Robust and Scalable Linked Data Reasoning Incorporating Provenance and Trust Annotations

52 Pages·2011·0.77 MB·English

RobustandScalableLinkedDataReasoningIncorporating
ProvenanceandTrustAnnotations
PieroA.Bonattia,AidanHoganb,AxelPolleresb,LuigiSauroa
aUniversita` diNapoli“FedericoII”,Napoli,Italy
bDigitalEnterpriseResearchInstitute,NationalUniversityofIreland,Galway
Abstract
Inthispaper,weleverageannotatedlogicprogramsfortrackingindicatorsofprovenanceandtrustduringreasoning,specifically
focussingontheuse-caseofapplyingascalablesubsetofOWL2RL/RDFrulesoverstaticcorporaofarbitraryLinkedData(Web
data).Ourannotationsencodethreefacetsofinformation:(i)blacklist:a(possiblymanuallygenerated)booleanannotationwhich
indicates that the referent data are known to be harmful and should be ignored during reasoning; (ii) ranking: a numeric value
derived by a PageRank-inspired technique—adapted for Linked Data—which determines the centrality of certain data artefacts
(such as RDF documents and statements); (iii) authority: a boolean value which uses Linked Data principles to conservatively
determine whether or not some terminological information can be trusted. We formalise a logical framework which annotates
inferences with the strength of derivation along these dimensions of trust and provenance; we formally demonstrate some
desirable properties of the deployment of annotated logic programming in our setting, which guarantees (i) a unique minimal
model (least fixpoint); (ii) monotonicity; (iii) finitariness; and (iv) finally decidability. In so doing, we also give some formal
resultswhichrevealstrategiesforscalableandefficientimplementationofvariousreasoningtasksonemightconsider.Thereafter,
wediscussscalableanddistributedimplementationstrategiesforapplyingourrankingandreasoningmethodsoveraclusterof
commodity hardware; throughout, we provide evaluation of our methods over 1 billion Linked Data quadruples crawled from
approximately 4 million individual Web documents, empirically demonstrating the scalability of our approach, and how our
annotationvalueshelpensureamorerobustformofreasoning.Wefinallysketch,discussandevaluateause-caseforasimple
repair of inconsistencies detectable within OWL 2 RL/RDF constraint rules using ranking annotations to detect and defeat the
“marginalview”,andinsodoing,inferanempirical“consistencythreshold”fortheWebofDatainoursetting.
Keywords: annotatedprograms;linkeddata;webreasoning;scalablereasoning;distributedreasoning;authoritativereasoning;owl2rl;
provenance;pagerank;inconsistency;repair
1. Introduction BasedontheResourceDescriptionFramework(RDF),
Linked Data emphasises four simple principles: (i)
The Semantic Web is no longer purely academic: use URIs as names for things (and not just doc-
in particular, the Linking Open Data project has fos- uments); (ii) make those URIs dereferenceable via
teredarichlodeofopenlyavailablestructureddataon HTTP; (iii) return useful and relevant RDF content
the Web, commonly dubbed the “Web of Data” [53]. upon lookup of those URIs; (iv) include links to other
datasets.Sinceitsinceptionfouryearsago,theLinked
Data community has overseen exports from corpo-
Emailaddresses:[email protected](PieroA.Bonatti), rate bodies (e.g., the BBC, New York Times, Free-
[email protected](AidanHogan),
base, Linked Clinical Trials [linkedct.org], Thomp-
[email protected](AxelPolleres),
[email protected](LuigiSauro). sonReuters[opencalais.com]etc.),governmentalbod-
PreprintsubmittedtoElsevier 02/06/2011
ies (e.g., UK Government [data.gov.uk], US Gov- However, reasoning over (even subsets) of the Web
ernment [data.gov], US National Science Founda- ofDataposestwomajorchallenges,withseriousimpli-
tion, etc.), community driven efforts (e.g., Wikipedia- cationsforreasoning:(i)scalability,whereonecanex-
basedexports[dbpedia.org],GeoNames,etc.),social- pectLinkedDatacorporacontainingintheorderofbil-
networking sites (e.g., MySpace [dbtune.org], flickr, lionsortensofbillionsofstatements(forthemoment);
Twitter[semantictweet.com],etc.)andscientificcom- (ii)tolerancetonoiseandinconsistency,wherebydata
munities(e.g.,DBLP,PubMed,UniProt),amongstoth- publishedontheWebareafflictedwithna¨ıveerrorsand
ers.1 disagreement, arbitrary redefinitions of classes/proper-
Interspersedwiththesevoluminousexportsofasser- ties, etc. Traditional reasoning approaches are not well
tional (or instance) data are lightweight schemata/on- positioned to tackle such challenges, where the main
tologiesdefinedinRDFS/OWL—whichwewillcollec- body of literature in the area is focussed on considera-
tively call vocabularies—comprising the terminologi- tions such as expressiveness, soundness, completeness
caldatawhichdescribeclassesandpropertiesandpro- andpolynomial-timetractability2,andmakessomeba-
vide a formalisation of the domain of discourse. The sic assumptions about the underlying data quality; for
assertional data describe things by assigning datatype example,tableaux-basedalgorithmsstrugglewithlarge
