Logical Tools for Handling Change in Agent-Based Systems

308 Pages·2010·1.85 MB·English

by  Dov M. Gabbay

Cognitive Technologies
ManagingEditors: D.M.Gabbay J.Siekmann
EditorialBoard: A.Bundy J.G.Carbonell
M.Pinkal H.Uszkoreit M.Veloso W.Wahlster
M.J.Wooldridge
AdvisoryBoard:
LuigiaCarlucciAiello AlanMackworth
FranzBaader MarkMaybury
WolfgangBibel TomMitchell
LeonardBolc JohannaD.Moore
CraigBoutilier StephenH.Muggleton
RonBrachman BernhardNebel
BruceG.Buchanan SharonOviatt
AnthonyCohn LuisPereira
Arturd’AvilaGarcez LuRuqian
LuisFarin˜asdelCerro StuartRussell
KoichiFurukawa ErikSandewall
GeorgGottlob LucSteels
PatrickJ.Hayes OlivieroStock
JamesA.Hendler PeterStone
AnthonyJameson GerhardStrube
NickJennings KatiaSycara
AravindK.Joshi MilindTambe
HansKamp HidehikoTanaka
MartinKay SebastianThrun
HiroakiKitano JunichiTsujii
RobertKowalski KurtVanLehn
SaritKraus AndreiVoronkov
MaurizioLenzerini TobyWalsh
HectorLevesque BonnieWebber
JohnLloyd
Forfurthervolumes:
http://www.springer.com/series/5216
·
Dov M. Gabbay Karl Schlechta
Logical Tools for Handling Change
in Agent-Based Systems
With 36 Figures and 8 Tables
123
Authors ManagingEditors
Prof.DovM.Gabbay Prof.DovM.Gabbay
DepartmentofComputerScience AugustusDeMorganProfessorofLogic
King’sCollegeLondon,Strand DepartmentofComputerScience
LondonWC2R2LS,UK King’sCollegeLondon
[email protected] Strand,LondonWC2R2LS,UK
and
DepartmentofComputerScience
Prof.Dr.Jo¨rgSiekmann
Bar-IlanUniversity
ForschungsbereichDeduktions-und
52900Ramat-Gan
Multiagentensysteme,DFKI
Israel
Stuhlsatzenweg3,Geb.43
and
66123Saarbru¨cken,Germany
ComputerScienceandCommunications
FacultyofSciences
UniversityofLuxembourg
6rueCoudenhove-Kalergi
1359Luxembourg
Prof.KarlSchlechta
Laboratoired’InformatiqueFondamentaledeMarseille
UMR6166
CNRS
and
Universite´deProvence,CMI
39,rueJoliot-Curie
13453MarseilleCedex13
France
[email protected]
[email protected]
CognitiveTechnologiesISSN1611-2482
ISBN978-3-642-04406-9 e-ISBN978-3-642-04407-6
DOI10.1007/978-3-642-04407-6
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2009939263
ACMComputingClassification(1998):I.2.3,F.4.1
(cid:2)c Springer-VerlagBerlinHeidelberg2010
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Preface
Nonmonotonic logics were created in 1970s by John McCarthy ([McC80]) and
others in order to model more closely common sense everyday reasoning. Such
logicsavoidexplicitmentioningofaverylargenumberofexceptionalcases.Thus,
formal nonmonotonic systems are logics which are closer to human reasoning. In
subsequent years, a multitude of formalisms were invented, some presented from
theprooftheoreticpointofviewandsomefromthesemanticpointofview.
Axiomatic systems for nonmonotonic consequence relations was one approach
([Gab85]) and tinkering and changing existing monotonic logics were other
approaches (many authors). Some tinkered with the semantics and some with the
prooftheory.Theresultwasawildlandscapeofdiverselogicsdrivenbyapplications
andamultitudeofformalrelationshipsamongthem.
Severalobservationshelpedbringsomeorderintothischaoticnetscape.Onthe
syntacticalside,itwasobservedbyDelgrande([Del87])thatanonmonotoniccon-
sequencerelationcorrespondstofirst-levelconditionals(noembeddedconditional
implications).LateritwasshownbyGabbaythatifwebring(fibre)anonmonotonic
consequence relation from the metalevel into the object level, we get conditionals
([Gab98]).Thuswehaveastrongconnectionbetweennonmonotonicconsequence
and conditionals. Also, Boutilier ([Bou90a]) made a connection with modal logic.
GardenforsandMakinson([MG91])madeaconnectionwiththerichareaofAGM
revisiontheory.
