Causal Action Theories and Satis ability Planning The University of

125 Pages·2007·0.91 MB·English

Causal Action Theories and Satis(cid:12)ability Planning
by
CharlesHudsonTurner,B.A.,M.L.I.S., M.S.C.S.
Copyright
Dissertation
by
PresentedtotheFacultyoftheGraduateSchoolof
CharlesHudsonTurner
TheUniversityofTexasatAustin
1998
inPartialFul(cid:12)llment
oftheRequirements
fortheDegreeof
DoctorofPhilosophy
The University of Texas at Austin
August1998
Causal Action Theories and Satis(cid:12)ability Planning
Approvedby
DissertationCommittee:
Thisdissertationisdedicatedtomywife,CarolGeorge,
andmyparents,CharlesandClariceTurner.
Causal Action Theories and Satis(cid:12)ability Planning
PublicationNo.
Acknowledgments
CharlesHudsonTurner,Ph.D.
TheUniversityofTexasatAustin,1998
IamdeeplygratefultoVladimirLifschitz,who,amongotherthings,madethiswork
possibleforme,andtoNormMcCain,whocarefullyandpatientlydiscussedwith
memanyoftheideasastheytookshape. Iamgratefulaswellforthehelpand Supervisor:VladimirLifschitz
encouragementofmanyotherfriends,teachersandcolleagues.
Thisdissertationaddressestheproblemofrepresentingandreasoningaboutcom-
monsenseknowledge of action domains. Untilrecently, most suchwork has sup-
pressed the notion of causality, despite its central role in everyday talking and
CharlesHudsonTurner
reasoning about actions. There is good reason for this. In general, causality is
adi(cid:14)cultnotion,bothphilosophicallyandmathematically. Nonetheless, itturns
TheUniversityofTexasatAustin
outthatactionrepresentationscanbemadenotonlymoreexpressivebutalsomath-
August1998
ematicallysimplerbyrepresentingcausalitymoreexplicitly.Thekeyistoformalize
onlyarelativelysimplekindofcausalknowledge: knowledgeoftheconditionsun-
derwhichfacts are caused. Inthe (cid:12)rstpartofthe dissertationwe dothisusing
inferencerulesandrule-basednonmonotonicformalisms. Asweshow,aninference
(cid:30)
rule canbeunderstoodtorepresenttheknowledgethatif(cid:30)iscausedthen is

caused. (Noticethatwedonotsay\(cid:30)causes .") Thisleadstosimpleandex-
pressiveactionrepresentationsinReiter'sdefaultlogic,arule-basednonmonotonic
formalism.Thisapproachalsoyieldsactiondescriptionsinlogicprogramming,thus
raisingthepossibility,atleastinprinciple,ofautomatedreasoningaboutactions
andplanning.Inthesecondpartofthedissertation,weintroduceanewmodalnon-
v vi
monotoniclogic|thelogicof\universalcausation"(UCL)|speci(cid:12)callydesignedfor
describingtheconditionsunderwhichfacts arecaused. We showthat UCLpro-
videsamoretraditionalsemanticaccountofthemathematicallysimpleapproachto
causalknowledgethatunderliesourcausaltheoriesofaction. Forinstance,instead
(cid:30)
oftheinferencerule ,wewritethemodalformulaC(cid:30)(cid:27)C ,whereCisamodal

Contents
operatorread as\caused." Inthe thirdpartof thedissertation, we showthat a
subsetofUCLiswell-suitedforautomatedreasoningaboutactions. Inparticular,
weshowthattheclassof\simple"UCLtheoriesprovidesanexpressivebasisfor
thecomputationallychallengingtaskofautomatedplanning. SimpleUCLtheories Acknowledgments v
haveaconcisetranslationintoclassicallogic,and,asweshow,theclassicalmodels
Abstract vi
ofthetranslationcorrespondtovalidplans. Thisenables\satis(cid:12)abilityplanning"
withcausalactiontheories,with\stateoftheart"performanceonlargeclassical
ListofTables xiii
planningproblems.
