Reasoning on Data

ReasoningonData:

TheOntology-MediatedQueryAnsweringProblem

Marie-Laure Mugnier

University of Montpellier

UNILOGSchool–Vichy–2018

KNOWLEDGEREPRESENTATIONANDREASONING(KR)

•  A field historically at the heart of Artificial Intelligence

•  Study formalisms (or languages) to

•  represent various kinds of human knowledge •  do reasoning on these representations

•  along the tradeoff expressivity / tractability of reasoning

à KR languages based on computational logic In this talk: classical first-order logic (FOL)

Major conferences: IJCAI, AAAI, KR

M.-L.Mugnier–UNILOGSchool– 2018 2

Part1:Basics

Knowledge bases, Ontologies Logical view of Queries and Data Main KR formalisms to represent and reason with ontologies Ontology-Mediated Query Answering

Part2:KRformalismsandalgorithmicapproaches

Part3:DecidabilityissuesintheexistenFalruleframework

OVERVIEWOFTHETUTORIAL


Knowledge Base (KB)

Reasoning Services

•  General knowledge on the application domain « Cats are Mammals »

•  Factual Knowledge Description of specific individuals, situations, ...

Félix is a Cat

Factbase, Database

Fundamental tasks

•  Checking the consistency of the KB

•  Computing answers to a query over the KB

...

KNOWLEDGEBASEDSYSTEMS

Ontology

Knowledge expressed in a KR language

Reasoning algorithms associated with the KR language


WHATISANONTOLOGY?

In computer science: a formal specification of the knowledge of a particular domain Ø  which allows for machine processing Ø  that relies on the semantics of knowledge

> automated reasoning

Such a specification consists of Ø  a vocabulary in terms of concepts and relations Ø  semantic relationships between these elements


EXAMPLESOFONTOLOGIES

¢  Medecine and life sciences : hundreds of available ontologies

�  general medical ontologies

SNOMED CT (400 000 terms) GALEN (> 30 000 terms)

�  specialized medical ontologies FMA (anatomy) NCI (cancer), ...

�  biology �  agronomy

¢  Information systems of large organizations and corporations


ATTHECOREOFONTOLOGIES:CONCEPTS/CLASSES

¢  Concept/class:acategoryofenSSes(objects)thatshareproperSes InFOL:unarypredicate:Cat,Mammal

¢  Instanceofclass:aspecificmemberofthisclass

¢  FundamentalrelaSononclasses:specializaFon(subsumpFon)(«isakindof»,«subclassof»)

SemanScs:everyinstanceofC1isalsoinstanceofC2

C1

C2

C1 subclass of C2

�x (C1(x) à C2(x)) �x (Cat(x) à Mammal(x))

In FOL: a term (variable or constant)


ATTHEHEARTOFONTOLOGIES:CONCEPTS/CLASSES

Disease

LungDisease

Pneumonia

InfecSousPneumonia

InfecSousDisease

BacterialDisease

ViralDisease

BacterialPneumonia

Legionella

InfecSousAgent

Bacteria Virus

OrganismDisorder

However, an ontology is not just a classification!

Concepts organized by specialization

(SNOMEDCT)


ONTOLOGIESAREMUCHMORETHANCLASSIFICATIONS

Vocabulary 1.  concepts / classes

2.  relations (between instances)

« An ontology specifies the vocabulary of an application domain and semantic relationships between the terms of the vocabulary»

+ semantic relationships between concepts

+ semantic relationships between relations

+ properties of relations

+ other axioms that more generally express domain knowledge

+ properties of concepts


RELATIONSBETWEENINSTANCES

Often these are binary relations (also called « roles » or « properties »)

subrelation of

SignatureofarelaSon:assignsamaximumconcepttoeachargument(«domain»and«range»inOWL)

�x�y (hasCA(x,y) à dueTo(x,y))

dueTo

hasCausaFveAgent

�x�y (hasCA(x,y) à Disease(x) � Organism(y))

Argument1:aDisorderArgument2:anOrganismor[...]

Argument1:aDiseaseArgument2:anOrganism


EXAMPLESOFOTHERFREQUENTTYPESOFAXIOMS

•  Necessaryand/orsufficientproperSesofconcepts (ex: BacterialDisease)

•  Properties of relations

•  NegaSveconstraints(disjointnessbetweenconcepts,relaSons,...)

Bacteria∩Virus=� �x(Bacteria(x)�Virus(x)à )�x(Bacteria(x)à¬Virus(x))

�x (BacterialDisease(x) → �y (Bacteria(y) � hasCausativeAgent(x,y))

A bacterial disease is caused by a bacteria

inverse relations: �x�y (hasPart(x,y) ⟷ isPartOf(y,x))

symmetry, transitivity, ...

functional relation: �x�y�z (isPartOf(x,y) � isPartOf(x,z) → y = z)


WHATKINDSOFLANGUAGESTOEXPRESSONTOLOGIES?

HierarchiesofclassesHierarchiesofbinaryrelaSons(called«properSes»)SignaturesoftheserelaSons(«domain»and«range»)

àOWLDLfragmentofRDFSchema(SemanScWeb)

DescripFonLogicsRule-basedlanguages Datalog,existenSalrules,

RDFdeducSverules,AnswerSetProgramming...

Very light languages

More expressive fragments of first-order logics

From a logical viewpoint: anontologyiscomposedofafinitesetofpredicates(toexpressconceptsandrelaSons)afinitesetof(closed)formulasoverthesepredicates oftheform�X(condiFon[X]àconclusion[X])


WHATAREONTOLOGIESGOODFOR?

•  provideacommonvocabulary

àitiseasiertoshareinformaSon(typicallybetweenexpertsofseveraldomains)

•  constrainthemeaningofterms

àforcestoexplicitnot-saidthingsandtoremoveambiguiSeshencelessmisunderstandings

•  todoautomatedreasoning,basisofhigh-levelservices

à findimplicitlinksbetweenpiecesofknowledgeà checktheconsistencyoftheKB,finderrorsinmodelingà enrichdataqueryanswering


«findallpaSentsaffectedbyalungdiseaseduetoabacteria»

Data

Query (SQL, SPARQL, MongoDB ...)

ONTOLOGY-MEDIATEDQUERYANSWERING(EX:MEDICALRECORDS)

Database(relaSonal,RDF,NoSQL,...)

??