(string) values for named attributes (datatype proper- bodies of assertional knowledge, and are tied by the
ties),namedrelationstootherthings(objectproperties), principleofexfalsoquodlibet—fromcontradictionfol-
andnamedclassifications(classes).Theterminological lowsanything—andthusstrugglewhenreasoningover
data describe these classes and properties, with RDFS possiblyinconsistentdata.3
and OWL providing the formal mechanisms to render In previous works, we presented our Scalable Au-
the semantics of these terms—in particular, their inter- thoritativeOWLReasoner(SAOR)whichappliesasub-
relation and prescribed intended usage. Importantly, set of OWL 2 RL/RDF rules over arbitrary Linked
best-practices encourage: (i) re-use of class and prop- Data crawls: in particular, we abandon completeness
erty terms by independent and remote data publishers; in favour of conducting “sensible” inferencing which
(ii)inter-vocabularyextensionofclassesandproperties; we argue to be suitable for the Linked Data use-
(iii)dereferenceabilityofclassesandproperties,return- case [36,38].4 We have previously demonstrated dis-
ingaformalRDFS/OWLdescriptionthereupon. tributed reasoning over ∼1b Linked Data triples [38],
Applicationsareslowlyemergingwhichleveragethis and have also presented some preliminary formalisa-
richveinofstructuredWebdata;however,thusfar(with tions of what we call “authoritative reasoning”, which
ofcourseafewexceptions)applicationshavebeenslow considersthesourceofterminologicaldataduringrea-
to leverage the underlying terminological data for rea- soning[36].TheSAORsystemisactivelyusedforma-
soning.Looselyspeaking,reasoningutilisestheformal terialising inferences in the Semantic Web Search En-
underlyingsemanticsofthetermsinthedatatoenable gine (SWSE) [37] which offers search and browsing
thederivationofnewknowledge.WithrespecttoLinked overLinkedData.5
Data,sometimesthereisonlysparsere-useof(termino- Inthispaper,welooktoreformalisehowtheSAOR
logical and/or assertional) terms between sources, and engineincorporatesthenotionsofprovenanceandtrust
so reasoning can be used to better integrate the data in which form an integral part of its tolerance to noise,
a merged corpus as follows: (i) infer and support the whereweseetheuseofannotatedlogicprograms[42]
semanticsofgroundequality(owl:sameAsrelationsbe- as a natural fit. We thus derive a formal logical frame-
tween individuals) to unite knowledge fractured by in- work for annotated reasoning in our setting, which en-
sufficient URI re-use across the Web; (ii) leverage ter- codesthreeindicatorsofprovenanceandtrustandwhich
minologicalknowledgetoinfernewassertionalknowl- thusallowsustoapplyrobustmaterialisationoverlarge,
edge,possiblyacrossvocabulariesandevenassertional static,LinkedDatacorpora.
documents; (iii) detect inconsistencies whereby one or
more parties may provide conflicting data—herein, we
2 In our scenario with assertional data from the Web in the order
focusonthelattertworeasoningtasks.
ofbillionsofstatements,evenquadraticcomplexityisprohibitively
expensive.
1 Seehttp://richard.cyganiak.de/2007/10/lod/for 3 Thereisongoingresearchdevotedtoparaconsistentlogicswhich
a comprehensive graph of such datasets and their inter-linkage; tackle this issue—e.g., see a recent proposal for OWL 2 [51]—but
however—andastheLinkedOpenNumbersproject(see[64,Figure theefficiencyofsuchapproachesisstillanopenquestion.
1]) has aptly evinced, and indeed as we will see ourselves in 4 For a high-level discussion on the general infeasibility of com-
later evaluation—not all of this current Web of Data is entirely pletenessforscenariossuchasours,see[31].
“compelling”. 5 http://swse.deri.org/
2
Further,inpreviousworkwenotedthatmanyincon- RDFConstant GiventhesetofURIreferencesU,the
sistenciesariseontheWebastheresultofincompatible setofblanknodesB,6 andthesetofliteralsL,theset
naming of resources, or na¨ıve publishing errors [35]: of RDF constants is denoted by C:=U∪B∪L. We
herein, we look at a secondary use-case for our an- alsodefinethesetofvariablesV whichrangeoverC.
notations which leverages OWL 2 RL/RDF constraint Herein,weuseCURIEs[5]todenoteURIs.Follow-
rules to detect inconsistencies, where subsequently we ingTurtlesyntax[2],useaasaconvenientshortcutfor
perform a simple “repair” of the Web knowledge-base rdf:type.Wedenotevariableswitha‘?’prefix.
using our annotations—particularly ranks—to identify
anddefeatthe“marginalview”presentintheinconsis- RDFTriple Atriplet:=(s,p,o)∈(U∪B)×U×C
tency. iscalledanRDFtriple,wheresiscalledsubject,ppred-
Morespecifically,we: icate, and o object. A triple t := (s,p,o) ∈ G,G :=
– introducesomenecessarypreliminaries(Section2); C×C×C is called a generalised triple [25], which
– discuss our proposed annotation values to repre- allows any RDF constant in any triple position: hence-
sent provenance and trust—blacklisting, authority, forth, we assume generalised triples unless explicitly
andranking—givingconcreteinstantiationsforeach stated otherwise. We call a finite set of triples G ⊂ G
(Section3); agraph.
