Table Of ContentAdvanced Information and Knowledge Processing
Series Editors
Professor Lakhmi Jain
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Professor Xindong Wu
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For other titles published in this series, go to
http://www.springer.com/4738
Michel Chein Marie-Laure Mugnier
Graph-based
Knowledge
Representation
Computational Foundations of Conceptual Graphs
(cid:65)(cid:66)(cid:67)
MichelChein Marie-LaureMugnier
LaboratoryofInformatics,Robotics,and LaboratoryofInformatics,Robotics,and
Micro-electronics(LIRMM) Micro-electronics(LIRMM)
FRANCE FRANCE
AI&KPISSN1610-3947
ISBN:978-1-84800-285-2 e-ISBN:978-1-84800-286-9
DOI10.1007/978-1-84800-286-9
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Preface
This book provides a definition and study of a knowledge representation and rea-
soningformalismstemmingfromconceptualgraphs,whilefocusingonthecompu-
tationalpropertiesofthisformalism.
Knowledge can be symbolically represented in many ways. The knowledge
representation and reasoning formalism presented here is a graph formalism –
knowledge is represented by labeled graphs, in the graph theory sense, and rea-
soning mechanisms are based on graph operations, with graph homomorphism at
thecore.
This formalism can thus be considered as related to semantic networks. Since
theirconception,semanticnetworkshavefadedoutseveraltimes,buthavealways
returnedtothelimelight.Theyfadedmainlyduetoalackofformalsemanticsand
the limited reasoning tools proposed. They have, however, always rebounded be-
causelabeledgraphs,schemasanddrawingsprovideanintuitiveandeasilyunder-
standablesupporttorepresentknowledge.
Thisformalismhasthevisualqualitiesofanygraphicmodel,anditislogically
founded.Thisisakeyfeaturebecauselogicshasbeenthefoundationforknowledge
representationandreasoningformillennia.Theauthorsalsofocussubstantiallyon
computationalfacetsofthepresentedformalismastheyareinterestedinknowledge
representationandreasoningformalismsuponwhichknowledge-basedsystemscan
be built to solve real problems. Since object structures are graphs, naturally graph
homomorphismisthekeyunderlyingnotionand,fromacomputationalviewpoint,
thismoorscalculustocombinatoricsandtocomputersciencedomainsinwhichthe
algorithmicqualitiesofgraphshavelongbeenstudied,asindatabasesandconstraint
networks.
The main notions are intuitively introduced from a knowledge representation
viewpoint, but the authors have also tried to give precise definitions of these no-
tionsalongwithcompleteproofsofthetheorems(readersonlyrequireafewsimple
mathematicalnotions,whicharesummarizedintheappendix).
Thisbookdoesnotpresentamethodologyforknowledgerepresentationnorde-
scribeactualapplications(exceptforonechapter)orsoftwaretools.Itpresentsthe-
oretical bases that merge numerous previous studies carried out in the conceptual
v
vi Preface
graphdomainsinceSowa’sseminalbookandlinksbasicreasoningissuestoother
fundamentalproblemsincomputerscience.
Inanutshell,theauthorshaveattemptedtoanswerthefollowingquestion:“How
far is it possible to go in knowledge representation and reasoning by representing
knowledgewithgraphs(inthegraphtheorysense)andreasoningwithgraphoper-
ations?”
Organization
Thebookisdividedinto3parts,13chaptersand1appendix.
Thefirstpartisdevotedtothekerneloftheformalism.InChap.2,Basicconcep-
tualGraphs(BGs)aredefined.BGsarestructuredbyasubsumptionrelationdefined
bythehomomorphismnotionbetweenBGs,aswellasbyelementaryspecialization
andgeneralizationoperations.InChap.3,SimpleconceptualGraphs(SGs)arein-
troduced.SGscanbesketchilydefinedasBGsaugmentedwithequality.Chapter4
is devoted to set and FOL semantics for SGs. The soundness and completeness
ofhomomorphismwithrespecttoentailmentrelationandFOLdeductionisproven.
Chapter5relatesBGhomomorphismwithhomomorphismsofotherstructures(e.g.,
hypergraphs,relationalstructures,conjunctivedatabasequeries)andsolvingacon-
straintnetwork.
The second part develops computational aspects of basic conceptual graphs.
Chapters 6, and 7 are devoted to algorithms for BG homomorphism. Chapter 8
presents other specialization and generalization operations that are not covered in
thefirstpart,e.g.,leastgeneralization,maximaljoinandotherextendedjoins.
