Formal Methods for Nonmonotonic and Related Logics: Vol II: Theory Revision, Inheritance, and Various Abstract Properties

511 Pages·2018·4.05 MB·English

by  Karl Schlechta

Karl Schlechta
Formal Methods
for Nonmonotonic
and Related Logics
Vol II: Theory Revision, Inheritance, and
Various Abstract Properties
Formal Methods for Nonmonotonic and Related Logics
Karl Schlechta
Formal Methods
for Nonmonotonic
and Related Logics
Vol II: Theory Revision, Inheritance,
and Various Abstract Properties
Karl Schlechta
CNRS, LIF UMR 7279
Aix-Marseille Université
Marseille, France
and Frammersbach, Germany

ISBN 978-3-319-89649-6 ISBN 978-3-319-89650-2 (eBook)
https://doi.org/10.1007/978-3-319-89650-2

Library of Congress Control Number: 2018960248

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Foreword
Thestudyofnonmonotonicconsequencerelationsbeganinthe1980s,andhasbeen
developing vigorously ever since. The literature is now so vast that it has become
difficultforgraduatestudentsandinvestigatorsaliketoobtainaclearandup-to-date
pictureofwhathasexactlyhasbeendoneandhowitallfitstogether.
This book, by one of the most eminent researchers in the field, fills much of the
gap. The first volume shows systematically how concepts and facts about many
nonmonotoniclogicsreflectthebehaviorofunderlyingsemanticstructures-setsof
classical models, equipped with relations of preference, operations of choice, and
related devices. The second volume traces the ways in which closely related tech-
niqueshavebeendevelopedtosupportneighboringareas–beliefchange,defeasible
inheritance,counterfactualconditionals,andmore.
Theauthor’smagisterialexpositionisaccompaniedbyintuitiveandheuristicasides,
andincludesmanysolvedexercisesthatwillbeofgreatassistancetothereader.Ev-
erymathematicallogicianworkinginthearea,orthinkingofgettingintoit,should
havethedoublevolumeonhandforconsultation,andanylibrarywithasectionon
mathematicallogicshouldhaveacopyforreference.
London,December2017 DavidMakinson
(LondonSchoolofEconomics)
V
Preface
Introduction
This book is a textbook, mainly discussing formal results and techniques for non-
monotonicandrelatedlogics.Therefore,motivationanddescriptionofthegeneral
context are quite limited, and given only when needed to understand the formal
material,ortoillustratethepathfromintuitiontoformalisation.
Thetextpresentsresultsandproofmethods,butalsotranslationsofmotivationaland
philosophicalconsiderationstoformalconstructions.Thisistrue,inparticular,for
Section 5.7 which contains hardly any formal results, but shows how to reconcile
various ideas in one formal construction. Section 7.3 is mostly about translating
intuitiverequirementsfordeonticlogicsintoformalproperties,too.Itisapersonal
book,inthesensethatittreatsmethodsandresultseitherused,or,morefrequently,
developedbytheauthor,sometimesincooperationwithcollegues.
Formoreintroductoryandmotivationalmaterial,thereaderisreferredtootherpub-
licationsbytheauthor,andinparticularto[Sch04].Itisstronglyadvisedtohavea
copyofthistextathand.
The intended audience is advanced students, but also researchers in the domain,
who want to have a compendium of methods developed by a colleague. The book
containsmanyexercises,thesolutionsareeitherintheappendix,orin[Sch04].
Manychaptersareratherindependentfromeachother,thenumbersofthechapters
give already a suggestion for reading. Chapter 3 however, contains many abstract
ideaswhichmayserveasintuitiveguidelines.(Itisputafterthechapteronprefer-
entialstructures,asthesestructuresgiveanexamplefromwhichthesizeconceptis
abstracted.)
VII
VIII Preface
PreviouslyPublishedMaterialandAcknowledgements
This text contains no new material - except the short Section 5.8. It is based on
previously published material, the following list will give the main sources, more
informationisgivenlocallyinthetext.
Inparticular,significantpartsinSection1.4,Section1.6,Section1.7,andSection
4.4werepublishedin[Sch04].
• Section 1.2 has evolved over time, mainly: [Sch92], [Sch96-1], [Sch00-1],
[Sch00-2],[GS08c],[GS09f].
• Section1.3:[Sch92],[Sch96-1],[Sch00-2].
• Section1.4:[Sch96-1],[Sch04],[GS08d],[GS10].
• Section1.5:[Sch96-1],[GS08a],[GS09f].