PaSentP:Diagnosis=«legionella»


DonnéesData

Query

Knowledge Base

AlegionellaisbacterialpneumoniaAbacterialpneumoniaisapneumoniaApneumoniaisalungdiseaseAbacterialpneumoniaiscausedbyabacteriaIfxiscausedbyythenxisduetoyIfthediagnosisofapaSentxcontainsadiseaseythenxisaffectedbyy

ONTOLOGY-MEDIATEDQUERYANSWERING

«findallpaSentsaffectedbyalungdiseaseduetoabacteria»

PaSentP:Diagnosis=«legionella»

Ontology


Knowledgebase

Query

Ontology

Addinganontologicallayerontopofdata

1-Enrichthevocabularyallowingtoabstractfromaspecificdatastorage

2-Infernewfacts,notexplicitelystored,allowingforincompletedatarepresentaSon

Data

ONTOLOGY-MEDIATEDQUERYANSWERING(OMQA)


Query

Ontology

Data

3–provideaunifiedviewofmulSplesources

Data Data



OMQAEXAMPLE:ONTOLOGICALKNOWLEDGE

AlegionellaisbacterialpneumoniaAbacterialpneumoniaisapneumoniaApneumoniaisalungdiseaseAbacterialpneumoniaiscausedbyabacteriaIfxiscausedbyythenxisduetoyIfthediagnosisofapaSentxcontainsadiseaseythenxisaffectedbyy

�x(Legionella(x)→BacterialPneumonia(x))

�x(BacterialPneumonia(x)→�y(hasCausaSveAgent(x,y)�Bacteria(y)))

�x�y(hasCausaSveAgent(x,y)→dueTo(x,y))

�x�y((Diagnosis(x,y)�Disease(y))→isAffectedBy(x,y))


FACTBASE

RelaSonalschema: finitesetRofrelaSonsàpredicates infinitedomainofvaluesàconstants

Factbase:usuallyasetofgroundatoms(ontheontologicalvocabulary)seenastheconjuncSonoftheseatoms

F={PaSent(P),Diagnosis(P,M),Legionella(M)}

ArelaFonaldatabasemaynaturallybeviewedasafactbase

InstanceofarelaSonr: finitesetoftuplesonràatomsonr

rakr1 akr2a1a2a1

a2a3a1

«Thediagnosisforthepa4entPislegionella»

Databaseinstance={instanceforeachrinR} àfactbase

{r(a1,a2),r(a2,a3),r(a1,a1)}


CONJUNCTIVEQUERIES(CQ)

« find all patients affected by a lung disease due to a bacteria»q(x) = ∃y ∃z (Patient(x) � isAffBy(x,y) � LungDisease(y) � dueTo(y,z) � Bacteria(z)) A CQ is an existentially quantified conjunction of atoms The free variables are the answer variables If closed formula: Boolean CQ DatalognotaSonans(x)ßPaSent(x),isAffBy(x,y),LungDisease(y),dueTo(y,z),Bacteria(z)Select-Join-ProjectqueriesinrelaSonalalgebra(SQL)SELECT...FROM…WHERE<joincondi4ons> SPARQL(semanScwebqueries)SELECT…WHERE<basicgraphpa=ern>


ANSWERSTOACONJUNCTIVEQUERY

¢  TheanswertoaBooleanCQqinFisyesifF�qyes=()

¢  LettheCQq(x1,...,xk).Atuple(a1,…,ak)ofconstantsisananswertoqwithrespecttoafactbaseFif F�q[a1,...,ak],whereq[a1,...,ak]isobtainedfromq(x1,...,xk)byreplacingeachxibyai

¢  LetFandqbeseenassetsofatoms.AhomomorphismhfromqtoFisa

mappingfromvariables(q)toterms(F)suchthath(q)F

F�q()iffqcanbemappedbyhomomorphismtoF(a1,…,ak)isananswertoq(x1,...,xk)w.r.t.Fiff

thereisahomomorphismfromqtoFthatmapseachxitoai


KEYNOTION:HOMOMORPHISM

q(x)=∃y(movie(y)∧play(x,y))

HomomorphismhfromqtoF:subsStuSonofvar(q)byterms(F)suchthath(q)F

F

h1:xàayàm1

h2:xàayàm2

h1(q)=movie(m1)∧play(a,m1)

h2(q)=movie(m2)∧play(a,m2)

movie(m1)movie(m2)movie(m3)actor(a)actor(b)actor(c)play(a,m1)play(a,m2)play(c,m3)

Answers:obtainedbyrestricSngthedomainsofhomomorphisms toanswervariables

x = a x = c

h3:xàcyàm3 h3(q)=movie(x0)∧play(c,m3)

movie(y)play(x,y)


« find all patients affected by a l u n g d i s e a s e d u e t o a bacteria »

q(x) = ∃y∃z(PaSent(x)�isAffectedBy(x,y)�LungDisease(y)�dueTo(y,z)�Bacteria(z))

Factbase = { Patient(P), Diagnosis(P,M), Legionella(M) } « The diagnosis for the patient P is legionella »

Legionellaspecialisa4onofLungDiseaseandBacterialDisease(andDisease)henceLungDisease(M) henceBacterialDisease(M),

Disease(M)

�x(BacterianDisease(x)→�y(hasCausaSveAgent(x,y)�Bacteria(y)))hencehasCausaSveAgent(M,b)andBacteria(b)

�x�y(hasCausaSveAgent(x,y)→dueTo(x,y))hencedueTo(M,b)

�x�y((Diagnosis(x,y)�Disease(y))→isAffectedBy(x,y))henceisAffectedBy(P,M) Answer : x = P

ONTHEOMQAEXAMPLE


AMOREGENERALSCHEMA

Query

Ontology

Data Data Data

Mappingsfromdatatofacts

{Databasequery⤳Facts}

Factbase

Conceptuallevel

DescripSonoftheapplicaSondomainwithahighabstracSonlevel

Queryusingthevocabularyoftheontology

Factbase(possiblyvirtual)usingthevocabularyoftheontology

Independentandheterogeneousdatasources

Theanswerstothequeryareinferredfromtheknowledgebase

«Ontology-BasedDataAccess»[Poggietal.,JoDS,2008]


MAPPINGS

q(x):�n�sPaSent_T(x,n,s) ⤳PaSent(x)q’(x):�n�sPaSent_T(x,n,s)�DiagnosSc_T(x,y)�y=«Legionella»

⤳�z(diagnosis(x,z)�legionella(z))

Patient_T [ID_PATIENT, NAME,SSN] Diagnosis_T[ID_PATIENT, DISORDER]

PaSent/1Diagnosis/2Legionella/1

Mapping:databasequery(X)⤳conjuncFonwithfreevariablesX

PaSent(P)Diagnosis(P,M)Legionella(M)

PaSent_T Diagnosis_T

......

id ssn disidname

⤳ P....

..

..

..

..

..

..

«Leg.»....

P....


Knowledgebase

Query

Ontology

Factbase


(Boolean)conjuncSvequeryq

TheoryOinasuitableFOLfragment

Setofgroundatoms(orexistenSallyclosedformula)F

Fundamental decision problem

O, F � q ?


