771