– describeaformalframeworkforannotatedprograms
which incorporate the above three dimensions of 2.2. Linked Data principles, Data Sources and
provenanceandtrust,includingformaldiscussionof Quadruples
constraintrules(Section4);
– describe our experimental setup and our 1 billion
In order to cope with the unique challenges of han-
tripleLinkedDatacorpus(Section5);
dlingdiverseandunverifiedWebdata,manyofourcom-
– describeourdistributed(i)implementationandeval-
ponentsandalgorithmsrequireinclusionofanotionof
uation of links-based ranking (Section 6), (ii) anno-
provenance: consideration of the source of RDF data
tatedreasoningforasubsetofOWL2RL/RDFrules
found on the Web. Tightly related to such notions are
(Section7),and(iii)ourinconsistencydetectionand
the best practices of Linked Data [3], which give clear
repairuse-case(Section8);
guidelines for publishing RDF on the Web. We briefly
– discuss issues relating to scalability and expressive-
discuss Linked Data principles and notions relating to
ness(Section9),renderrelatedworkinthefield(Sec-
provenance.7
tion10),andconclude(Section11).
Linked Data Principles The four best practices of
LinkedDataareasfollows[3]:
– (LDP1)useURIstonamethings;
2. Preliminaries – (LDP2) use HTTP URIs so that those names can be
lookedup;
In this section, we provide some necessary prelim- – (LDP3) provide useful structured information when
inaries relating to (i) RDF (Section 2.1); (ii) Linked a look-up on a URI is made – loosely, called deref-
Dataprinciplesanddatasources(Section2.2);(iii)rules erencing;
andatoms(Section2.3);(iv)generalisedannotatedpro- – (LDP4)includelinksusingexternalURIs.
grams (Section 2.4) (v) terminological data given by
RDFS/OWL (Section 2.5); and (vi) OWL 2 RL/RDF Data Source We define the http-download function
rules(Section2.6).Weattempttopreservenotationand get : U → 2G as the mapping from a URI to an RDF
terminologyasprevalentintheliterature. graph(setoffacts)itmayprovidebymeansofagiven
HTTP lookup [21] which directly returns status code
200OKanddatainasuitableRDFformat;thisfunction
6 We interpret blank-nodes as skolem constants, as opposed to
2.1. RDF existential variables. Also, we rewrite blank-node labels to ensure
uniquenessperdocument,asprescribedin[30].
7 Note that in a practical sense, all HTTP-level functions
We briefly give some necessary notation relating to
{get,redir,redirs,deref} are set at the time of the crawl, and are
RDFconstantsandRDFtriples;cf.[30]. boundedbytheknowledgeofourcrawl.
3
also performs a rewriting of blank-node labels (based variablesofAyieldsB);wemayalsosaythatB isan
on the input URI) to ensure uniqueness when merging instanceofA;ifB isground,wesaythatitisaground
RDF graphs [30]. We define the set of data sources instance.Similarly,ifwehaveasubstitutionθsuchthat
S⊂UasthesetofURIsS:={s∈U|get(s)(cid:54)=∅}. Aθ =Bθ,wesaythatθisaunifierofAandB;wede-
note by mgu(A,B) the most general unifier of A and
RDF Triple in Context/RDF Quadruple An ordered B which provides the “minimal” variable substitution
pair(t,c)withatriplet=(s,p,o),c∈Sandt∈get(c) (uptovariablerenaming)requiredtounifyAandB.
is called a triple in context c. We may also refer to (s,
p,o,c)asanRDFquadrupleorquadqwithcontextc. Rule AruleR isgivenasfollows:
H ←B ,...,B (n≥0)
1 n
HTTP Redirects/Dereferencing A URI may provide
where H,B ,...,B are atoms, H is called the
1 n
a HTTP redirect to another URI using a 30x response
head(conclusion/consequent)andB ,...,B thebody
code [21]; we denote this function as redir : U → U 1 n
(premise/antecedent). We use Head(R) to denote the
which may map a URI to itself in the case of failure
head H of R and Body(R) to denote the body
(e.g., where no redirect exists)—note that we do not
B ,...,B of R. Our rules are range restricted – or
1 n
needtodistinguishbetweenthedifferent30xredirection
safe [60]: like Datalog, the variables appearing in the
schemes,andthatthisfunctionwouldimplicitlyinvolve,
headofeachrulemustalsoappearinthebody.
e.g., stripping the fragment identifier of a URI [4]. We
The set of all rules that can be defined over atoms
denotethefixpointofredirasredirs,denotingtraversal
usingan(arbitrarybutfixed)infinitesupplyofvariables
of a number of redirects (a limit may be set on this
willbedenotedbyRules.Arulewithanemptybody
traversal to avoid cycles and artificially long redirect
is considered a fact; a rule with a non-empty body is
paths).Wedefinedereferencingasthefunctionderef :=
calledaproper-rule.Wecallafinitesetofsuchrulesa
get◦redirswhichmapsaURItoanRDFgraphretrieved
programP.
with status code 200 OK after following redirects, or
Like before, a ground rule is one without variables.
whichmapsaURItotheemptysetinthecaseoffailure.
We denote with Ground(R) the set of ground instan-
tiations of a rule R and with Ground(P) the ground
2.3. AtomsandRules instantiationsofallrulesoccurringinaprogramP.
In this section, we briefly introduce some notation ImmediateConsequenceOperator Wegivethe(clas-
as familiar from the field of Logic Programming [48], sical)immediateconsequenceoperatorC ofaprogram
P
which acts as a generalisation of the aforementioned P underinterpretationI as:
RDFnotation.
