Revision, Acceptability and Context: Theoretical and Algorithmic Aspects

396 Pages·2010·2.67 MB·English

by  Dov M. Gabbay

Cognitive Technologies
ManagingEditors: D.M.Gabbay J.Siekmann
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Dov M. Gabbay Odinaldo T. Rodrigues
Alessandra Russo
Revision, Acceptability
and Context
Theoretical and Algorithmic Aspects
123
Prof.Dr.DovM.Gabbay Dr.OdinaldoT.Rodrigues
King’sCollegeLondon King’sCollegeLondon
Dept.ComputerScience Dept.ComputerScience
LondonWC2R2LS LondonWC2R2LS
UnitedKingdom UnitedKingdom
[email protected] [email protected]
Dr.AlessandraRusso
ImperialCollegeLondon
Dept.Computing
180Queen’sGate
LondonSW72BZ
UnitedKingdom
[email protected]
ManagingEditors
Prof.Dr.DovM.Gabbay Prof.Dr.Jo¨rgSiekmann
AugustusDeMorganProfessorofLogic ForschungsbereichDeduktions-und
King’sCollegeLondon Multiagentensysteme,DFKI
Dept.ComputerScience Stuhlsatzenweg3,Geb.43
LondonWC2R2LS 66123Saarbru¨cken,Germany
UnitedKingdom
CognitiveTechnologiesISSN1611-2482
ISBN978-3-642-14158-4 e-ISBN978-3-642-14159-1
DOI10.1007/978-3-642-14159-1
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2010932026
ACMComputingClassification(1998):I.2,F.4,H.2
(cid:2)c Springer-VerlagBerlinHeidelberg2010
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Preface
We are happy to present our new book to the community. Although this is a book
on revision theory, it uses general methodologies which are applied in other areas
oflogic.Wewouldliketohighlightthesemethodologiesinthispreface.
1. theideaoflogicbytranslation
GivenanewlogicalsystemL,whosepropertieswewanttoinvestigate,wetrans-
lateLintoawell-knownlogic,usuallyclassicallogicC,modallogicorintuition-
isticlogic.Westudythepropertieswewantinthetargetlogicandthentranslate
back into L. Our approach of revision by translation (see Chapter 7) is an in-
stanceofthatmethodology.Wecanalsodointerpolationbytranslation.Whatis
mostwellknowninthecommunityissemanticsbytranslationanddecidability
andundecidabilitybytranslation.Thisisacommonwayofobtainingsemantics
ordecidabilityresultsforlogicalsystems.
2. meta-level/object-levelmovement
Given a logic L,there may be some manipulations of L done in the meta-level.
Such notions can be provability or consequence, e.g., A(cid:2)B or deletion from a
database.Wemaywishtobringthesenotionstotheobjectlevel.Thiscanbea
complicatedtask.Weallknowabouttheproblemofbringingtheproofpredicate
intotheobjectlevel.Inthisbookwebringintheoperationofdeletion.Another
lesser known result is the bringing in to the object level of a non-monotonic
consequencerelation|∼,andwhatwegetwhenwedoitcorrectlyisaconditional
intheobjectlanguage.
3. context
The use of context can determine how we use our logic. The context is a label
inLabelledDeductiveSystemsanditisaveryrichlabelinsituationsemantics.
Inrevisiontheory,itaffectsthealgorithmsweusetorevise,andinmanycasesit
mighttellusnottoreviseatallandinsteadtotrytoobtainmoredataorsimply
livewiththeinconsistencyinacarefulway.
v
vi Preface
Acknowledgements We would like to acknowledge feedback and collaborations with Krysia
Broda, Fa´tima Cristina Canazaro Dargam, Artur d’Avila Garcez, Anthony Hunter, Lu´ıs Lamb,
Je´roˆmeLang,DavidMakinson,SanjayModgil,GabriellaPigozzi,KarlSchlechtaandJohnWoods.
ThismanuscriptwaswrittenusingLATEX.WeusedPaulTaylor’scommutativediagramspack-
ageforsomeofthediagrams.JaneSpurrkindlyhelpedinthesolutionofmanyformattingissues.
London, D.M.Gabbay
September2009 O.T.Rodrigues
A.Russo
Contents
1 BackgroundandOverview ...................................... 1
1.1 IntroductoryDiscussion ..................................... 1
1.2 Focusingon‘Acceptability’Ratherthan‘Inconsistency’ .......... 5
1.3 OverviewofThisBook...................................... 7
1.4 NotationUsedinThisBook.................................. 8
1.5 BasicMathematicalNotions ................................. 10
References..................................................... 11
2 IntroducingRevisionTheory .................................... 13
2.1 AGMBeliefRevision ....................................... 13
2.2 KatsunoandMendelzon’sCharacterisation ..................... 20
2.3 CounterfactualStatementsandtheRamseyTest ................. 21
2.4 Grove’sSystemsofSpheres.................................. 25
2.5 EpistemicEntrenchment..................................... 28
2.6 Discussion ................................................ 30
2.7 ActionUpdates ............................................ 30
2.8 GeneralisingtheConceptofRevision.......................... 33
2.9 IteratingtheRevisionOperation .............................. 37
2.9.1 DarwicheandPearl’sPostulatesforIteratedRevisions ..... 37
2.9.2 Lehmann’sApproach:BeliefRevision,Revised........... 40
2.10 CompromiseRevision....................................... 41
2.11 ControlledRevision ........................................ 45
2.12 RevisionbyTranslation ..................................... 47
2.13 AGeneralSettingforAlgorithmicRevision .................... 49
2.14 OutlineofThisBook ....................................... 52
References..................................................... 52
3 StepwiseRevisionOperations ................................... 55
3.1 Introduction ............................................... 55