The third part pools the kernel extensions. All of these extensions are provided
with logically sound and complete reasoning mechanisms based on graph homo-
morphism.Chapter9focusesonnestedconceptualgraphs,whicharehierachically
structuredgraphs.InChap.10,theimportantrulenotion,whichallowsrepresenta-
tionofimplicitknowledge,forinstance,isstudied.Theintroductionofrulesgives
thepowerfulnessofacomputabilitymodeltotheformalism.Positiveandnegative
constraints are considered in Chap. 11, and the BG family of models, combining
facts,rulesandconstraints,ispresented.Chapter12isdevotedtonegationincon-
ceptualgraphs.Thelastchapterpresentssemanticannotations—acurrentlyfavored
andpotentiallyfruitfulapplication.
Finally, the Appendix summarizes the basic mathematical notions used in the
book.
Eachchapterbeginswithanoverviewofitscontent.Itendswithbibliographical
notes devoted mainly to studies in the conceptual graph domain. Concerning the
references,theauthorschosetocitestudiesfromotherdomainsthroughoutthetext
andprovidetheconceptualgraphreferencesinthebibliographicalnotes.
Preface vii
Implementation
This book is about a KR formalism that can serve as a theoretical foundation for
knowledge-based systems. A distinction is made between a KR formalism (e.g., a
fragment of first order logic) and a KR programming language (e.g., PROLOG).
EvenwhenaKRprogramminglanguageisbasedonaKRformalism,theKRlan-
guagepresentsvariationstotheKRformalism(limitations,e.g.,Hornclausesorex-
tensions,secondorderfeatures,etc.),andithasaconcretesyntax,containsspecific
programmingconstructs(e.g.,thecutinPROLOG)andisequippedwithsoftware
tools. Between a KR formalism and a programming language implementing this
formalism,theremaybeKRtoolsandplatforms.Thisisthecurrentsituationofthe
graph-based KR formalism presented in the book. CoGITaNT [cog01a] contains
alibraryofC++classesthatimplementmostofthenotionsandreasoningmecha-
nisms presented here. CoGITaNT also contains a graphical interface and a client–
serverarchitecture.CoGITaNTthusallowsaprogrammertobuildknowledge-based
systems grounded on the formalism presented in this book. Let us also mention
COGUI [cog01b], a graphical user interface dedicated to the construction of a
knowledgebaseandwhichintegratesCoGITaNT.
Audience
The book is intended for computer science students, engineers and researchers.
Some of the materials presented in this book have been used for several years at
different academic levels, ranging from AI courses for graduate students to pro-
fessional and research master’s level courses. The mathematical prerequisites are
minimal,andtheAppendixoutlinesthemathematicalnotionsusedinthebook.
Herearesomesuggestionsforreadingpathsofthisbookdependingonreaders’
specificinterests.
Chapter1(introduction),Chap.2(basicconceptualgraphs),Chap.3(simplecon-
ceptualgraphs),thetwofirstsectionsofChap.4(setandfirstorderlogicsemantics)
arethecorechaptersandshouldbeconsideredasthebasis.
Forprogrammingandalgorithmicpurposesthefollowingmaterialscanbeadded
to the base: Chap. 6 (basic algorithms for BG homomorphism), the last section
ofChap.5(relationshipswithconstraintprogrammingtechniques),Chap.7(tech-
niquesfortreesandothertractablecases),Chap.8(algorithmsforotherspecializa-
tion/generalization operations, especially maximal joins), Chap. 10 (rule process-
ing),andChap.12(algorithmsforprocessingBGswithatomicnegation).
For modeling purposes the following parts can be added to the base: Chap. 9
(nested conceptual graphs), Chap. 10 (definition and use of rules), Chap. 11 (defi-
nitionanduseofconstraintsandtheircombinationwithrules),Chap.12(definition
anduseofatomicnegation)andChap.13(semanticannotations).
For more theory-oriented readers, expressivity, decidability and complexity re-
sults,aswellasstatingequivalencewithproblemsofotherdomains,arepresented
throughoutthebook,exceptinthelastchapter.
viii Preface
Acknowledgments
WeareindebtedtoJohnSowa,whoisthefoundingfatherofconceptualgraphs,and
thepresentbookwouldnothaveexistedwithouthispioneeringwork[Sow84].