C : 2Facts →2Facts
P
I (cid:55)→{Head(R)θ |R∈P and
Atom Atoms are of the form p(e ,...,e ) where
1 n ∃I(cid:48) ⊆I s.t.θ =mgu(Body(R),I(cid:48))}
e ,...,e areterms(likeDatalog,functionsymbolsare
1 n
disallowed) and where p is a predicate of arity n—we Intuitively, the immediate consequence operator maps
denote the set of all such atoms by Atoms. This is a fromasetoffactsI tothesetoffactsitdirectlyentails
generalisation of RDF triples, for which we employ a with respect to the program P—note that C (I) will
P
ternary predicate T where our atoms are of the form retain the facts in P since facts are rules with empty
T(s,p,o)—for brevity, we commonly omit the ternary bodiesandthusunifywithanyinterpretation,andnote
predicate and simply write (s,p,o). An RDF atom of thatforourpurposesC ismonotonic—theadditionof
P
this form is synonymous with a generalised triple pat- factsandrulestoaprogramcanonlyleadtoasuperset
ternwherevariablesofthesetVareallowedinanypo- ofconsequences.
sition).Agroundatom—orsimplyafact—isonewhich SinceourrulesareasyntacticsubsetofDatalog,C
P
doesnotcontainvariables(e.g.,ageneralisedtriple);we has a least fixpoint—denoted lfp(C )—whereby fur-
P
denote the set of all facts by Facts—a generalisation ther application of C will not yield any changes, and
P
ofG.A(Herbrand)interpretationI isafinitesubsetof which can be calculated in a bottom-up fashion, start-
Facts—ageneralisationofagraph. ing from the empty interpretation ∆ and applying iter-
Letting A and B be two atoms, we say that A sub- atively C [66] (here, convention assumes that P con-
P
sumesB—denotedA(cid:46)B—ifthereexistsasubstitution tainsthesetofinputfactsaswellasproperrules).De-
θ of variables such that Aθ = B (applying θ to the fine the iterations of C as follows: C ↑ 0 := ∆; for
P P
4
allordinalsα,C ↑(α+1):=C (C ↑α);sinceour Restricted immediate consequences In the gener-
P P P
rulesareDatalog,thereexistsanαsuchthatlfp(C )= alised annotation framework, the restricted immediate
P
C ↑ α for α < ω, where ω denotes the least infinite consequenceoperatorofanannotatedprogramP isde-
P
ordinal—i.e.,theimmediateconsequenceoperatorwill finedasfollows,whereσ rangesoversubstitutions:
reach a fixpoint in countable steps [61], thereby giv- (cid:8)
R (I)(A):=lub ρ|(A:ρ←B :µ ,...,B :µ )σ
ing all ground consequences of the program. We call P 1 1 n n
lfp(CP) the least model, which is given the more suc- ∈Ground(P)andI |=(Bi:µi)σ
(cid:9)
cinctnotationlm(P). for(1≤i≤n) .
Example: Take a program P which com-
2.4. Generalisedannotatedprograms
prises of the annotated rule shown in Equa-
tion 1 and the annotated fact Parent(sam):0.6.
In generalised annotated programs [42] the set of Then, R (I)(Father(sam)) = 0.3. Instead, let’s
P
truthvaluesisgeneralisedtoanarbitraryuppersemilat- say that P also contains Father(sam):0.5. Then
tice T, that may represent—say—fuzzy values, incon- R (I)(Father(sam))=lub{0.5,0.3}=0.5. (cid:51)
P
sistencies, validity intervals (i.e. time), or a confidence
value,tonamebutafew. Importantly, various properties of RP have been for-
mallydemonstratedforgeneralisedannotatedprograms
in [42], where we will reuse these results later for our
(Generalised) Annotated rules Annotated rules are
own (more specialised) annotation framework in Sec-
expressionslike
tion 4; for example, R has been shown to be mono-
P
H:ρ←B1:µ1,...,Bn:µn tonic,butnotalwayscontinuous[42].
whereeachµ canbeeitheranelementofT oravariable
i
rangingoverT;ρcanbeafunctionf(µ ,...,µ )over
1 n 2.5. Terminologicaldata:RDFS/OWL
T. Programs, Ground(R) and Ground(P) are defined
analogouslytonon-annotatedprograms.
Aspreviouslydescribed,RDFS/OWLallowforpro-
Example: Consider the following simple example of
viding terminological data which constitute definitions
a (generalised) annotated rule where T corresponds to
of classes and properties used in the data. A detailed
a set of confidence values in the interval [0,1] of real
discussion of RDFS/OWL is out of scope, but the dis-
numbers:
tinction of terminological and assertional data—which
Father(?x):(0.5×µ)←Parent(?x):µ. (1) wenowdescribe—isimportantforourpurposes.First,
werequiresomepreliminaries.9
This rule intuitively states that something is a Father
with (at least) half of the confidence for which it is a
Parent.8 (cid:51)
Meta-class We consider a meta-class as a
class specifically of classes or properties; i.e., the
Restricted interpretations So-called restricted inter- members of a meta-class are themselves either
pretations map each ground atom to a member of T. classes or properties. Herein, we restrict our no-
A restricted interpretation I satisfies A:µ (in symbols, tion of meta-classes to the set defined in RDF(S)
I |= A:µ) iff I(A) ≥ µ, where ≥ is T’s ordering. and OWL specifications, where examples include
T T
Now I satisfies a rule like the above iff either I satis- rdf:Property,rdfs:Class,owl:Class,owl:Restric-
fiestheheadorI doesnotsatisfysomeoftheannotated tion, owl:DatatypeProperty, owl:FunctionalProp-
atomsinthebody. erty, etc. Note that rdfs:Resource, rdfs:Literal,
Example: Take the annotated rule from Equation 1, e.g.,arenotmeta-classes,sincetheirmembersneednot
and let’s say that we have a restricted interpretation I beclassesorproperties.