ListofFigures xiv
Chapter1 Introduction 1
1.1 TheFrameProblemandNonmonotonicity. . . . . . . . . . . . . . . 1
1.2 CommonsenseInertiaasMinimalChange . . . . . . . . . . . . . . . 3
1.2.1 TheYaleShootingProblem . . . . . . . . . . . . . . . . . . . 3
1.2.2 PossibleNextStates . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 StateConstraintsandStaticCausalLaws . . . . . . . . . . . . . . . 5
1.4 ACausalAccountofCommonsenseInertia . . . . . . . . . . . . . . 6
1.5 CausallyPossibleWorldsandUniversalCausation . . . . . . . . . . 8
1.6 AutomatedReasoningaboutActionsandSatis(cid:12)abilityPlanning . . 10
1.7 OutlineofDissertation. . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter2 LiteratureSurvey 14
2.1 TheSituationCalculus. . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 NonmonotonicFormalisms. . . . . . . . . . . . . . . . . . . . . . . . 17
vii viii
2.3 ActionTheoriesinLogicalFormalisms . . . . . . . . . . . . . . . . . 19 Chapter4 ProofsforPrecedingChapter 88
2.4 High-LevelActionLanguages . . . . . . . . . . . . . . . . . . . . . . 21 4.1 SplittingaDefaultTheory. . . . . . . . . . . . . . . . . . . . . . . . 88
2.5 PossibleNextStatesandTheoryUpdate. . . . . . . . . . . . . . . . 24 4.1.1 SplittingSets . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.6 CausalTheoriesofAction . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1.2 SplittingSequences. . . . . . . . . . . . . . . . . . . . . . . . 92
4.2 ProofofSplittingSetTheorem . . . . . . . . . . . . . . . . . . . . . 93
Chapter3 InferenceRulesinCausalActionTheories 27
4.3 ProofofSplittingSequenceTheorem . . . . . . . . . . . . . . . . . . 101
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 ProofofCorrespondenceTheoremandReachabilityCorollary. . . . 106
3.2 FourExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5 Proof of LP Correspondence Theorem, LP Reachability Corollary,
3.3 ACausalDe(cid:12)nitionofPossibleNextStates . . . . . . . . . . . . . . 40
andVividDomainsTheorem . . . . . . . . . . . . . . . . . . . . . . 126
3.3.1 PreliminaryDe(cid:12)nitions . . . . . . . . . . . . . . . . . . . . . 40
3.3.2 PossibleNextStates: RuleUpdate . . . . . . . . . . . . . . . 42 Chapter5 ALogicofUniversalCausation 132
3.3.3 RuleUpdateandMinimalChange . . . . . . . . . . . . . . . 43 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.3.4 ExplicitDe(cid:12)nitionsinRuleUpdate. . . . . . . . . . . . . . . 45 5.2 PropositionalUCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.4 TheActionLanguageAC . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2.1 SyntaxandSemantics . . . . . . . . . . . . . . . . . . . . . . 137
3.4.1 SyntaxofAC . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.4.2 SemanticsofAC . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3 PossibleNextStatesandInertiainUCL . . . . . . . . . . . . . . . . 139
3.4.3 AnExampleACDomainDescription. . . . . . . . . . . . . . 53 5.3.1 InferenceRulesinUCL . . . . . . . . . . . . . . . . . . . . . 139
3.4.4 RemarksontheActionLanguageAC . . . . . . . . . . . . . 56 5.3.2 TwoEmbeddingsofRuleUpdateinUCL . . . . . . . . . . . 142
3.5 RepresentingActionsinDefaultLogic . . . . . . . . . . . . . . . . . 65 5.3.3 AThirdEmbedding:CommonsenseInertiainUCL . . . . . 143
3.5.1 ReviewofDefaultLogic . . . . . . . . . . . . . . . . . . . . . 66 5.4 UCLandDefaultLogic . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.5.2 EmbeddingPossibleNextStatesinDefaultLogic. . . . . . . 67 5.4.1 ReviewofDisjunctiveDefaultLogic . . . . . . . . . . . . . . 146