Webeganworkingonconceptualgraphsin1992,andovertheyearsmanyofour
colleaguesandstudentshavecontributed,inonewayoranother,tothiswork.
Wewouldliketothank:
Friendsandcolleagueswhohavesupportedourapproachovertheyears:Franz
Baader, Marie-Christine Rousset, Pierre Marquis, Fritz Lehmann, Daniel Kayser,
MichelHabib,GuyChaty;
Colleaguesorstudentswithwhomwehavedirectlycollaboratedontopicscov-
ered in this book: Jean-Franc¸ois Baget, Boris Carbonneill, Olivier Cogis, David
Genest, Olivier Guinaldo, Ollivier Haemmerle´, Gwen Kerdiles, Michel Lecle`re,
AnnePreller,EricSalvat,Genevie`veSimonet,RallouThomopoulos;
Colleaguesfromtheconceptualgraphcommunitywithwhomwehavehadfruit-
ful discussions: Tru Cao, Fritjof Dau, Gerard Ellis, Jean Fargues, Bikash Gosh,
FritzLehmann,RobertLevinson,DicksonLukose,GuyMineau,JohnSowa,Mark
Willems,VilasWuwongse.
Our approach has benefited from numerous collaborations on applied research
projects, and we would like to thank: Christian Baylon, Alain Bonnet, Bernard
Botella, Olivier Corby, Patrick Courounet, Rose Dieng, Steffen Lalande, Philippe
Martin,PatrickTaillibert.
We acknowledge Jean-Franc¸ois Baget, David Genest, Alain Gutierrez, Michel
Lecle`re, Khalil Ben Mohamed, Nicolas Moreau, Fatiha Sa¨ıs, Eric Salvat, who re-
viewedpartsofthisbook,withspecialthanks toGenevie`ve Simonet(someofher
commentscouldbetheseedsforanewbook...).
WewouldalsoliketothankDavidManleyforrectifyingourFranglaisandBev-
erleyFordandherteamatSpringerLondonfortheirsupport.
Obviously,allremainingerrorsareourfaultalone.Wewelcomecorrectionsand
suggestionsonallaspectsofthebook;pleasesendthesetousat:
[email protected]@lirmm.fr.Awebsitecompan-
iontothebookcanalsobequeriedat:
http://www.lirmm.fr/gbkrbook
LaBoissie`re, MichelCheinandMarie-LaureMugnier
March2008
Contents
1 Introduction................................................... 1
1.1 KnowledgeRepresentationandReasoning ..................... 1
1.1.1 Knowledge-BasedSystems............................ 2
1.1.2 RequirementsforaKnowledgeRepresentationFormalism.. 3
1.2 ConceptualGraphs ......................................... 8
1.2.1 BasicNotions ....................................... 8
1.2.2 SubsumptionandHomomorphism...................... 9
1.2.3 FormalSemantics.................................... 11
1.2.4 FullCGs ........................................... 12
1.3 AGraph-BasedApproachtoKR.............................. 13
1.3.1 Motivations ......................................... 13
1.3.2 ExtensionsoftheBasicFormalism ..................... 14
1.3.3 SeveralApproachestoCGs............................ 16
PartI Foundations:BasicandSimpleConceptualGraphs
2 BasicConceptualGraphs ....................................... 21
2.1 DefinitionofBasicConceptualGraphs(BGs)................... 22
2.1.1 Vocabulary ......................................... 22
2.1.2 BasicConceptualGraphs.............................. 25
2.1.3 SubBGsandPseudoBGs .............................. 29
2.2 BGHomomorphism ........................................ 30
2.2.1 SubsumptionandHomomorphism...................... 30
2.2.2 BijectiveHomomorphismsandIsomorphisms ............ 33
2.2.3 BGQueriesandAnswers ............................. 34
2.3 BGSubsumptionProperties ................................. 35
2.3.1 SubsumptionPreorder ................................ 35
2.3.2 IrredundantBGs..................................... 37
2.4 GeneralizationandSpecializationOperations ................... 40
2.4.1 ElementaryGeneralizationOperationsforBGs ........... 40
2.4.2 GeneralizationandHomomorphism..................... 42
ix
x Contents
2.4.3 ElementarySpecializationOperations ................... 45
2.4.4 SpecializationandHomomorphism ..................... 48
2.5 NormalBGs............................................... 49
2.5.1 DefinitionofNormalBGs............................. 49
2.5.2 ElementaryOperationsforNormalBGs ................. 52
2.6 ComplexityofBasicProblems ............................... 54
2.7 BibliographicNotes ........................................ 56
3 SimpleConceptualGraphs...................................... 59
3.1 Introduction ............................................... 60
3.2 Vocabulary ................................................ 62
3.3 SimpleConceptualGraphs(SGs) ............................. 66