OVERVIEWOFTHELECTURE

Part1:Basics


Outline of description logics – Horn DLs Existential Rules Materialization approach (forward chaining) Query rewriting approach (related to backward chaining)



DESCRIPTIONLOGICS

¢  AfamilyofKRlanguagesforrepresenSngandreasoningwithontologies

¢  MostlycorrespondtodecidablefragmentsofFOL

(relatedtomodalproposiSonallogic,theguardedfragmentofFOL,...)¢  Variable-freesyntax

¢  Usedtobecalled«conceptlanguages»:fromconceptandrolenames(unaryandbinarypredicates)andasetofconstructorsdefinecomplexconcepts(morerecently:complexroles)

¢  Anontologyisasetofaxiomsthatstateinclusionsbetweenconcepts (andbetweenroles)


DESCRIPTIONLOGICS:BUILDINGBLOCKS(SYNTAX)Vocabulary

Atomicconcepts: Human,Parent,Student…(unarypredicates)Atomicroles: parentOf,siblingOf,… (binarypredicates)

Complexconceptsandrolescanbebuiltusingasetofconstructors(whichdependsoneachparFcularDL)

conjuncSon(П),disjuncSon(�),negaSon(¬)HumanП¬Parent Female�Male

restrictedformsofexistenSalanduniversalquanSficaSon(�,�)

∃parentOf.(FemaleПStudent) �parentOf.Male inverseofarole(-),composiSonofroles(o) ∃parentOf- parentOfoparentOf


DESCRIPTIONLOGICS:BUILDINGBLOCKS(SEMANTICS)

ToeachconceptisassignedaFOLsentencewithfreevariablex Human Human(x)

HumanП¬Parent Human(x)�¬Parent(x)∃parentOf.(FemaleПStudent) ∃y(parentOf(x,y)�Female(y)

�Student(y))�parentOf.Female �y(parentOf(x,y)àFemale(y))ToeachroleisassignedaFOLsentencewith2freevariablesxandy parentOfoparentOf ∃z(parentOf(x,z)�parentOf(z,y))


DESCRIPTIONLOGICS:KNOWLEDGEBASE

KnowledgeBase=TBox(ontology)+ABox(factbase)Tbox:axiomsoftheformC1�C2∀x(fol(C1)àfol(C2))

orr1�r2 ∀x∀y(fol(r1)àfol(r2))

Abox:setofgroundfacts parentOf(A,B),Female(A),…

Human�Male�Female ∀x(Human(x)àMale(x)�Female(x))Adult�¬Child ∀x(Adult(x)∧Child(x)à⊥)Parent��parentOf ∀x(Parent(x)à�yparentOf(x,y))HappyFather��parentOf.Female ∀x(HP(x)à(�y(parentOf(x,y)àFemale(y))Human��parentOf-.Human ∀x(Human(x)à�y(parentOf(y,x)∧Human(y)))parentOfoparentOf�ancestorOf ∀x∀y(�z(parentOf(x,z)∧parentOf(z,y))à

ancestorOf(x,y)


DESCRIPTIONLOGICS:STANDARDREASONINGTASKSStandardreasoningtasksonaKB(T,A)

¢  ConceptsubsumpSon T�C�D?¢  ConceptsaSsfiability isCsaSsfiablew.r.t.T ?¢  KBsaSsfiability is(T,A)saSsfiable?

¢  Instancechecking (T,A)�C(b),wherebisaconstant?

ConceptsubsumpSon T�C�Diff(T, {C(a),¬D(a)})unsaSsfiableConceptsaSsfiability CsaSsfiablew.r.t.T iff(T, {C(a)})saSsfiable

Instancechecking (T,A)�C(b)iff(T,A�{¬C(b)})unsaSsfiable

AllthesetaskscanbeexpressedintermsofKB(un)saSsfiabilityprovidedthattheconstructorsintheconsideredDLallowforit

Queryansweringbeyondinstancechecking?cannotbereducedtothestandardreasoningtasks


EVOLUTIONOFDLS

¢  Concepts:

¢  TBoxaxioms:onlyconceptinclusions

SaSsfiabilityandinstancecheckinginALCare:EXPTIME-completeincombinedcomplexitycoNP-completeindatacomplexity

Evenworseifweaddinverseroles:2EXPTIME-completeincombinedcomplexity

StandardexpressiveDL ALC


TWOCOMPLEXITYMEASURESFORQUERYANSWERINGPROBLEMS

Combined complexity (usual complexity measure) The input is O, F and q

Data complexity

The input is F (O and q supposed to be fixed)

This distinction comes from database theory: the size of the query is negligible compared to the size of the data

Problem: Given a KB = (O, F), with O the ontology and F the factbase, and a query q, is q entailed by the KB?

E.g.,qBooleanCQ,FfactbaseDoesF�q?

NP-complete(combined)PTime(data)


EVOLUTIONOFDLS

¢  Concepts:

¢  TBoxaxioms:onlyconceptinclusions

SaSsfiabilityandinstancecheckinginALCare:EXPTIME-completeincombinedcomplexitycoNP-completeindatacomplexity

Evenworseifweaddinverseroles:2EXPTIME-completeincombinedcomplexity

TwofactorsledtotheevoluFonofdescripFonlogics:1.  pracScaluse(e.g.SNOMEDCT):peoplemostlyuseconjuncSonandexistenSal

quanSficaSon2.  complexitytoohighforqueryansweringproblems

StandardexpressiveDL ALC


NEWDLSWITHLOWERCOMPLEXITY

DL-LiteR EL

Commonfeature:nodisjuncSon(no«true»negaSon) ThenasaSsfiableKBhasauniquecanonicalmodelM:

ForanyBooleanCQq,KB�qiffMisamodelofqReasoningtechniquesfortheselighterDLsareverysimilartoforwardorbackwardchaininginrule-basesystems

where

where LargeTBoxesClassificaSon

LargeABoxesQueryanswering


COMPLEXITYINTRODUCEDBYDISJUNCTIONORNEGATION

q(): �x�y (Blue(x) � on(x,y) � Other(y))

CBA

T: T � Blue � Other A: Blue(A), Other(C), on(A,B), on(B,C)

To answer q, we have to consider two cases: in each model of the KB, either Blue(B) or Other(B) holds

Similarly if we replace T by: ¬Blue � Other (equivalentaxiom)

KB (T,A)

Note that Other � ¬Blue is harmless: it is just a disjointness constraint


INSUMMARY

DL ontology (TBox) has axioms of the form

∀x (fol(C1) à fol(C2))

∀x∀y (fol(r1) à fol(r2)) wherefol(r)isapathofatomicrolesortheirinverses

With the new DLs: left and right parts of the implication are both existentially quantified conjunctions of atoms

called « Horn description logics »

DLs essentially satisfy the tree model property:

if a KB is satisfiable then it has a « tree-shaped » model


WHY«HORNDLS»ONANEXAMPLE

ELAxiom

Letusskolemize(uandvresp.replacedbyf1(x)andf2(x)):weobtainasetof3Hornclauses(withskolemterms)

HencethenameHorndescripFonlogics

FOLtransla4on

prenexform


X,Y,Z:setsofvariables

∀x(actor(x)à∃zplay(x,z))

EXISTENTIALRULES

∀X∀Y(Body[X,Y]à∃ZHead[X,Z])

any positive conjunction (without functional symbols except constants)

Key point: ability to assert the existence of unknown entities

Crucial for representing ontological knowledge in open domains

See « value invention » in databases

∀x ∀y ( siblingOf(x,y) à ∃ z (parentOf(z,x) ∧ parentOf(z,y)) )

we often simplify by omitting universal quantifiers


DATA/FACTS

m1m2?x

RelaSonaldatabase RDF

rdf:type

am1am2c?x

ex:actor

AbstracFoninfirst-orderlogic(FOL)

∃x(movie(m1)∧movie(m2)∧movie(x)actor(a)∧actor(b)∧actor(c)play(a,m1)∧play(a,m2)∧play(c,x))

Etc.

ex:play Movie PlayActor

abc

ex:b

rdf:type

rdf:type

ex:movie

rdf:type ex:a ex:m1

ex:c

rdf:type

ex:m2

ex:play

WegeneralizeheretheclassicalnoSonofafactbyallowingexistenSalvariablesfact/factbase=existenFallyclosedconjuncFonofatoms

rdf:type

......