3.5.3 EmbeddingACinDefaultLogic . . . . . . . . . . . . . . . . 69 5.4.2 UCLandDisjunctiveDefaultLogic. . . . . . . . . . . . . . . 147
3.5.4 TheYaleShootingProbleminDefaultLogic . . . . . . . . . 74 5.5 EmbeddingACinUCL . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.6 LogicProgramsforRepresentingActions . . . . . . . . . . . . . . . 79 5.6 FlatandDe(cid:12)niteUCLTheories. . . . . . . . . . . . . . . . . . . . . 155
3.6.1 ReviewofLogicProgramming . . . . . . . . . . . . . . . . . 79 5.6.1 FlatUCLTheories . . . . . . . . . . . . . . . . . . . . . . . . 155
3.6.2 LP-SimpleACDomainDescriptions . . . . . . . . . . . . . . 80 5.6.2 De(cid:12)niteUCLTheories . . . . . . . . . . . . . . . . . . . . . . 156
3.6.3 LP-SimpleACDomainDescriptionsasLogicPrograms . . . 82 5.7 (More)CausalTheoriesofActioninUCL . . . . . . . . . . . . . . . 158
3.6.4 MakingVividACDomainDescriptionsLP-Simple . . . . . . 85 5.7.1 L(F;A;T)Languages . . . . . . . . . . . . . . . . . . . . . . 159
ix x
5.7.2 L(F;A;T)DomainDescriptions . . . . . . . . . . . . . . . . 159 6.5.3 ExperimentalResults . . . . . . . . . . . . . . . . . . . . . . 209
5.7.3 ExpressivePossibilities. . . . . . . . . . . . . . . . . . . . . . 165 6.6 ProofofMainProposition . . . . . . . . . . . . . . . . . . . . . . . . 212
5.8 ASubsetofUCLinCircumscription . . . . . . . . . . . . . . . . . . 171
Chapter7 ConcludingRemarks 215
5.9 UCLandLin'sCircumscriptiveActionTheories. . . . . . . . . . . . 174
5.9.1 Lin'sCircumscriptiveCausalActionTheories . . . . . . . . . 174 Bibliography 219
5.9.2 Lin'sCircumscriptiveActionTheoriesinUCL . . . . . . . . 177
Vita 233
5.9.3 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.10 UCLandAutoepistemicLogic . . . . . . . . . . . . . . . . . . . . . 183
5.11 UCLwithQuanti(cid:12)ers . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.12 NonpropositionalCausalTheoriesinUCL . . . . . . . . . . . . . . . 188
5.12.1 Lifschitz'sNonpropositionalCausalTheories . . . . . . . . . 188
5.12.2 Second-OrderCausalTheoriesinUCL . . . . . . . . . . . . . 189
Chapter6 Satis(cid:12)abilityPlanningwithCausalActionTheories 192
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.2 PlanningwithL(F;A;T)DomainDescriptions . . . . . . . . . . . . 194
6.2.1 CausallyPossiblePlans . . . . . . . . . . . . . . . . . . . . . 195
6.2.2 Su(cid:14)cientPlans . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.2.3 ExecutablePlans . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.2.4 ValidPlans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6.2.5 DeterministicPlans . . . . . . . . . . . . . . . . . . . . . . . 198
6.3 Satis(cid:12)abilityPlanningwithL(F;A;T)DomainDescriptions. . . . . 201
6.3.1 SimpleDomainDescriptions. . . . . . . . . . . . . . . . . . . 201
6.3.2 SimpleDomainDescriptionsYieldValidPlans . . . . . . . . 203
6.4 Satis(cid:12)abilityPlanningProgram . . . . . . . . . . . . . . . . . . . . . 204
6.5 LargePlanningProblems . . . . . . . . . . . . . . . . . . . . . . . . 205
6.5.1 BlocksWorldProblems . . . . . . . . . . . . . . . . . . . . . 205
6.5.2 LogisiticsPlanningProblems . . . . . . . . . . . . . . . . . . 209
xi xii
List of Tables List of Figures
6.1 Satis(cid:12)ability Planning with Causal Action Theories. Sizes are for 3.1 DefaulttheoryforExample1.. . . . . . . . . . . . . . . . . . . . . . 31