On the semantical side, it was observed by Shoham ([Sho87b]) and later sys-
tematically developed by Kraus, Lehmann, and Magidor, ([KLM90]) that prefer-
ential models can yield many axiomatic nonmonotonic systems. This approach
was studied mathematically by Schlechta (summarized in [Sch04]) for a num-
ber of completeness and incompleteness results. In a certain way, most of the
semantical work can be seen as continuing the tradition begun by Kripke (e.g.
[Kri59]), to create semantics by adding structure to a set of classical models.
However, despite all the above observations, numerous connections and repre-
sentations results, the landscape of nonmonotonic logics still was not properly
organised.
The present research monograph goes further and begins to unite more logics,
not only nonmonotonic ones but also modal logic, counterfactuals, and others,
using a novel tool, that of reactive diagrams. Reactive diagrams allow “higher”
v
vi Preface
nonmonotonicity,asthearrowsofpreferentialstructures,cancellingmodelsasrep-
resentingnon-normalstates,cannowbecancelledthemselves.
Thus,thisbookisafirststeptowardsproperlyandfundamentallyorganisingthe
field.Itsystematizeslogicalpropertiesandseparatesthemfromunderlyingabstract
semantical properties. It bases many of them on simple rules of working with a
notionofsizeinanabstractsemanticsandshowsthepowerofreactivityinstructural
semantics,breakingthestrongcoherenceconditionsofpreferentialmodels.
Therearehoweverproblemsanddifferences:
We show the limitations of traditional size semantics in our investigation of
defeasible inheritance; ongoing research will strive to close this gap, using
againtheideaofreactivity.
Weunderlinethedifferencesbetweenthesemanticsofwhatis(e.g.in“normal
worlds”)andofwhatshouldbe(e.g.inthe“bestworlds”ofdeonticlogic).
Weemphasizetheimportanceofclosureconditionsofthedomain,inparticular
ofclosureunderfiniteunions,aproblemlargelyoverlookeduptonow.
WewishtothankD.Makinsonformanyveryinterestingdiscussions.
London,UK DovM.Gabbay
Marseille,France KarlSchlechta
August2009
Contents
1 IntroductionandMotivation ..................................... 1
1.1 Programme................................................ 1
1.2 ShortOverviewoftheDifferentLogics ........................ 2
1.2.1 NonmonotonicLogics................................ 2
1.2.2 TheoryRevision .................................... 3
1.2.3 TheoryUpdate...................................... 4
1.2.4 DeonticLogic ...................................... 6
1.2.5 CounterfactualConditionals........................... 6
1.2.6 ModalLogic........................................ 7
1.2.7 IntuitionisticLogic .................................. 8
1.2.8 InheritanceSystems.................................. 8
1.2.9 ASummarizingTablefortheSemantics................. 8
1.3 ADiscussionofConcepts.................................... 10
1.3.1 BasicSemanticEntities,TruthValues,andOperators...... 10
1.3.2 AlgebraicandStructuralSemantics..................... 12
1.3.3 RestrictedOperatorsandRelations ..................... 15
1.3.4 CopiesinPreferentialModels ......................... 16
1.3.5 FurtherRemarksonUniversalityofRepresentationProofs . 17
1.3.6 |∼intheObjectLanguage? ........................... 17
1.3.7 VariousConsiderationsonAbstractSemantics ........... 20
1.3.8 AComparisonwithReiterDefaults..................... 21
1.4 IBRS ..................................................... 21
1.4.1 DefinitionandComments............................. 21
1.4.2 ThePowerofIBRS .................................. 25
1.4.3 Abstract Semantics for IBRS and Its Engineering
Realization......................................... 26
2 BasicDefinitionsandResults ..................................... 31
2.1 AlgebraicDefinitions ....................................... 31
2.2 BasicLogicalDefinitions .................................... 34
2.3 BasicDefinitionsandResultsforNonmonotonicLogics .......... 35
vii
viii Contents
3 AbstractSemanticsbySize....................................... 53
3.1 TheFirst-OrderSetting...................................... 53
3.2 GeneralSizeSemantics...................................... 54
3.2.1 Introduction ........................................ 54
3.2.2 MainTable ......................................... 55
3.2.3 CoherentSystems ................................... 64
3.2.4 SizeandPrincipalFilterLogic......................... 69
4 PreferentialStructures–PartI ................................... 73
4.1 Introduction ............................................... 73
4.1.1 Remarks on Nonmonotonic Logics and Preferential
Semantics.......................................... 73
4.1.2 BasicDefinitions .................................... 77
4.2 PreferentialStructuresWithoutDomainConditions .............. 87
4.2.1 GeneralDiscussion .................................. 87
4.2.2 DetailedDiscussion.................................. 94
5 PreferentialStructures–PartII ..................................119
5.1 SimplificationsbyDomainConditions,LogicalProperties ........119
5.1.1 Introduction ........................................119
5.1.2 SmoothStructures ...................................119