whichsatisfiesParent(sam):0.6.Now,forI tosatisfy
the given rule, it must also satisfy Father(sam):0.3 9 As we are dealing with Web data, we refer to the OWL 2 Full
suchthatI(Father(sam))≥0.3. (cid:51) languageandtheOWL2RDF-basedsemantics[18]unlessexplic-
itly stated otherwise. Note that a clean and intuitive definition of
terminologicaldataissomewhatdifficultforRDFSandparticularly
8 This example can be trivially converted to RDF by instead con- OWLFull.Weinsteadrelyona‘shibboleth’approachwhichidenti-
sidering the ternary atoms (?x, a, ex:Parent) and (?x, a, fiesmarkersforwhatweconsidertobeRDFS/OWLterminological
ex:Father). data.
5
Meta-property A meta-property is one which has a where the T , 0 ≤ i ≤ m atoms in the body (T-atoms)
i
meta-class as its domain. Again, we restrict our no- are all those that can only have terminological ground
tionofmeta-propertiestothesetdefinedinRDF(S)and instances,whereastheA ,1≤i≤natoms(A-atoms),
i
OWL specifications, where examples include rdfs:- canhavearbitrarygroundinstances.WeuseTBody(R)
domain,rdfs:subClassOf,owl:hasKey,owl:inverse- andABody(R)torespectivelydenotethesetofT-atoms
Of, owl:oneOf, owl:onProperty, owl:unionOf, etc. and the set of A-atoms in the body of R. Herein, we
Note that rdf:type, owl:sameAs, rdfs:label, e.g., do presumethattheT-atomsandA-atomsofourrulescan
nothaveameta-classasdomain,andarenotconsidered bedistinguishedandreferencedasdefinedabove.
meta-properties. Example: LetR denotethefollowingrule
EX
(?x,a,?c2)←(?c1,rdfs:subClassOf,?c2),(?x,a,?c1)
Terminologicaldata Wedefinethesetofterminolog- WhenwritingT-splitrules,wedenoteTBody(R )by
EX
icaltriplesastheunionofthefollowingsetsoftriples: underlining: the underlined T-atom can only be bound
(i) triples with rdf:type as predicate and a meta- by a triple with the meta-property rdfs:subClassOf
classasobject; as RDF predicate, and thus can only be bound by a
(ii) tripleswithameta-propertyaspredicate; terminologicaltriple.Thesecondatominthebodycan
(iii) triples forming a valid RDF list whose head is be bound by assertional or terminological triples, and
theobjectofameta-property(e.g.,alistusedfor soisconsideredanA-atom. (cid:51)
owl:unionOf,owl:intersectionOf,etc.);
(iv) triples which contribute to an all-disjoint-classes
orall-disjoint-propertiesaxiom.10
T-ground rule A T-ground rule is a set of rule in-
Notethatthelastcategoryofterminologicaldataisonly
stances for the T-split rule R given by grounding
required for special consistency-checking rules called
TBody(R). We denote the set of such rules for a pro-
constraints: i.e., rules which check for logical contra-
gram P and a set of facts I as GroundT(P,I), defined
dictions in the data. For brevity, we leave this last cat-
as:
egory of terminological data implicit in the rest of the
(cid:8)
GroundT(P,I):= Head(R)θ ←ABody(R)θ |R∈P
paper, where owl:AllDisjointClasses and owl:All-
DisjointProperties can be thought of as “honorary and∃I(cid:48) ⊆I s.t.θ =mgu(TBody(R),I(cid:48))(cid:9).
meta-classes”includedincategory1,owl:memberscan
TheresultisasetofruleswhoseT-atomsaregrounded
bethoughtofasan“honorarymeta-property”included
bytheterminologicaldatainI.