clausaltheoriesobtained,vialiteralcompletion,fromcausalaction 3.2 LogicprogramforExample1.. . . . . . . . . . . . . . . . . . . . . . 33
theories(aftersimpli(cid:12)cation).Timeinsecondsusingthesatis(cid:12)ability 3.3 DefaulttheoryforExample2.. . . . . . . . . . . . . . . . . . . . . . 35
solverrelsat onaSparcstation5.. . . . . . . . . . . . . . . . . . . . 211 3.4 LogicprogramforExample3.. . . . . . . . . . . . . . . . . . . . . . 37
6.2 Kautz and Selman Problem Descriptions. Here we establish the 3.5 LogicprogramforExample4.. . . . . . . . . . . . . . . . . . . . . . 39
benchmarks|theresultsfortheclausaltheoriesusedin[KS96],with 3.6 Standardelementsofthetranslation(cid:14).. . . . . . . . . . . . . . . . . 71
solutiontimesobtainedinthesamemannerasinTable6.1. . . . . . 211 3.7 ACdomaindescriptionD1. . . . . . . . . . . . . . . . . . . . . . . . 72
6.3 ProvingPlansOptimal: Satis(cid:12)abilityPlanningwithCausalAction 3.8 Translation(cid:14)(D1)ofACdomaindescriptionD1. . . . . . . . . . . . 73
Theories. Here, in each case, the domain descriptionincludes one 3.9 DefaulttheoryY1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
timesteplessthanneededforasolution. Timereportedisnumber 3.10 DefaulttheoryY2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
ofsecondsrequiredforsolverrelsat todetermineunsatis(cid:12)ability.. . 212 3.11 DefaulttheoryY3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1 UCLtranslationofdefaulttheoryforExample1. . . . . . . . . . . . 150
5.2 SimplerUCLtheoryforExample1. . . . . . . . . . . . . . . . . . . 150
5.3 UCLtheoryforExample2. . . . . . . . . . . . . . . . . . . . . . . . 151
5.4 UCLtheoryforExample3. . . . . . . . . . . . . . . . . . . . . . . . 151
5.5 AnotherUCLtheoryforExample1. . . . . . . . . . . . . . . . . . . 153
5.6 AnotherUCLtheoryforExample2. . . . . . . . . . . . . . . . . . . 153
5.7 AnotherUCLtheoryforExample3. . . . . . . . . . . . . . . . . . . 154
5.8 L(F;A;T)descriptionD3ofLin'sSuitcasedomain.. . . . . . . . . . 164
xiii xiv
5.9 L(F;A;T)descriptionD4ofCoinTossdomain.. . . . . . . . . . . . 166
5.10 L(F;A;T)descriptionD5ofDominosdomain. . . . . . . . . . . . . 168
5.11 L(F;A;T)descriptionD6ofPendulumdomain.. . . . . . . . . . . . 170
5.12 Lin'sSuitcasedomaininsecond-orderUCL. . . . . . . . . . . . . . . 188
6.1 Exampleinput(cid:12)lefortheplanningsystem: thePendulumdomain.. 204 Chapter 1
6.2 PlanningsessionwithPendulumdomain. . . . . . . . . . . . . . . . 206
6.3 Characterizationoflargeblocksworldproblemsfrom[KS96]. . . . . 206
6.4 Input(cid:12)leforBlocksWorldD. . . . . . . . . . . . . . . . . . . . . . . 207 Introduction
6.5 Input(cid:12)leforLogisticsC. . . . . . . . . . . . . . . . . . . . . . . . . 210
JohnMcCarthyinhis1959paper\ProgramswithCommonSense"[McC59]pro-
posedthatresearchersinarti(cid:12)cialintelligencetrytoformalizeandautomatecom-
monsensereasoningaboutactions. Thechallengeistoobtaincorrectconclusions
abouttheoutcomesofactionsonthebasisofconcisedeclarativerepresentationsof
commonsenseknowledgeaboutactiondomains. Thishasproveddi(cid:14)cult.
Itiswidelyremarkedthatthenotionofcausalityplayslittleornorolein
descriptionsoftheworldinthephysicalsciences. Thesamehasbeengenerallytrue
ofproposedformalizationsofreasoningaboutaction,despitethecentralroleplayed
bycausalnotionsineverydaydiscourseandthoughtaboutactions.Thisdissertation
belongstoalineofrecentworkinvestigatingtheadvantagesofconsideringcausality
moreexplicitly.