...

...

m_id a_id m_id a_id

_:x


LABELLEDHYPERGRAPH/GRAPHREPRESENTATION¢  Afactorasetoffactscanbeseenasasetofatoms

movie(m1),movie(m2),movie(x),actor(a),actor(b),actor(c),play(a,m1),play(a,m2),play(c,x)àhenceahypergraphoritsassociatedbiparFte(mulF-)graph

pvie

12

34

p(x,y,a,x),r(x,y)

•  one(labelled)nodeperterm•  one(labelled)nodeperatom(~hyperedge)•  totallyorderededges

1

2

a

r

x

y


movievie

m1 m2

a b c

movievie

actorvie

playvie

movievie

1 1 1

1 11

2 2 2

1 1 1

Ifpredicatesareatmostbinary:atomnodescanbereplacedbylabelsanddirectededges

a

m1 m2

b c

play

actor actoractor

movie movie movie

actor

actor

play

play

movie(m1),movie(m2),movie(x),actor(a),actor(b),actor(c),play(a,m1),play(a,m2),play(c,x)


GRAPHHOMOMORPHISMS(1)

•  LetG1=(V1,E1)toG2=(V2,E2)beclassicalgraphs.HomomorphismhfromG1toG2: mappingfromV1toV2s.t.

foreveryedge(u,v)inE1,(h(u),h(v))isinE2

mapsto mapsto


a

m1 m2

b c play

actor actoractor

movie movie movie




•  Iftherearelabels:theyhavetobe``kept’’aswell

play

movie

q F





•  Iftherearelabels:theyhavetobe``kept’’aswell

play 1

2

1 movie

qF

movievie

m1 m2

a b c

movievie

actorvie

playvie

movievie

1 1 1

1 11

2 2 2

1 1 1

actor

actor

play

play


∀x ∀y ( siblingOf(x,y) à ∃ z (parentOf(z,x) ∧ parentOf(z,y)) )

graph graph

P

P 2

S

1

2

2

1

1

GRAPHVIEWOFEXISTENTIALRULES

∀X∀Y(Body[X,Y]à∃ZHead[X,Z])

s

p

p

Theruleheadhas2kindsofvariables:-fronFer:sharedwiththebody-existenFal(new``blank’’nodes)

x x

y

y

zz


F = siblingOf(a,b)

R = ∀x ∀y (siblingOf(x,y) à ∃ z (parentOf(z,x) ∧ parentOf(z,y)))

F�=∃z0(siblingOf(a,b)∧parentOf(z0,a)∧parentOf(z0,b))

s

p

p

a

b

s

RisapplicabletoFifthereisahomomorphismhfrombody(R)toF

xàayàb

ApplyingRtoFw.r.t.hproducesF∪h(head(R))whereanewvariableiscreatedforeachexistenSalvariableinR

a

b

s

p

p

GENERATIONOFFRESH(UNKNOWN)INDIVIDUALS

R F


EXISTENTIALRULEFRAMEWORK(LOGICAL/GRAPHICAL)

q(x)=∃y(movie(y)∧play(x,y))

Data/Facts

« Pure » existential rules

Equality rules

Negative Constraints

Conjunctive Queries

∀x ( actor(x) à ∃ z (movie(z) ∧ play(x,z)) )

∀x ∀y ∀z ( movie(y) ∧ director(x,y) ∧ director(z,y) à x = z )

∀x ( movie(x) ∧ person(x) à ⊥ )

movie(m1) play(a,m1) play(c, x) ...


MULTIPLETHEORETICALFOUNDATIONS

Datalog(70-80s)

ConceptualGraphs[Sowa1984]

logical translation

+ « value invention »

��-rules,existenFalRules[Baget+IJCAI2009]

Datalog+/- [Cali+PODS2009]

[CheinMugnier1992,2009]

•  Samelogicalformas«Tuple-GeneraSngDependencies»(TGDs)longstudiedinrelaSonaldatabases

LightweightDescripSonLogics,e.g.OWL2tractableprofilesMoregenerally,HornDLs

+ «unrestricted cycles » on variables + unbounded arity


u  TheFOLtranslaSonofHornDLsyieldsexistenSalrulesu  ExistenSalrulesarestrictlymoreexpressive:

siblingOf(x,y)à∃z(parentOf(z,x)∧parentOf(z,y))cannotbeexpressedinmostDLsbecauseofthe«cycleonvariables»(needsrolecomposi4on: s�pop)

u  Theunboundedpredicatearityallowsformoreflexibility:àdirecttranslaSonofdatabaserelaSonsàaddingcontextualinformaSoniseasy(provenance,trust,etc.)

EXISTENTIALRULESAREMOREEXPRESSIVETHANHORN-DLS

s p

p

x

y

z

Morecomplexinterac4onsbetweenvariablescannotbeexpressedatallinDLs

Unsurprisingly,thisaddedexpressivityhasacost


¢  Fundamentaldecisionproblem

Input:K=(F,R)knowledgebase qBooleanconjuncSvequeryQuesSon:isqentailedbyK ?¢  Thisproblemisnotdecidable

f.i.[BeeriVardiICALP1981]onTGDsevenwithasinglerule[Baget&al.KR2010]à  find«decidable»classesofrules

withgoodexpressivity/tractabilitytradeoff

EXISTENTIALRULEFRAMEWORK

Data/Facts

« Pure » existential rules

Equality rules

Negative Constraints

Conjunctive Queries


atomic body

frontier-1

weakly- guarded

weakly frontier-guarded

datalog

guarded

weaklyacyclic

acyclic Graph of Rule Dependencies

wa-GRD jointly- acyclic

frontier- guarded

jointly-fg

sticky-join

w-sticky-join

sticky

w-sticky

1970s

2003 2004

Since 2008

(PARTIAL)MAPOFDECIDABLECLASSES

DL-LiteR EL


FUNDAMENTALNOTIONSFORREASONINGINFOL(�,∧)

¢  BacktotheposiFveconjuncFveexistenFalfragmentofFOL:FOL(�,∧)

¢  Allowstoexpressfactsand(Boolean)conjuncFvequeries¢  Suchformulascanbeseenassetsofatoms,labelledgraphs,relaSonal

structures,...¢  HomomorphismisafundamentalnoSoninthisfragment:

�  AninterpretaSonI isamodelofasentencefiffthereisahomomorphism

fromftoI

�  OnecandefinehomomorphismsbetweeninterpretaSons.Then:IfI1mapstoI2then,foranyf,I1modeloff�I2modeloff

�  Toaformulaf,weassignitsisomorphicmodelM(f)(akacanonicalmodel)