3.2 QuantitativeMeasurementofChange.......................... 55
3.2.1 MinimalChangeandtheFunctiond .................... 56
vii
viii Contents
3.3 QualitativeMeasurementsofChange .......................... 63
3.4 RepresentationIssues ....................................... 65
3.5 RevisionofFormulae ....................................... 67
3.6 PropertiesoftheRevisionOperator◦ ......................... 70
r
3.7 OtherBeliefChangeOperators ............................... 77
3.7.1 BeliefContraction ................................... 77
3.7.2 ConsolidatingInformationontheBeliefBase ............ 82
3.8 ComparisonwithOtherBeliefChangeOperators ................ 85
3.9 OperatorsforReasoningAbouttheEffectsofActions............ 88
3.9.1 UpdatesofSentencesviaDistanced .................... 89
3.9.2 PropertiesoftheActionUpdateOperator ................ 93
3.9.3 ActionUpdatesofSentencesviaDistancediff ............ 96
3.9.4 AmbiguousActionUpdates ........................... 98
3.9.5 TakingCausalityintoAccount .........................100
References.....................................................102
4 IteratingRevision ..............................................105
4.1 Introduction ...............................................105
4.2 MotivatingStructureforBeliefRevision .......................107
4.3 IterationoftheRevisionOperation ............................108
4.4 PrioritisedDatabases........................................111
4.4.1 PropertiesoftheRevisionsofPDBs ....................115
4.4.2 DiscussionaboutIterationofRevision...................119
4.5 StructuredDatabases........................................124
4.6 ApplicationsandExamples ..................................126
4.7 RelatedWork..............................................128
4.7.1 PrioritisedBaseRevision..............................128
4.7.2 OrderedTheoryPresentations..........................131
4.8 UsingAdditionalInformationforActionUpdates ...............132
References.....................................................136
5 StructuredRevision ............................................139
5.1 IdentifyingInconsistency ....................................139
5.2 ReasoningwithPartialPriorities ..............................148
5.2.1 DegreeofConfidence/ReliabilityoftheSource ...........149
5.2.2 Linearisations .......................................165
5.3 Clustering.................................................166
5.4 ApplicationsinSoftwareEngineering .........................168
5.4.1 RequirementsSpecification............................168
5.4.2 AnExamplewiththeLightControlSystem ..............169
References.....................................................175
Contents ix
6 AlgorithmicContextRevision ...................................177
6.1 Introduction ...............................................177
6.2 AbductiveRevision.........................................177
6.2.1 IntroducingLDSfor⇒ ...............................180
6.2.2 Goal-DirectedAlgorithmfor⇒ ........................183
6.2.3 DiscussionontheAbductionProcedure .................187
6.2.4 AbductionAlgorithmfor⇒ ...........................192
6.2.5 AbductionforIntuitionisticLogic ......................194
6.3 CompromiseRevision ......................................197
6.3.1 IntroducingCompromiseRevisionfor⇒ ................197
6.3.2 ComparisonwithAGMRevision .......................210
6.4 ControlledRevision ........................................214
6.4.1 ProofTheory........................................216
6.4.2 PoliciesforInconsistency .............................217
6.4.3 Conclusions.........................................221
References.....................................................222
7 RevisionbyTranslation.........................................223
7.1 Introduction ...............................................223
7.2 BeliefRevisionforModalLogic..............................225
7.2.1 AnOverviewofPropositionalModalLogics .............225
7.2.2 HilbertSystemsforModalLogics ......................230
7.2.3 TranslationoftheModalLogicK intoClassicalLogic .....233
7.3 RevisinginŁukasiewicz’FinitelyMany-ValuedLogicŁ .........235
n
7.3.1 Łukasiewicz’FinitelyMany-ValuedLogicŁ ............235
n
7.3.2 TranslatingŁukasiewicz’Many-ValuedLogicŁ into
n
ClassicalLogic ......................................241
7.3.3 RevisioninŁukasiewicz’Many-ValuedLogic(Ł )........244
n
7.4 RevisinginAlgebraicLogics.................................246
7.4.1 TranslatingAlgebraicLogicintoClassicalLogic..........246
7.4.2 RevisioninAlgebraicLogics ..........................248
7.5 IntroducingBelnap’sFour-ValuedLogic .......................249
7.5.1 BeliefRevisioninBelnap’sFour-ValuedLogic ...........249
7.6 Belnap’sFour-ValuedLogic..................................251
7.6.1 AxiomatisingBelnap’sFour-ValuedLogic ...............251
7.6.2 EntailmentinBelnap’sFour-ValuedLogic ...............253
7.6.3 GeneralisingBelnap’sNotionofEntailmenttoInfinite
Theories............................................255
7.6.4 TranslatingBelnap’sLogicintoClassicalLogic...........259
7.6.5 RevisinginBelnap’sFour-ValuedLogic .................265
7.7 ConclusionsandDiscussions.................................268
References.....................................................269