1.1 The Frame Problem and Nonmonotonicity
Afundamentaldi(cid:14)cultyinreasoningaboutaction|theso-called\frameproblem"|
was recognized and named by McCarthy and Hayes in their 1969 paper \Some
PhilosophicalProblemsfromthe StandpointofArti(cid:12)calIntelligence"[MH69]. A
xv 1
naturalstrategyformakingactionrepresentationsconciseistofocusondescribing 1.2 Commonsense Inertia as Minimal Change
thechangescausedbyanaction,whileleavingimplicitourknowledgeoffactsun-
In most proposalsfor reasoning about action, the commonsense law of inertia is
a(cid:11)ectedbytheaction. Aboutfactsuna(cid:11)ectedbyanaction,weassumethatthey
understoodaccordingtoaprincipleofminimalchange. Roughlyspeaking,theidea
simplypersist,accordingtoa\commonsenselawofinertia."Thus,generallyspeak-
is to capture the assumption that things change as little as possible, while also
ing,theframeproblemistheproblemofmakingthecommonsenselawofinertia
re(cid:13)ectingourknowledgeofwhatdoeschange.
mathematicallyprecise.
Itisclearthatsolutionstotheframeproblemwillbenonmonotonic: thatis,
1.2.1 TheYaleShootingProblem
incontrasttoclassicallogic,conclusionsmaybelostwhenpremisesareadded. For
In1986McCarthyproposedaformalizationofcommonsenseknowledgeaboutac-
example,consideranactiondomaindescriptioninvolvingtwopropositional(cid:13)uents,
tionsinwhichthecommonsenselawofinertiaisunderstoodaccordingtoaprinciple
PandQ,andasingleactionA. (Apropositional(cid:13)uentisapropositionwhosevalue
ofminimalchange[McC86]. Essentially,McCarthysaidthatchangeisabnormal,
dependsontime.) SupposeyouaretoldthatP andQareinitiallyfalse,andthat
andheusedtechnicalmeans|namely,circumscription(introducedin[McC80])|to
AmakesP true. YouareexpectedtoconcludenotonlythatP wouldbecometrue
select models of his action theory inwhichthat kindof abnormality is minimal.
ifAwereperformed,butalsothatQwouldnot. Nowsupposethatyouaretold
Thatis,hepreferredmodelsinwhichthingschangeaslittleaspossible.
inadditionthatAmakesQtrue. YoushouldnolongerconcludethatQwouldbe
HanksandMcDermottfamouslyexposedafundamentaldi(cid:14)cultywithMc-
falseafterA;instead,QwouldbetrueafterA.
Carthy's proposal, by introducing a counterexample widely known as the \Yale
Althoughthepreviousinformalexampledemonstratesthatsolutionstothe
Shooting" domain [HM87]. The essential elements can be described as follows.
frameproblemwillbenonmonotonic, itdoeslittletosuggestthatsuchsolutions
Thereisaturkey(Fred)andagun. Ifthegunisloaded,shootingitkillsFred. The
may besubtleor di(cid:14)cultto (cid:12)nd. Nonetheless, thisseems so farto bethe case,
questionisthis: IfFredisinitiallyaliveandthegunisinitiallyloaded,willFred
particularlyasweattempttorepresentmoreelaboratekindsofdomainknowledge.
bedeadaftertheactionsWait andShoot areperformedinsequence? Clearlythe
Forinstance,inthisdissertationweareinterestednotonlyinhowtorepresentthe
answershouldbeyes. Unfortunately,McCarthy'sformalizationcouldnotpredict
\direct" e(cid:11)ects of actions, butalso inhow to represent \background knowledge"
this.
concerningrelationshipsbetween(cid:13)uents,inordertocorrectlyinferthe\indirect"
Thefundamentaldi(cid:14)cultywithMcCarthy's1986proposalisthatitmini-
e(cid:11)ectsofactions. Forexample,youmightbetoldnotonlythatAmakesP true,
mizeschangeglobally(i.e.acrossallsituations). IntheintendedmodelsoftheYale
butalsothatQcanbemadetruebymakingP true. Youshouldagainconclude
Shootingdomain,no(cid:13)uentschangeasaresultoftheWaitaction|inparticular,the
thatQwouldbetrueafterA.
gunremainsloaded|andthenFredbecomesdeadasaresultoftheShoot action.