5.1.3 RankedStructures ...................................123
5.1.4 TheLogicalPropertieswithDefinabilityPreservation .....124
5.2 A-RankedStructures........................................125
5.2.1 RepresentationResultsforA-RankedStructures..........125
5.3 Two-SequentCalculi........................................131
5.3.1 Introduction ........................................131
5.3.2 PlausibilityLogic ...................................131
5.3.3 ACommentontheWorkbyArieliandAvron............133
5.4 BlurredObservation–AbsenceofDefinabilityPreservation .......134
5.4.1 Introduction ........................................134
5.4.2 GeneralandSmoothStructuresWithoutDefinability
Preservation ........................................138
5.4.3 RankedStructures ...................................143
5.5 TheLimitVariant...........................................145
5.5.1 Introduction ........................................145
5.5.2 TheAlgebraicLimit .................................147
5.5.3 TheLogicalLimit ...................................149
6 HigherPreferentialStructures....................................157
6.1 Introduction ...............................................157
6.2 TheGeneralCase...........................................162
6.3 DiscussionoftheTotallySmoothCase.........................167
6.4 TheEssentiallySmoothCase.................................169
6.5 TranslationtoLogic ........................................173
Contents ix
7 DeonticLogicandHierarchicalConditionals .......................175
7.1 SemanticsofDeonticLogic ..................................175
7.1.1 IntroductoryRemarks ................................175
7.1.2 BasicDefinitions ....................................179
7.1.3 PhilosophicalDiscussionofObligations.................186
7.1.4 ExaminationoftheVariousCases ......................193
7.1.5 WhatIsAnObligation? ..............................197
7.1.6 Conclusion .........................................198
7.2 ACommentonWorkbyAqvist...............................198
7.2.1 Introduction ........................................198
7.2.2 ThereAre(AtLeast)TwoSolutions ....................199
7.2.3 Outline ............................................201
7.2.4 Gm(cid:3)AImpliesG(cid:3)A(Outline) ......................205
7.3 HierarchicalConditionals....................................205
7.3.1 Introduction ........................................205
7.3.2 FormalModellingandSummaryofResults..............209
7.3.3 Overview ..........................................211
7.3.4 ConnectionswithOtherConcepts ......................212
7.3.5 Formal Results and Representation for Hierarchical
Conditionals........................................214
8 TheoryUpdateandTheoryRevision ..............................219
8.1 Update....................................................219
8.1.1 Introduction ........................................219
8.1.2 HiddenDimensions..................................219
8.2 TheoryRevision............................................227
8.2.1 IntroductiontoTheoryRevision .......................227
8.2.2 BoothRevision .....................................236
8.2.3 RevisionandIndependence ...........................244
8.2.4 PreferentialModellingofDefaults .....................249
8.2.5 RemarksonIndependence ............................250
9 AnAnalysisofDefeasibleInheritanceSystems......................251
9.1 Introduction ...............................................251
9.1.1 Terminology........................................251
9.1.2 InheritanceandReactiveDiagrams.....................251
9.1.3 ConceptualAnalysis .................................257
9.2 IntroductiontoNonmonotonicInheritance......................258
9.2.1 BasicDiscussion ....................................258
9.2.2 Directly Sceptical Split Validity Upward Chaining
Off-PathInheritance .................................262
9.2.3 ReviewofOtherApproachesandProblems..............269
9.3 DefeasibleInheritanceandReactiveDiagrams ..................270
9.3.1 SummaryofOurAlgorithm ...........................270
x Contents
9.3.2 Overview ..........................................272
9.3.3 CompilationandMemorization ........................273
9.3.4 ExecutingtheAlgorithm..............................273
9.3.5 Signposts ..........................................274
9.3.6 BeyondInheritance ..................................274
9.4 Interpretations .............................................275
9.4.1 Introduction ........................................275
9.4.2 InformalComparisonofInheritancewiththeSystems
P and R ...........................................276
9.4.3 InheritanceasInformationTransfer.....................277
9.4.4 InheritanceasReasoningwithPrototypes ...............281
9.5 Detailed Translation of Inheritance to Modified Systems
ofSmallSets .............................................. 284
9.5.1 Normality ..........................................284
9.5.2 SmallSets..........................................285
Bibliography.......................................................295
Index .............................................................299