in category 2, and the respective RDF lists included in
Example: Consider the T-split rule R from
category3. EX
Finally, we do not consider triples involving “user- the previous example. Now let IEX := { (foaf:-
Person, rdfs:subClassOf, foaf:Agent), (foaf:-
defined”meta-classesormeta-propertiesascontributing
Agent, rdfs:subClassOf, dc:Agent) }. Here,
totheterminology,whereinthefollowingexample,the
first triple is considered terminological, but the second GroundT({REX},IEX) = { (?x, a, foaf:Agent) ←
tripleisnot.11 (?x, a, ?foaf:Person); (?x, a, dc:Agent) ← (?x, a,
?foaf:Agent)}. (cid:51)
(ex:inSubFamily,rdfs:subPropertyOf,
rdfs:subClassOf)
(ex:Bos,ex:inSubFamily,ex:Bovinae)
T-splitprogramandleastfixpoint Herein,wegivean
overviewofthecomputationoftheT-splitleastfixpoint
T-splitrule AT-splitruleR isgivenasfollows:
for a program P, which is broken up into two parts:
H ←A1,...,An,T1,...,Tm (n,m≥0) (2) (i) the terminological least fixpoint, and (ii) the asser-
tional least fixpoint. Let PF := {R ∈ P | Body(R) =
10Thatis,tripleswithrdf:typeaspredicateandowl:AllDis- ∅} be the set of facts in P,12 let PT∅ := {R ∈
jointClassesorowl:AllDisjointPropertiesasobject,
P | TBody(R) (cid:54)= ∅,ABody(R) = ∅}, let P∅A :=
tripleswhosepredicateisowl:membersandwhosesubjectunifies
with the previous category of triples, and triples forming a valid {R ∈ P | TBody(R) = ∅,ABody(R) (cid:54)= ∅}, and let
RDF list whose head unifies with the object of such an owl:- PTA := {R ∈ P | TBody(R) (cid:54)= ∅,ABody(R) (cid:54)=
memberstriple. ∅}. Clearly, P = PF ∪ PT∅ ∪ P∅A ∪ PTA. Now,
11In particular, we require a data-independent method for distin-
guishingterminologicaldatafrompurelyassertionaldata,suchthat
we only allow those meta-classes/-properties which are known a- 12Ofcourse,PF canrefertoaxiomaticfactsand/ortheinitialfacts
priori. givenbyaninputknowledge-base.
6
let TP := PF ∪PT denote the initial (terminologi- S/OWL meta-classes do not appear in a position other
cal) program containing ground facts and T-atom only thanasthevalueforrdf:type;and(ii)thatowl:sameAs
rules, and let lm(TP) denote the least model for the doesnotaffectconstantsintheterminology.13
terminological program. Now, let AP := lm(TP) ∪
P∅A∪GroundT(PTA,lm(TP))denotethesecond(as-
3. Annotationvalues
sertional) program containing all available facts and
rules with empty or grounded T-atoms. Now, we can
Na¨ıvelyconductingmaterialisationwrt.non-arbitrary
givetheleastmodeloftheT-splitprogramP aslm(AP)
rulesoverarbitrary,non-verifieddatamergedfrommil-
for AP derived from P as above—we more generally
lions of sources crawled from the Web broaches many
denotethisbylmT(P).
obvious dangers. In this section, we discuss the anno-
In [38], we showed that the T-split least fixpoint is
tation values we have chosen to represent the various
completewrt.thestandardvariant(giventhatourrules
dimensionsofprovenanceandtrustweuseforreason-
aremonotonic)andthattheT-splitleastfixpointiscom-
ing over Linked Data, which have been informed by
plete with respect to the standard fixpoint if rules re-
ourpastexperiencesinreasoningoverarbitraryLinked
quiring assertional knowledge do not infer unique ter-
Data.
minologicalknowledgerequiredbytheT-splitprogram
(i.e.,theassertionalprogramAP doesnotgeneratenew
terminologicalfactsnotavailabletotheinitialprogram 3.1. Blacklisting
TP).
Despite our efforts to create algorithms which auto-
matically detect and mitigate noise in the input data,
2.6. OWL2RL/RDFrules
it may often be desirable to blacklist input data based
on some criteria: for example, data from a certain do-
OWL 2 RL/RDF [25] rules are a partial axiomati- main may be considered likely to be spam, or certain
sation of the OWL 2 RDF-Based Semantics which is triple patterns may constitute common publishing er-
applicable for arbitrary RDF graphs, and thus is com- rors which hinder the reasoning process. We currently
patible with RDF Semantics [30]. The atoms of these do not require the blacklisting function, and thus con-
rules comprise primarily of ternary predicates encod- sideralltriplestobenotblacklisted.However,suchan
ing generalised RDF triples; some rules have a special annotationhasobvioususesforbypassingnoisewhich
head denoted false which indicate that an instance of cannototherwisebeautomaticallydetected.
the body is inconsistent. All such rules can be consid- One particular use-case we have in mind for includ-
ered T-split where we use the aforementioned criteria ing the blacklisting annotation relates to the publica-
forcharacterisingterminologicaldataandsubsequently tion of void values for inverse-functional properties:
T-atoms. the Friend Of A Friend (FOAF)14 vocabulary offers
As we will further discuss in Section 7, full materi- classesandpropertiesfordescribinginformationabout
alisation wrt. the entire set of OWL 2 RL/RDF is in- people,organisations,documents,andsoforth:itiscur-
feasible in our use-case; in particular, given a large A- rently one of the most widely instantiated vocabular-
Box of arbitrary content, we wish to select a subset of ies in Linked Data [37, Appendix A]. FOAF includes
the OWL 2 RL/RDF profile which is linear with re- a number of inverse-functional-properties which allow
specttothatA-Box;thus,weselectasubsetO2R− of for identifying (esp.) people in the absence of agree-
OWL 2 RL/RDF rules where |ABody(R)| ≤ 1 for all ment upon URIs, e.g.: foaf:homepage, foaf:mbox,
R ∈ O2R− [36]—we provide the full ruleset in Ap- foaf:mboxsha1sum. However, FOAF exporters on the
pendix A. Besides ensuring that the growth of asser- Webcommonlydonotrespecttheinverse-functionalse-
tional inferences is linear wrt. the A-Box—and as we manticsoftheseproperties;oneparticularlypathogenic
will see in later sections—our linear profile allows for case we encountered was exporters producing empty
near-trivialdistributionofreasoning,aswellasvarious
otheroptimisationtechniques(see[38,32]). 13Note that the OWL 2 RL/RDF eq-rep-* rules can cause incom-
WithrespecttoOWL2RL/RDF,in[32]weshowed pleteness by condition (ii), but herein, we do not support these
that the T-split least fixpoint is complete assuming (i) rules. Further, note that in other profiles (such as RDFS and pD*)
axiomatic triples may be considered non-standard—however, none
nonon-standardusage,wherebyrdf:type,rdf:first,
of the OWL 2 RL/RDF axiomatic triples (see Table A.1) are non-
rdf:rest and the RDFS/OWL meta-properties do not standard.