McCarthycallsthedeathofFredabnormal,andisinprinciplewillingtotradethe
deathofFredforotherpossibleabnormalities.Thus,theglobalminimizationpolicy
2 3
issatis(cid:12)edbyanomalousmodelsinwhichthegunbecomesunloadedasaresultof 1.3 State Constraints and Static Causal Laws
the Wait action, and then no (cid:13)uents change as a result of the Shoot action|in
Stateconstraintsareformulasofclassicallogicthataresaidtoholdineverypossible
particular,FredremainsaliveaftertheShoot action.1
stateofanactiondomain. Traditionally,stateconstraintshavebeenusedtoderive
ThisaccountoftheYaleShootingproblemsuggeststhatitiswrongtomin-
indirecte(cid:11)ects,or\rami(cid:12)cations,"ofactions. Adaptingawidelyfamiliarexample
imizechangeglobally,butdoesnotshowthattheprincipleofminimalchangewill
(from[Bak91]),let'sassumethatyoucanmakeFredstopwalkingbymakinghimnot
neverdo.
alive.SoifshootingFredkillshim,youcanmakehimstopwalkingbyshootinghim:
notwalkingisarami(cid:12)cationofshooting. Traditionally,thebackgroundknowledge
1.2.2 PossibleNextStates
usedinthisexamplehasbeenexpressedbythestateconstraint
Theprincipleofminimalchangecancarryusalongwayifappliedmorecarefully.
Theessence oftheframeproblemcanbeformulatedasfollows. Givenaninitial :Alive(cid:27):Walking: (1.1)
stateoftheworldandadescriptionofthee(cid:11)ectsofanactionwhenperformedin
Thisstateconstraintisequivalentinclassicallogictoitscontrapositive
thatstate,wemustsaywhichstatesoftheworldmayresult|sofarasweknow|
aftertheactionisperformed. Wesayade(cid:12)nitionofthiskindidenti(cid:12)es\possible Walking(cid:27)Alive:
nextstates." Winslett[Win88]proposedade(cid:12)nitionofpossiblenextstatesinwhich Thisistroublesomebecauseitisclearthatthecausalrelationitselfisnotcontra-
thecommonsenselawofinertiaiscapturedmathematicallyasthestraightforward positive. Thatis,roughlyspeaking,althoughyoucanmakeFredstopwalkingby
requirementthatthingschangeaslittleaspossible(whilestillsatisfyingthee(cid:11)ects
killinghim, it does not follow from this that you can bringFred back to life by
oftheaction). makinghimwalk.
Thissimpleidea(invariousguises)hasledtoconsiderableprogress. Infact,
Recently, a number of researchers have argued that state constraints are
it seems that a good deal of the widely-remarked technical di(cid:14)culty in work on
inadequateforrepresentingbackgroundknowledgeinactiondomains,becausethey
reasoningaboutactioncanbeattributedtotheneedto(cid:12)ndmathematicalmeans do not adequately represent causal relations [Gef90, Elk92, BH93, Bar95, Lin95,
forcapturingthissimplede(cid:12)nitionofpossiblenextstateswithindescriptionsthat
MT95b,Thi95a,Gus96]. Inthisdissertationweexploretheuseof\staticcausal
aremorecomplexprimarilybecausetheyencompassmorethananinitialsituation, laws"ofthekindintroducedbyMcCainandTurnerin[MT95b]:ifa(cid:13)uentformula(cid:30)
anactionandaresultingsituation. (Seeforexample[Bak91],orChapter4ofthis iscausedtobetrue,thena(cid:13)uentformula isalsocausedtobetrue. Fromsuch
dissertation.)2
acausallawitfollowsthatonecanmake truebymaking(cid:30)true. Italsofollows
1Accordingtotheinformaldescriptiongivenhere,therewillbeanotherclassofmodels,inwhich thatineverypossiblestate, istrueif(cid:30)is. Thatis,thestateconstraint(cid:30)(cid:27)
FreddiesasaresultoftheWaitaction(whilethegunremainsloaded). Asithappens,thiskind
ofanomalousmodelisruledoutinMcCarthy'sstyleofformalization,whichforcesanabnormality statesoftheworld.KatsunoandMendelzonemphasizeasimilarpointintheirin(cid:13)uentialpaper\On
withrespecttoAlivewheneverShootisperformedwithaloadedgun,evenifFredisalreadydead. theDi(cid:11)erenceBetweenUpdatingaKnowledgeBaseandRevisingIt"[KM91].Wewillalludetothis
2InpassingweremarkthatWinslettalsoemphasizedasecondcrucialelementinthepossible pointseveraltimesinthisdissertation,sinceitalsohelpsexplaintechnicaldi(cid:14)cultiesencountered
nextstatessetting.Reasoningaboutactionisbyitsnatureamatterofreasoningaboutcomplete insomeformalizationsofactions.
4 5