MODELISOMORPHICTOAFOL(�,∧)FORMULA

ToaformulafinFOL(�,∧),weassignitsisomorphicmodelM(f) alsocalledcanonicalmodel

f=�x�y�z(p(x,y)∧p(y,z)∧r(x,z,a))

M(f): D={dx,dy,dz,a} pM(f)={(dx,dy),(dy,dz)} rM(f)={(dx,dz,da)}

ThecanonicalmodelM(f)isuniversal:forallM’modeloff,M(f)mapstoM’

foranyfandginFOL(�,∧),g�fiffM(g)isamodeloffifffmapstog


ADDINGRANGE-RESTRICTED(=DATALOG)RULESTOFACTS

K = (F, R) whereRisasetofrange-restrictedrules(i.e.,var(head)var(body))

Fisafactbase(ruleswithanemptybody):groundatomsByapplyingrulesfromRstarSngfromF,auniqueresultisobtained:thesaturaFonofF(denotedbyF*)F*isfinitesincenonewvariableiscreatedF*isacore(noredundancies)

F

q

R

F*

TheniceproperSesofFOL(�,∧)arekept:

F*isauniversalmodelofK Hence:foranyCQq,K ⊨ q iff q maps to F*


KNOWLEDGEBASESWITHEXISTENTIALRULES

K = (F, R) whereRisasetofexistenFalrulesFisafactbase(ruleswithanemptybody):existenSalconjuncSonsofatoms

Mainchange:F*canbeinfinite

R=person(x)à�yhasParent(x,y)∧person(y)

∧person(y0)∧hasParent(a,y0)

∧person(y1)∧hasParent(y0,y1) Etc.

F=person(a)

butitremainsauniversalmodelhence K ⊨ q iff q mapstoF*


F

q

R

« bottom-up » « chase » (TGDs)

APPROACH1TORULES:FORWARDCHAINING/MATERIALISATION

K= (F, R)

K ⊨ q iff q maps by homomorphism to F*

Pros: materialisation offline, then online query answering is fast Cons: volume of the materialisation

needs writing access rights to the data not feasible if data is distributed among several databases not adapted if data change frequently

F*


EXAMPLE(MATERIALIZATION)

x = a y = m1 x = a y = m2 x = b y = z0 x = c y = x0

movie(m1)movie(m2)movie(x0)movieActor(a)movieActor(b)play(a,m1)play(a,m2)play(c,x0)

∀x(movieActor(x)à∃z(movie(z)∧play(x,z)))

q(x) = ∃ y (movie(y) ∧ play(x, y)) « find those who play in a movie »

SaturaFon

movie(z0) play(b,z0)


« top-down » decomposition into 2 steps [DL-Lite]

APPROACH2TORULES:BACKWARDCHAINING/QUERYREWRITING

R

Rewriting into a set of CQs, seen as a union of conjunctive queries (UCQ) and more generally into a « first-order » query (core SQL query)

F

q K= (F, R)

Query rewriting is independant from any factbase. For any F, F,R�qiffF�Q(i.e.,ifQisaUCQ:thereisqi�QwithF�qi)

Q

Pros: independent from the data Cons: rewriting done at query time, easily leads to huge and unusual queries


EXAMPLE

Rewq(x) = ∃ y (movie(y) ∧ play(x, y)) � movieActor(x) Query rewriting

x = a y = m1 x = a y = m2 x = c y = x0

x = a x = b

movie(m1)movie(m2)movie(x0)movieActor(a)movieActor(b)play(a,m1)play(a,m2)play(c,x0)

∀x(movieActor(x)à∃z(movie(z)∧play(x,z)))

q(x) = ∃ y (movie(y) ∧ play(x, y)) « find those who play in a movie »


BACKWARD CHAINING SCHEME!

UnificaSonbyaunifieru(ofq’andh’)

Body Headq’

QueryrewriSng

Body

Basicstep:Queryq RuleR

Newquery

h’

Direct rewriting of q with R and u = u(q \ q’) � u(body(R))


BASICPROPERTIES(1)

LetF2beobtainedfromF1bytheapplicaSonofRuleRLetaqueryQ1thatmapstoF2byahomomorphismthatusesatleastoneatom

broughtbyRThenthereisQ2,adirectrewriSngofQ1withR,suchthatQ2mapstoF1

F1 F2application of R

Q1

h1

Q2

h2

direct rewriting with R

and h1usesF2\F1

The reciprocal property holds


BASICPROPERTIES(2)

F1 F2application of R

Q1

h1

Q2

h2

direct rewriting with R

LetQ2beadirectrewriSngofQ1withRuleRLetF1beafactbasesuchthatQ2mapstoF2ThenthereisanapplicaSonofRtoF1thatproducesF2suchthatQ2mapstoF1


ForanyconjuncSvequeryq,foranyfactbaseF,foranysetofrules:thereisahomomorphismfromqtoF’,whereF’isobtainedfromFbyaruleapplicaSonsequenceoflength≤niffthereisahomomorphismfromq’toF,whereq’isobtainedfromqbyarewriSngsequenceoflength≤n

EQUIVALENCEDERIVATION/REWRITINGSEQUENCES

F1 F2

Q1

h1

Q2

h2

F1 F2

Q1

h1

Q2

h2

s.t. h1 usesF2\F1


TAKINGINTOACCOUNTEXISTENTIALVARIABLESINRULEHEADS(1)

¢  WewantacompletesetofsoundrewriSngs(setofCQs):qis.t.foranyF,ifF⊨qithenF,R⊨q

R=person(x)à�yhasParent(x,y)q=hasParent(v,w),denSst(w)u={x�v,y�w}rew(q,R,u)=qi=person(v),denSst(w)

qiisunsound:F=person(Maria),denSst(Giorgos)F⊨qihowever(F,{R})doesnotentailq

(1)IfwinqisunifiedwithanexistenSalvariableofR,thenallatomsinwhichwoccurmustbepartoftheunificaSon


TAKINGINTOACCOUNTEXISTENTIALVARIABLESINRULEHEADS(2)

R=p(x)à�z1�z2r(x,z1),r(x,z2),s(z1,z2)q=r(v,w),s(w,w)u={x�v,z1�w,z2�w}rew(q,R,u)=qi=p(v)(2)AnexistenSalvariableofRcannotbeunifiedwithanotherterminhead(R)

qiisunsound:F=p(a)F⊨ qi however(F,{R})doesnotentailq


PIECE-UNIFIER(FORBOOLEANCQS)

Apiece-unifieruofq’�qandh’head(R)isasubsStuSonofvar(q’+h’)byterms(q’+h’)[ifxisunchanged,wewriteu(x)=x]suchthat:¢  u(q’)=u(h’)

¢  existenSalvariablesofh’areunifiedonlywithvariablesofq’thatdonotoccur in(q\q’)(i.e.,ifxisexistenSalandu(x)=u(t),thentisavariableofq’andnotof(q\q’))

Queryq RuleR

q’ body head

h’

variablessharedbyq’and(q\q’)

ToextendthenoSontogeneralCQs:universalvariablescannotbeunifiedwithanswervariables