appear other than as a predicate in the data, and RDF- 14http://xmlns.com/foaf/0.1/
7
strings or values such as ‘mailto:’ for foaf:mbox mandatedbythatvocabularyandrecursivelyreferenced
when users omitted specifying their email. Similarly, vocabularies for that term. Thus, once a publisher in-
we encountered many corresponding erroneous values stantiates a class or property from a vocabulary, only
for the foaf:mboxsha1sum property—representing a thatvocabularyanditsreferencesshouldinfluencewhat
SHA1 encoded email value—referring to the SHA1 inferencesarepossiblethroughthatinstantiation.
hashes of ‘mailto:’ and the empty string [34]. These Firstly, we must define the relationship between a
values—-caused by na¨ıve publishing errors—lead to class/property term and a vocabulary, and give the no-
quadratic spurious inferences equating all users who tionofterm-levelauthority.WeviewatermasanRDF
omitted an email to each other. Although for reasons constant, and a vocabulary as a Web document. From
of efficiency we currently do not support the rele- Section 2.2, we recall the get mapping from a URI
vant OWL 2 RL/RDF rules which support the seman- (a Web location) to an RDF graph it may provide by
ticsofinverse-functionalproperties,theblacklistingan- meansofagivenHTTPlookupandtheredirsmapping
notation could be used to negate the effects of such for following the HTTP redirects of a URI. Also, let
pathogenicvalues—essentially,itservesasapragmatic bnodes(G) denote the set of blank nodes appearing in
lastresort. the RDF graph G. Now, we denote a mapping from a
sourceURItothesetoftermsitspeaksauthoritatively
forasfollows:17
3.2. Authoritativeanalysis
auth: S→2C
InourinitialworksonSAOR[36]—apragmaticrea-
s(cid:55)→{c∈U|redirs(c)=s}∪bnodes(get(s))
sonerforLinkedData—weencounteredapuzzlingdel-
ugeofinferenceswhichwedidnotinitiallyexpect.We where a Web source is authoritative for URIs which
found that remote documents sometimes cross-define dereference to it and the blank nodes it contains; e.g.,
terms resident in popular vocabularies, changing the the FOAF vocabulary is authoritative for terms in its
inferences authoritatively mandated for those terms. namespace since it follows best-practices and makes
Forexample,wefoundonedocument15 whichdefines its class/property URIs dereference to an RDF/XML
owl:Thing to be a member of 55 union classes—thus, documentdefiningtheterms.Notethatnodocumentis
materialisation wrt. OWL 2 RL/RDF rule cls-uni [25, authoritativeforliterals.
Table6]overanymemberofowl:Thingwouldinfer55 Tonegatetheeffectsofnon-authoritativeterminolog-
additionalmembershipsfortheseobsoleteandobscure icalaxiomsonreasoningoverWebdata(asexemplified
unionclasses.Wefoundanotherdocument16 whichde- above), we add an extra condition to the T-grounding
finesninepropertiesasthedomainofrdf:type—again, of a rule: in particular, we only require amendment to
anything defined to be a member of any class would ruleswherebothTBody(R)(cid:54)=∅andABody(R)(cid:54)=∅.
be inferred to be a member of these nine properties.
Even aside from “cross-defining” core terms, popular
Authoritative T-ground rule Let varsTA(R)⊂V de-
vocabularies such as FOAF were also affected [36]—
note the set of variables appearing in both TBody(R)
suchpracticeleadtothematerialisationofanimpracti-
and ABody(R). Now, we define the set of authorita-
cal bulk of arguably irrelevant data (which would sub-
tiveruleinstancesforaprogramP,RDFgraphG,and
sequentlyburdentheconsumerapplication).
sourcesas:18
Tocounter-actremotecontributionsabouttheseman-
tics of terms, we introduced a more conservative form G(cid:92)roundT(P,G,s):={Head(R)θ ←ABody(R)θ |
of reasoning called authoritative reasoning [36] which
R∈P
criticallyexaminesthesourceofterminologicalknowl-
and∃G(cid:48) ⊆Gs.t.θ =mgu(TBody(R),G(cid:48))
edge.Wenowre-introducetheconceptofauthoritative
reasoningfrom[36],hereinprovidingmoredetailedfor- andifTBody(R)(cid:54)=∅∧ABody(R)(cid:54)=∅
(cid:9)
malismsandupdateddiscussion. then∃v ∈varsTA(R)s.t.θ(v)∈auth(s)
Ourauthoritativereasoningmethodsarebasedonthe
intuition that a publisher instantiating a vocabulary’s
17Even pre-dating Linked Data, dereferenceable vocabulary
term (class/property) thereby accepts the inferencing
terms were encouraged; cf. http://www.w3.org/TR/2006/
WD-swbp-vocab-pub-20060314/
15http://lsdis.cs.uga.edu/˜oldham/ontology/ 18Here we favour RDF graph notation as authority applies only
wsag/wsag.owl in the context of Linked Data (but could be trivially generalised
16http://www.eiao.net/rdf/1.0 throughtheauthfunction).
