arithmetic and modularity in declarative languages for knowledge

190 Pages·2014·1.36 MB·English

ARITHMETIC AND MODULARITY IN DECLARATIVE
LANGUAGES FOR KNOWLEDGE REPRESENTATION
by
ShahabTasharrofi
M.Eng.,SharifUniversityofTechnology,2008
B.Eng.,IranUniversityofScienceandTechnology,2004
A THESIS SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DoctorofPhilosophy
inthe
SchoolofComputingScience
FacultyofAppliedSciences
(cid:13)c ShahabTasharrofi 2013
SIMONFRASERUNIVERSITY
Fall2013
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reproducedwithoutauthorizationundertheconditionsfor“FairDealing.”
Therefore,limitedreproductionofthisworkforthepurposesofprivatestudy,
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APPROVAL
Name: ShahabTasharrofi
Degree: DoctorofPhilosophy
TitleofThesis: Arithmetic and Modularity in Declarative Languages for Knowl-
edgeRepresentation
ExaminingCommittee: Dr.AndreiBulatov,AssociateProfessor,
ComputingScience,SimonFraserUniversity
Chair
Dr.EugeniaTernovska,AssociateProfessor,
ComputingScience,SimonFraserUniversity
SeniorSupervisor
Dr.DavidG.Mitchell,AssociateProfessor,
ComputingScience,SimonFraserUniversity
Supervisor
Dr.UweGla¨sser,Professor,
ComputingScience,SimonFraserUniversity
Supervisor
Dr.OliverSchulte,AssociateProfessor,
ComputingScience,SimonFraserUniversity
SFUExaminer
Dr.MiroslawTruszczynski,ExternalExaminer,
ProfessorofComputerScience,
UniversityofKentucky
DateApproved: December16,2013
ii
Partial Copyright Licence