EXAMPLE

R=twin(x,y)à�zmotherOf(z,x)∧motherOf(z,y)

q=motherOf(v,w)∧motherOf(v,t)∧Female(w)∧Male(t)?



piece-unifieru1 = {z � v,x� w, y � w}

rewrite(q,R,u1)=motherOf(v,t) ∧Female(w)∧Male(w) ∧ twin(w,w)



piece-unifieru2 = {z � v,x � w,y � t}

rewrite(q,R,u2)=twin(w,t)∧Female(w)∧Male(t)

Ifwerewriteagainthisquerywecouldremovethefirstatom


WHATIFWESKOLEMIZEDRULES?

qiisunsound:F=person(Maria),denSst(Giorgos)F⊨qihowever(F,{R})doesnotentailq

R=person(x)à�yhasParent(x,y)q=hasParent(v,w),denSst(w)u = { x � v, y � w } rew(q,R,u)=qi=person(v),denSst(w)

Skolem(R)=person(x)àhasParent(x,f(x))

ClassicalmostgeneralunifierofhasParent(x,f(x))andhasParent(v,w):v�xandw�f(x)rew(q,R,u)=denSst(f(x))�person(x) whichcannotbeunifiedwitharulehead

(wouldnotbekeptintheouputsinceitcontainsaskolemfuncSon

Wecouldskolemizetherulesandrelyonusualm.g.u.thenkeeponlyrewriSngswithoutskolemfuncSon

butthiswouldcreateuselessintermediaterewriSngs


WHY«PIECES»?

Apieceisaunitofknowledgebroughtbyarule:¢  FronFervariables(andconstants)actascutpointstodecomposeruleheads

intopieces(«minimalnon-emptysubsetsgluedbyexistenSalvariables»)

R=b(x)à�y�zp(x,y)∧p(y,z)∧p(z,x)∧q(x,x) x y

z

¢  Arulewithkpiecescanbedecomposedintokrules,oneforeachpiece,whilekeepingthesamebody

¢  Itcannotbefurtherdecomposed(exceptbyintroducingnewpredicates)

b(x)à�y�z p(x,y)∧p(y,z)∧p(z,x)

b(x)àq(x,x)


DECOMPOSITIONOFRULESINTOATOMICHEADRULES(1)

R:b(x)à�y�z p(x,y)∧p(y,z)∧p(z,x) rule with single-piece head

Decomposition into rules with atomic head by introducing a fresh predicate

R0:b(x)à�y�z pR(x,y,z)R1:pR(x,y,z)àp(x,y)R2:pR(x,y,z)àp(y,z)R3:pR(x,y,z)àp(z,x)

We lose the structure of the head •  much less efficient query rewriting

•  may even lead to lose the property of having a finite universal model

(if the set of rules has this property)


DECOMPOSITIONOFRULESINTOATOMICHEADRULES(2)

R:p(x,y)→�zp(y,z),p(z,y)

F:p(a,b)

a b z0 ...F1

F2

z1

z2

F2≡ F1 (F2 mapstoF1 ) henceF*≡ F1

Finiteuniversalmodel

AÅerdecomposiSonintoatomicheadrules:

R0:p(x,y)→�zpR(y,z)R1:pR(y,z)àp(y,z)R2:pR(y,z)àp(z,y)

a b z0 ...F1

F2

z1

z2

F2F1

Nofiniteuniversalmodel


OVERVIEWOFTHELECTURE

Part1:Basics



UndecidabilityofthefundamentalproblemGenericproperSesthatensuredecidability

Main«concrete»decidableclassesofexistenSalrules


However,here:queryrewriSngwithRisfiniteforanyq

SATURATIONMAYNOTHALT

R=person(x)àhasParent(x,y)∧person(y)

∧person(y0)∧hasParent(a,y0)

∧person(y1)∧hasParent(y0,y1)

F=person(a)

NoredundanciesareaddedTheKBhasnofiniteuniversalmodel


R=friend(u,v)∧friend(v,w)àfriend(u,w)

q=friend(Giorgos,Maria)

q1=friend(Giorgos,v0)∧friend(v0,Maria)

q2=friend(Giorgos,v1)∧friend(v1,v0)∧friend(v0,Maria)

q2’=friend(Giorgos,v0)∧friend(v0,v1)∧friend(v1,Maria)

q2andq2’areequivalent

Etc.q3=friend(Giorgos,v2)∧friend(v2,v1)∧friend(v1,v0)∧friend(v1,Maria)

QUERYREWRITINGMAYNOTHALT

However,here:saturaSonwithRisfiniteforanyF

There are cases where both processes do not halt (even if the factbase is known)

Thereisaninfinitenumberofnon-redundantrewriSngs


UNDECIDABILITYOFTHEFUNDAMENTALPROBLEM

FundamentaldecisionproblemInput:K=(F,R)knowledgebase,qBooleanconjuncSvequeryQuesSon:isqentailedbyK?

This problem is undecidable (only semi-decidable)

E.g. proof by reduction from the word problem in a semi-Thue system

There is a one-step derivation from a word w to w’ if there is a rule wi à wj in G, and w = w1wiw2, w' = w1wjw2 w’ is derived from w if

there is a (finite) sequence of one-step derivations from w to w’

Input: a set G of rules of the form wi à wj, 2 words w0 and wf

Question: is it possible to derive (exactly) wf from w0 using the rules in G?


REDUCTIONFROMTHEWORDPROBLEMFromG,w0andwfwebuildaKB(F, R)andaBooleanCQq

Vocabulary constants:thelekersoccuringinG,w0andwf +twospecialconstantsBandE binarypredicates:succandval

FactbaseF=T(w0,B,E)

SetofrulesRisobtainedbytranslaSngeachrulewiàwjintotheexistenSalrule �x �y (T(wi,x,y)à T(wj,x,y))

Toawordw=a1...anweassignthefollowinggraphT(w,x,y) wheretheziareexistenSalvariablesandx,yarefree

x y

Queryq=T(wf,B,E)

Key:anywordwderivablefromw0withGcorrespondstoapathT(w, B, E) inthesaturaSonofFbyR,andreciprocally


finite saturation

bounded treewidth saturation

finite UCQ rewriting

atomic body (linear)

frontier-1

weakly- guarded


datalog

guarded

weaklyacyclic

acyclic GRD

wa-GRD jointly- acyclic

frontier- guarded

jointly-fg

sticky-join

w-sticky-j

sticky

weakly-sticky

(PARTIAL)MAPOFDECIDABLECASES

DL-Lite

EL


GENERICPROPERTIESTHATENSUREDECIDABILITY

ThreegenerickindsofproperSesensuringdecidability:-  SaturaSonbyForwardChaininghaltsforanyfactbase

(«finiteexpansionset»,fes)

-  QueryrewriSnghaltsforanyconjuncSvequery(«finiteunificaSonset»,fus,orUCQ-rewritability)

-  SaturaSonbyForwardChainingmaynothaltbutforanyfactbasethegeneratedfactshaveatree-likestructure(«boundedtreewidthset»,bts)

NoneoftheseproperSesisrecognizable[Baget+KR10]buttheseproperSesprovidegenericalgorithmicschemes


No existential variables

MainClasseswithFiniteSaturaFon(fes)