3.4 GeneralizationandSpecializationOperations ................... 69
3.5 StandardandNormalSGs ................................... 71
3.6 Coref-Homomorphism ...................................... 72
3.7 AntinormalFormandHomomorphism......................... 76
3.8 BibliographicNotes ........................................ 80
4 FormalSemanticsofSGs ....................................... 83
4.1 ModelSemantic ........................................... 84
4.1.1 ModelsofSGs ...................................... 84
4.1.2 SoundnessandCompletenessof(coref)Homomorphism ... 87
4.2 LogicalSemantic........................................... 89
4.2.1 TheFOLSemanticΦ ................................ 89
4.2.2 ModelSemanticandΦ ............................... 91
4.3 Positive,Conjunctive,andExistentialFragmentofFOL .......... 92
4.3.1 OrderedLanguageandL-Substitution................... 93
4.3.2 SoundnessandCompletenessRevisited.................. 94
4.3.3 RelationshipsBetweenSGsandFOL(∧,∃) .............. 95
4.3.4 AnotherFOLSemantic:Ψ ............................ 98
4.4 Note on the Relationships Between Description Logics
andConceptualGraphs......................................100
4.5 BibliographicNotes ........................................104
5 BGHomomorphismandEquivalentNotions ......................105
5.1 ConceptualGraphsandConceptualHypergraphs ................107
5.1.1 ConceptualHypergraphs ..............................107
5.1.2 DifferentKindsofVocabularies ........................110
5.2 Graphs ...................................................113
5.2.1 FromGraphstoBGs .................................114
5.2.2 FromBGstoGraphs .................................117
5.3 RelationalStructuresandDatabases ...........................119
5.3.1 RelationalStructuresandBGs .........................119
5.3.2 ConjunctiveQueriesandBGs..........................120
5.4 ConstraintSatisfactionProblem ..............................124
Contents xi
5.4.1 DefinitionofCSP....................................124
5.4.2 FromCSPtoBGs....................................127
5.4.3 FromBGstoCSP....................................128
5.5 BibliographicNotes ........................................132
PartII ComputationalAspectsofBasicConceptualGraphs
6 BasicAlgorithmsforBGHomomorphism.........................135
6.1 AlgorithmsforBGHomomorphisms ..........................135
6.1.1 BasicBacktrackAlgorithms ...........................136
6.1.2 BacktrackImprovements..............................141
6.2 ConstraintProcessing .......................................151
6.2.1 APanoramaofConstraintProcessingTechniques .........151
6.2.2 Arc-Consistency.....................................154
6.2.3 ForwardChecking ...................................159
6.3 LabelComparison..........................................161
6.3.1 BasicDataStructuresandAlgorithms ...................163
6.3.2 RelatedProblems ....................................164
6.3.3 TreeOrders .........................................166
6.3.4 PartitioninChains ...................................167
6.3.5 Lattices ............................................168
6.4 BibliographicNotes ........................................169
7 TractableCases................................................171
7.1 Introduction ...............................................171
7.1.1 AboutTractability ...................................172
7.1.2 TheStructureoftheTargetBGisofNoHelp.............172
7.2 TractabilityBasedontheMultigraph-AcyclicityoftheSourceBG .174
7.2.1 AcyclicMultigraphsandTrees.........................174
7.2.2 BGsTriviallyLogicallyEquivalenttoAcyclicBGs........184
7.3 TractabilityBasedontheHypergraph-AcyclicityoftheSourceBG.185
7.3.1 UseofaJoinTree....................................187
7.3.2 ConstructionofaJoinTree ............................191
7.3.3 EquivalencewiththeExistentialConjunctiveGuarded
Fragment ...........................................193
7.4 GeneralizationsofGraph-AcyclicityandHypergraph-Acyclicity ...198
7.4.1 GraphsandTreewidth ................................198
7.4.2 HypergraphsandHypertreewidth.......................200
7.4.3 ExpressivityResults..................................201
7.5 WhatAboutLabels? ........................................202
7.6 ComplementaryNotes ......................................204