iii
Abstract
Thepastdecadehaswitnessedthedevelopmentofmanyimportantdeclarativelanguagesforknowl-
edgerepresentationandreasoningsuchasanswersetprogramming(ASP)languagesandlanguages
that extend first-order logic. Also, since these languages depend on background solvers, the recent
advancements in the efficiency of solvers has positively affected the usability of such languages.
Thisthesisstudiesextensionsofknowledgerepresentation(KR)languageswitharithmeticalopera-
torsandmethodstocombinedifferentKRlanguages.
With respect to arithmetic in declarative KR languages, we show that existing KR languages
sufferfromahugedisparitybetweentheirexpressivenessandtheircomputationalpower. Therefore,
we develop an ideal KR language that captures the complexity class NP for arithmetical search
problemsandguaranteesuniversalityandefficiencyforsolvingsuchproblems.
Moreover,weintroduceaframeworktolanguage-independentlycombinemodulesfromdiffer-
ent KR languages. We study complexity and expressiveness of our framework and develop algo-
rithms to solve modular systems. We define two semantics for modular systems based on (1) a
model-theoretical view and (2) an operational view on modular systems. We prove that our two
semanticscoincideandalsodevelopmechanismstoapproximateanswerstomodularsystemsusing
the operational view. We augment our algorithm these approximation mechanisms to speed up the
processofsolvingmodularsystem.
We further generalize our modular framework with supported model semantics that disallows
self-justifying models. We show that supported model semantics generalizes our two previous
model-theoretical and operational semantics. We compare and contrast the expressiveness of our
framework under supported model semantics with another framework for interlinking knowledge
bases,i.e.,multi-contextsystems,andprovethatsupportedmodelsemanticsgeneralizesandunifies
different semantics of multi-context systems. Motivated by the wide expressiveness of supported
models,wealsodefineanewsupportedequilibriumsemanticsformulti-contextsystemsandshow
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thatsupportedequilibriumsemanticsgeneralizesprevioussemanticsformulti-contextsystems. Fur-
thermore, we also define supported semantics for propositional programs and show that supported
model semnatics generalizes the acclaimed stable model semantics and extends the two celebrated
propertiesofrationalityandminimalityofintendedmodelsbeyondthescopeoflogicprograms.
Keywords: KnowledgeRepresentationandReasoning,DeclarativeProblemSolving,Built-inArith-
metic,Modularity,Language-independence,SupportedSemantics,StableModelSemantics,Multi-
contextSystems
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Tomybeautifulwife,Hanie,forallheremotionalsupport
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“Puremathematicsconsistsentirelyofassertionstotheeffectthat,ifsuchandsuchapropositionis
trueofanything,thensuchandsuchanotherpropositionistrueofthatthing. Itisessentialnotto
discusswhetherthefirstpropositionisreallytrue,andnottomentionwhattheanythingis,of
whichitissupposedtobetrue... Ifourhypothesisisaboutanything,andnotaboutsomeoneor
moreparticularthings,thenourdeductionsconstitutemathematics. Thusmathematicsmaybe
definedasthesubjectinwhichweneverknowwhatwearetalkingabout,norwhetherwhatweare
sayingistrue. Peoplewhohavebeenpuzzledbythebeginningsofmathematicswill,Ihope,find
comfortinthisdefinition,andwillprobablyagreethatitisaccurate.”
—PrinciplesofMathematics,InternationalMonthly,vol. 4, BERTRAND RUSSELL,1901
vii
Acknowledgments
First and foremost, I would like to offer my utmost gratitude towards my senior supervisor, Dr.
Eugenia Ternovska for guiding me and supporting me with her insightful vision. Not only has
Eugeniaalwaysbeenasourceofinspiration,but,also,shehasinstilledinmethetwomostessential
skillsforaresearcher: findingworthwhilequestionsandsettingyourgoalscorrectly.
IwouldalsoliketoexpressmygratitudetowardstheothertwomembersofmyPhDcommittee:
Dr. DavidMitchellandDr. UweGla¨sser. DavidhasalwaysbeenfullofinterestingideasandUwe
hashelpedmediscoverinterestingrelatedworksinthefieldofSoftwareEngineering.
Moreover,IfeelobligedtoextendmysincerestgratitudetowardsthetwoexaminersofmyPhD
research: Dr. Oliver Schulte, the internal examiner and Dr. Miroslaw Truszczynski, my external
examinerfortheirhelpfulcommentsonmywork.
Furthermore,Iwanttothankmymanycollaboratorsduringtheseyearsthatinclude: Xiongnan
(Newman) Wu, Amir Aavani, Pashootan Vaezipoor, Alireza Ensan, Maarten Marien, etc. Without
them,manyofthediscussionsthatdirectlyaffectedthisthesiswouldhavesimplynotexisted.
Last but not the least, I would like to thank my family and my wife for believing in me and
helpingmeinwhateverwaypossiblethroughtheseyears.
ShahabTasharrofi
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Contents
Approval ii
PartialCopyrightLicense iii
Abstract iv
Dedication vi
Quotation vii
Acknowledgments viii
Contents ix
ListofFigures xiii
1 Introduction 1
1.1 Extendingthelanguageofasolver . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Combininglanguagesandsolvers . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Background 14
2.1 ModelExpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Multi-contexSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Built-inArithmeticinModelExpansion 18
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 OurGoals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.2 PreviousCloselyRelatedWork . . . . . . . . . . . . . . . . . . . . . . . 20
ix
3.1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 MotivatingExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 CapturingandNon-ExpressibilityResultsforPracticalKRLanguages . . . . . . . 31
3.4.1 CapturingResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.2 Non-expressibilityResultsunderComplexityAssumptions . . . . . . . . . 34
3.4.3 UncoditionalInexpressibilityResults . . . . . . . . . . . . . . . . . . . . 40
3.4.4 SafetyinASP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5 LogicPBINT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5.1 Bellantoni-CookCharacterizationofPTIME . . . . . . . . . . . . . . . . 47
3.5.2 NP⊆PBINTMX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5.3 PBINTMX⊆NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.6 PBINTastheBasisforaModelingLanguage . . . . . . . . . . . . . . . . . . . . 50
3.7 RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 ModularModelExpansion 56
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.1 ModelExpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.2 PartialStructuresandExtensions. . . . . . . . . . . . . . . . . . . . . . . 63
4.3 ModularSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.1 TheAlgebraofModularSystems . . . . . . . . . . . . . . . . . . . . . . 66
4.3.2 Model-theoreticSemanticsforModularSystems . . . . . . . . . . . . . . 68
4.3.3 FixpointSemanticsforModularSystems . . . . . . . . . . . . . . . . . . 70
4.4 ExpressivePower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5 ComputingModelsofModularSystems . . . . . . . . . . . . . . . . . . . . . . . 82
4.5.1 NaiveModularModelExpansionAlgorithm . . . . . . . . . . . . . . . . 82
4.5.2 PartialStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5.3 RequirementsontheModules . . . . . . . . . . . . . . . . . . . . . . . . 85
4.5.4 RequirementsontheSolver . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.5.5 LazyModularModelExpansionAlgorithm . . . . . . . . . . . . . . . . . 91
4.6 CaseStudies: ExistingFrameworks . . . . . . . . . . . . . . . . . . . . . . . . . 93
x