Datalog

Acyclic position dependency graph

Weak-acyclicity

Acyclic Graph of Rule Dependencies Acyclic

GRD

Joint-acyclicity

Acyclic existential dependency graph

GRD with fes strongly connected components

fes-GRD

[BagetKR�04]

[BagetKR�04]

[Deutsch+ICDT�03][Fagin+ICDT03]

[Krötzsch+IJCAI�11]

Position dependency graph: nodes are positions in predicates edges show how existential variables are propagated

Graph of rule dependencies: nodes are rules edges express that a rule may lead to trigger a rule

M.-L. Mugnier – UNILOG School – 2018 81

WEAK-ACYCLICITY

R1:p(x)→�yr(x,y)�q(y)R2:r(x,y)→p(x)

PosiFondependencygraphnodes:posiSons(p,i)inpredicatesedges:foreachfronServariablexinposiSon(p,i)inarulebody -anedgefrom(p,i)toeachposiSon(q,j)ofxintherulehead

-aspecialedgefrom(p,i)toeachposiSonofanexistenSalintheruleheadRisweakly-acyclicifitsposiSongraphcontainsnocircuitwithaspecialedge(*)

weaklyacyclic

notweaklyacyclicspecialedge(p,1)à(r,1)duetoR1edge(r,1)à(p,1)duetoR2

R1:p(x)→�y�zr(x,y)�r(y,z)�r(z,x)R2:r(x,y)�r(y,x)→p(x)


ACYCLICGRAPHOFRULEDEPENDENCYGraphofRuleDependenciesnodes:therulesedges:anedgefromRitoRjifanapplicaSonofRimayleadtotriggeranew

applicaSonofRj(«RjdependsonRi»)DependencycanbeeffecSvelycomputedbycheckingifthereisapiece-unifierof

body(Rj)andhead(Ri)

Theseexamplesshowthatweak-acyclicityandacyclicGRDareincomparablecriteriaCommongeneralizaSonsofthesetwonoSonshavebeendefined

CyclicGRDsinceR1andR2dependoneachother

R1:p(x)→�yr(x,y)�q(y)R2:r(x,y)→p(x)

R1:p(x)→�y�zr(x,y)�r(y,z)�r(z,x)R2:r(x,y)�r(y,x)→p(x)


E.g. inclusion dependencies, necessary properties of concepts / relations

MainClasseswithFiniteQueryRewriFng(fus)

Atomic- body Sticky Domain-

restricted

Sticky-join Each head atom contains

all or none of the body variables

E.g. concept product Elephant(x) ∧ Mouse(y) à bigger-than(x,y)

[Cali+PVLDB2010]

[Baget+IJCAI�09][Baget+IJCAI�09]

[Cali+RR�10]

= linear Datalog+ [Cali+PODS2009]

Body restricted to a single atom

Restricts multiple occurrences of body variables that do not occur in all head atoms

Each head atom contains all the body variables

E.g. Human(x) àparentOf(y,x) ∧Human(y) is atomic-body, sticky and domain-restricted


WidthofatreedecomposiSon=maxnumberofnodesinabag(minus1)Treewidthofagraph=minwidthoveralldecomposiSontreesofthisgraph

r

a

p(a,b)q(b,z0)r(a,b,t0)p(b,t0)q(t0,z1)r(b,t0,t1)p(t0,t1)

DecomposiFontree1)eachnode(term)appearsinabag2)eachhyperedge(atom)hasallitsnodesinabag3)foreachnodex,thesubgraphinducedbythebagscontainingxisconnected

b

r

t0

z0 ab

node

hyperedge

DecomposiFonTree/Treewidth

p p p

q q

z1

t1bz0 abt0

t0z1 bt0t1

p(a,b)

q(b,z0)

r(a,b,t0) p(b,t0)

r(b,t0,t1) p(t0,t1)

q(t0,z1)


The decidability proof does not provide a halting algorithm (relies on the bounded treewidth model property [Courcelle90])

R is bts if the forward chaining with R generates facts with bounded treewidth: i.e., for any factbase F, there is an integer b s.t. any factbase R -derived from F has treewidth bounded by b

F

BoundedTreewidthoftheDerivedFacts(bts)

EssenSally[CaliGoklobKiferKR’08]

fes (finite saturation) is included in bts (bound given by the number of terms in the finite saturation)


An atom in the body guards all the body variables

Guard only the frontier

Guard only affected variables (i.e.possibly mapped to new existentials)

Guard only affected variables from the frontier

Frontier: variables shared by the body and the head

SomeRecognizablebts(andnotfes)ClassesofRules

guarded [Cali+KR’08]

weakly guarded

[Cali+KR�08]The frontier has size 1

[Baget+KR�10]

frontier guarded

[Baget+KR�10]

weakly frontier guarded

[Baget+IJCAI�09]

frontier 1

r(x,y) ∧ r(y,z) ∧ s(x,y,z) à �u r(y,u) ∧ r(z,u) r(x,y) ∧ r(y,z) ∧ r(x,z) à �u r(z,u)

r(x,y) ∧ r(y,z) à r(y,u) ∧ r(z,u)

datalog

These classes are moreover « greedy bts » => a halting algorithm [Baget+IJCAI�11]M.-L. Mugnier – UNILOG School – 2018 87

Greedybts

R1 = p(x,y) à ∃z p(y,z) R2 = p(x,y) ∧ q(x,z) à ∃t r(x,y,t) ∧ p(y,t)

F = p(a,b) ab

bz0 abt0

t0z1 bt0t1

p(a,b)

q(b,z0)r(a,b,t0)p(b,t0)

r(b,t0,t1)p(t0,t1)q(t0,z1)

R1 R2

R1 R2

Greedy construction of a decomposition tree of derived facts

with bounded width

Etc.


The«Greedybts»Property[Baget+ IJCAI�11]

T0 T0 = terms(F) + {constants} F

T0 ∪ var(h(H))

B

H

h

Derived facts Decomposition tree

All bags contain T0 F

h(H)

Foranyfactbase,foreachruleapplicaSon,fronServariablesnotbeingmappedtoiniSaltermsarejointlymappedtovariablesoccurringinatomsaddedbyasinglepreviousruleapplicaSon


MainIdeasoftheAlgorithmforgbts(1)

1.  Bagpakern={homomorphismsfrompartofarulebodyto«currentfact»

thatusesometermsofthebag}

•  Aruleisapplicabletothecurrentfactbaseiffabagpakerncontainsitsbody•  FCcanbeperformedonthedecoratedtree

2.  EquivalencerelaSononbags

OnlyonebagperequivalenceclassisdevelopedTheothernodesareblocked

Boundednumberofequivalenceclassesàfinite«fullblockedtree»T*

BuildafinitedecomposiSontreethatencodesapotenSallyinfinitefact


MainIdeasoftheAlgorithmforgbts(2)

[Baget+IJCAI2011]qaddedasarule«qàmatch»

qisentailediffmatchoccursinabagpa=erni.e.,qmapsbyhomomorphismtoatoms(T*)

[Thomazo+KR2012]offline/onlineseparaSon

(1)compilaSon:treeT*builtindependentlyfromanyquery

(2)querying:anyqisentailediffitmapsby*-homomorphismtoT* i.e.qmapsbyhomomorphismtoabounded«development»ofT*

QuerythisfinitedecomposiSontree


DataComplexityofgbtsClasses

guarded

weakly guarded

frontier guarded

weakly frontier guarded

frontier 1

Datalog (fes)

ExpTime-c

PTime-c

Previousalgorithmisworst-caseopSmalongbtsfordata/combinedcomplexity.