• Section1.6andSection1.7:[Sch04].
• Section1.8:[Sch92],[Sch95-3],[Sch97-2],[BLS99],[Sch99],[SGMRT00].
• Chapter2:[GS08b],[GS09f].
• Section3.1:[Sch97-4].
• Section3.2hasevolvedovertime,mainly:[GS08c],[GS09f].
• Section3.3:[Sch95-1],[Sch97-2].
• Section 4.2 has evolved over time, mainly: [SLM96], [LMS01], [Sch04],
[GS09f].
• Section4.3:[Sch91-1],[SLM96],[LMS01],[Sch04],[GS08f],[GS08h].
• Section4.4:[DS99],[SD01],[Sch04].
• Section5.2andSection5.3:[Sch97-2].
• Section5.4:[Sch93].
• Section5.5:[GS08f].
• Section5.6:[Sch90].
• Section5.7:[GS16].
• Chapter6:[GS09c],[GS10],[GS16].
• Chapter7:[GS10].
• Chapter8:[GS16].
• Section9.1:[SM94].
• Section9.2:[Sch91-2],[Sch97-2].
• Section9.3:[Sch95-2],[Sch97-2].
• Section9.4:[GS16].
Preface IX
I would like to thank my co-authors, and also Ronan Nugent of Springer, for his
verypatientandhelpfulcomments.
Frammersbach2017 KarlSchlechta
Contents (Volume II)
VolumeII TheoryRevision,Inheritance,andVariousAbstractProperties
4 TheoryRevisionandSums ......................................339
4.1 Introduction ...............................................339
4.2 BasicDefinitionsandOverview ..............................340
4.2.1 Introduction ........................................340
4.2.2 TheAGMApproach .................................341
4.2.3 AlgebraicProperties .................................344
4.2.4 LogicalProperties ...................................345
4.2.5 ConnectionsBetweenAlgebraicandLogicalProperties ...346
4.2.6 TablesforTheoryRevision ...........................348
4.3 TheoryRevision ...........................................350
4.3.1 Introduction ........................................350
4.3.2 RevisionbyDistance ................................351
4.3.2.1 Introduction ................................351
4.3.2.2 TheAlgebraicResults ........................351
4.3.2.3 TheLogicalResults .........................363
4.3.2.4 ThereIsNoFiniteCharacterization ............367
4.3.3 TheLimitCase .....................................370
4.3.4 RevisionandDefinability .............................371
4.3.4.1 “SoftCharacterisation”–TheAlgebraicResult ..372
4.3.4.2 TheLogicalResult ..........................374
XI
XII Contents(VolumeII)
4.3.5 Update ............................................375
4.3.5.1 Introduction ................................375
4.3.5.2 HiddenDimensions ..........................375
4.3.6 TheoryRevisionandProbability .......................384
4.3.6.1 Introduction ................................384
4.3.6.2 EpistemicPreferenceRelations ................386
4.3.6.3 Measuring Theories, and an Outlook for a
DifferentTreatmentofTheoryRevision .........392
4.3.7 RevisionandIndependence ...........................394
4.3.7.1 ProblemandBackground .....................394
4.3.7.2 Factorisation ...............................396
4.3.7.3 FactorisationandHammingDistance ...........401
4.3.8 ExtensionoftheMultipleRelationsIdeaof[BCMG04]to
theinfinitecase .....................................402
4.3.8.1 Introduction ................................402
4.3.8.2 TheFrameworkof[BCMG04] ................405
4.3.8.3 ConstructionandProof .......................407
4.4 SumsandtheFarkasAlgorithm ..............................417
4.4.1 Introduction ........................................417
4.4.1.1 TheGeneralSituationandtheFarkasAlgorithm ..417
4.4.1.2 UpdatebyMinimalSums .....................418
4.4.1.3 “Between”and“Behind” .....................420
4.4.2 TheFarkasAlgorithm ................................420
4.4.3 RepresentationforUpdatebyMinimalSums ............422
4.4.3.1 Introduction ................................422
4.4.3.2 AnAbstractResult ..........................423
4.4.3.3 Representation ..............................425
4.4.3.4 ThereIsNoFiniteRepresentationforOurType
ofUpdatePossible ..........................428
4.4.4 “Between”and“Behind” .............................430
4.4.4.1 ThereIsNoFiniteRepresentationfor“Between”
and“Behind” ...............................431
5 DefeasibleInheritanceTheory...................................435
5.1 Introduction ...............................................435