CanbespecializedtobeopSmalonthesegbtssubclassesM.-L. Mugnier – UNILOG School – 2018 92

FES GBTS

FUS

linear

frontier-1

weakly- guarded


Datalog

guarded weaklyacyclic

aGRD jointly- acyclic frontier-

guarded

jointly-fg

sticky-join

w-sticky-join

sticky

weakly-sticky

DL-Lite

EL

MFA

glut-fg (BTS)

domain- restricted

super-weak- acyclic


CONCLUSION

•  Reasoningwithontologiesisbecomingcentralinmanydata-centricapplicaSons

•  SolidtheoreScalfoundaSonswitharangeofontologicalformalisms

thatoffervarioustradeoffexpressivity/complexity

•  Ongoingresearch•  Gobeyond(unionsof)conjuncSvequeries,e.g.combinethemwith

navigaSonalquerieslikeregularpathqueries•  NewqueryrewriSngtechniquesthattargetmorepowerful

langages,e.g.Datalog•  NewqueryansweringtechniquesthatcombinematerialisaSonand

queryrewriSng•  StudytheinteracSonoftheontologywithmappings,whichiskey

toefficientqueryansweringoverheterogeneousdata•  RepresenSngandreasoningwithtemporalandspaSaldata

•  Dealingwithdatainconsistencies ....M.-L.Mugnier–UNILOGSchool– 2018 94

(Small) Bibliography BienvenuM.,LeclèreM.,Mugnier,M.-L.andRousset,M.-C., Reasoning with Ontologies, chapter6,volume1in«AguidedtourofarSficialintelligenceresearch»,Springer,toappear.IntroducFonstoseveralaspectsofontology-mediatedqueryansweringwithdescripFonlogicsorexistenFalrulesintheReasoningWebsummerschoolbooks:inparScular:Bienvenu,M.andOrSz,M.(2015).Ontology-mediatedqueryansweringwithdatatractabledescripSonlogics.11thInternaSonalReasoningWebSummerSchool,volume9203ofLNCS,pages218–307.Springer.Mugnier,M.andThomazo,M.(2014).AnintroducSontoontology-basedqueryansweringwithexistenSalrules.10thInternaSonalReasoningWebSummerSchool,volume8714ofLNCS,pages245–278.Springer.Goklob,G.,Orsi,G.,Pieris,A.,andSimkus,M.(2012).DataloganditsextensionsforsemanScwebdatabases.10thInternaSonalReasoningWebSummerSchool,volume7487ofLNCS,pages54–77.Springer.

ThesesynthesesprovidefurtherreferencesM.-L.Mugnier–UNILOGSchool– 2018 95

APPENDIX:FURTHERDETAILS

¢  FundamentaldefiniSonsandproperSesfortheFOL(�,∧)fragment

¢  Piece-unifiers


INTERPRETATIONS/MODELS(1)

¢  VocabularyV=(P,C),where P=finitesetofpredicates C=setofconstants

¢  InterpretaFonI=(DI ,.I)ofV,where DI≠ø (domain) forallcinC,cIinDI

forallpinPwitharityk,pIDIk

¢  Furthermore,uniquenameassumpSon:forallcanddinC,cI≠dI

¢  SimplifyingassumpSon(inlinewiththeuniquenameassumpSon):CDIandforallcinC,cI=c

V = ( {p/2, r/3 }, {a, b} ) I: DI = {a, b, d1} pI = { (b, a), (b, d1), (d1, b) }

rI = { (d1, d1, a) }

¢  Iisamodeloff(builtonV)iffistrueinI


INTERPRETATIONS/MODELS(2)

¢  LetfinFOL(�,∧).Iisamodeloffiff thereisamappingvfromterms(f)toDIsuchthat

forallp(e1,...,ek)inf,(v(e1),...,v(ek))inpI

I: DI = {a, b, d1} pI = {(b, a), (b, d1), (d1, b)} rI = {(d1, d1, a)} f = �x�y�z ( p(x, y) ∧ p(y, z) ∧ r(x, z, a) )

v: x↦ d1 y↦ b z↦ d1

¢  InterpretaSonscanbeseenassetsofatoms

(withelementsfromD\Cseenasvariables)

p(b,a), p(b,x1), p(x1,b), r(x1, x1,a)¢  I isamodeloffiffthereisahomomorphismfromftoI


HOMOMORPHISMSAGAINANDAGAIN

¢  OnecandefinehomomorphismsbetweeninterpretaFons¢  Wehave:

IfI1mapsI2then,foranyf,I1modeloff�I2modeloff

¢  ToaformulafinFOL(�,∧),weassignitsisomorphicmodelM(f) (alsocalledcanonicalmodel)

f=�x�y�z(p(x,y)∧p(y,z)∧r(x,z,a))

M(f): D={dx,dy,dz,a} pM(f)={(dx,dy),(dy,dz)} rM(f)={(dx,dz,da)}


NICESEMANTICPROPERTIESOFFOL(�,∧)

¢  ThecanonicalmodelM(f)isuniversal,i.e.,forallM’modeloff,M(f)mapstoM’

Proof:LetM’modeloff.Then,fmapstoM’.SinceM(f)isomorphictof,M(f)mapstoM’

¢  g⊨f(i.e.,everymodelofgisamodeloff)ifffmapsbyhomomorphismtoM(g)ifffmapsbyhomomorphismtogProof:⇒Assumeg⊨f.InparScularM(g)isamodeloff,hencefmapstoM(g)

⇐AssumefmapstoM(g).SinceM(g)isuniversal:foranyM’modelofg,fmapstoM’,i.e.,M’isamodeloff,henceg⊨f


WHY«PIECES»?(CONT’D)–PIECE-UNIFIERS

¢  UnificaSonmust«map»partsofqaccordingto«pieces»thatcanbeprovidedbyaruleapplicaSon

(otherwiseitisunsoundoruseless) body head

body head

specializa4onofthefron4er

homomorphism

ThetermsofqunifiedwiththefronSer(orwithconstants)cutqinto«pieces»⇒ enSre«pieces»ofqmustbemappedtothepiecesoftherule

R

q


PIECE-UNIFIER:ALTERNATIVEDEFINITION

Letu1:fronSer(R)àfronSer(R)�constants(u1isaspecializaSonofthefronSerofR)Letu2beahomomorphismfromq’�qtou1(head(R))Cutpoints:termsofq’mappedtou1(fron4er(R))

ortoconstantsThecutpointscutqinto«pieces»u1+u2isapiece-unifierifq’iscomposedofpieces

body head

body head

u1

u2


Reasoning on Data

Documents

Transcript of Reasoning on Data