8
whereauthoritativeruleinstancesaresynonymouswith (?x, owl:hasValue, ?y), (?x, owl:onProperty, ?p), we
authoritatively T-ground rules and where the notion of would expect ?x to be ground by a blank-node skolem
authoritative rule instances for a program follows nat- andthusexpecttheinstancetocomefromonegraph.)
urally. The additional condition for authoritativeness
statesthatifarulecontainsbothT-atomsandA-atoms
3.3. Links-basedranking
in the body (ABody(R) (cid:54)= ∅∧TBody(R) (cid:54)= ∅), then
the unifier must substitute atleast one variable appear-
There is a long history of links-based analysis
inginbothABody(R)andTBody(R)(avariablefrom
over Web data—and in particular over hypertext
thesetvarsTA(R))foranauthoritativetermfromsource
documents—where links are seen as a positive vote
s (a constant from the set auth(s)) for the resulting T-
for the relevance or importance of a given document.
ground rule to be authoritative. This implies that the
SeminalworksexploitingthelinkstructureoftheWeb
sourcesmustspeakauthoritativelyforatleastoneterm
for ranking documents include HITS [44] and PageR-
that will appear in the body of each proper T-ground
ank [8]. Various approaches (e.g., [1,16,33,23,15,29])
rulewhichitsterminologygenerates,andsocannotcre-
look at incorporating links-based analysis techniques
ate new assertional rules which could apply over arbi-
for ranking RDF data, with various end-goals in mind,
trary assertional data not mentioning any of its terms.
mostcommonly,prioritisationofinformationalartefacts
Weillustratethiswithanexample.
inuserresult-views;however,suchanalyseshavebeen
Example: Take the T-split rule R as before where
EX
appliedtootheruse-cases,includingworkbyGue´retet
varsTA(REX)={?c1}representingthesetofvariables
al.[28]whichusesbetweennesscentralitymeasuresto
in both TBody(R ) and ABody(R ). Let I be
EX EX EX
identify potentially weak points in the Web of Data in
thegraphfromsources,wherenowforeachsubstitution
termsofmaintainingconnectednessintegrity.
θ,theremustexistv ∈varsTA(R )suchthatsspeaks
EX
Herein,weemploylinks-basedanalysiswiththeun-
authoritativelyforθ(v).Inthiscase,
derlying premise that higher ranked sources contribute
– s must speak authoritatively for the URI foaf:Per-
more“trustworthy”data:inourcase,wewouldexpect
son—forwhich?c1issubstituted—fortheT-ground
a correlation between the (Eigenvector) centrality of a
rule(?x,a,foaf:Agent)←(?x,a,?foaf:Person)to
source in the graph, and the quality of data that it pro-
beauthoritative,
vides. Inspired in particular by the work of Harth et
– analogously,smustspeakauthoritativelyfortheURI
al. [29] on applying PageRank to RDF, we implement
foaf:Agent—againforwhich?c1issubstituted—for
atwo-stepprocess:(i)wecreatethegraphoflinksbe-
theT-groundrule(?x,a,dc:Agent)←(?x,a,foaf:-
tween sources, and apply a standard PageRank calcu-
Agent)tobeauthoritative.
lation over said graph to derive source ranks; (ii) we
Inotherwords,thesourcesservingtheT-factsinI
EX
propagatesourcerankstothetriplestheycontainusing
must be the FOAF vocabulary for the above rules to
authoritative. (cid:51) asimplesummationaggregation.Wenowdiscussthese
twostepsinmoredetail.
For reference, we highlight variables in varsTA(R)
withboldfaceinTableA.4.
3.3.1. Creatingandrankingthesourcegraph
(ItisworthnotingthatforruleswhereABody(R)and
Creating the graph of interlinking Linked Data
TBody(R) are both non-empty, authoritative instantia-
sources is non-trivial, in that the notion of a hyperlink
tionoftherulewillonlyconsiderunifiersforTBody(R)
does not directly exist. Thus, we must extract a graph
which come from one source: however, in practice for
sympathetictoLinkedDataprinciplesandcurrentpub-
OWL 2 RL/RDF this is not so restrictive: although
lishingpatterns.
TBody(R) may contain multiple atoms, in such rules
Recalling the Linked Data principles enumerated in
TBody(R)usuallyreferstoanatomicaxiomwhichre-
Section 2.2, according to LDP4, links should be spec-
quires multiple triples to represent—indeed, the OWL
ified simply by using external URI names in the data.
2 Structural Specification19 enforces usage of blank-
These URI names should dereference to an RDF de-
nodes and cardinalities on such constructs to ensure
scriptionofthemselvesaccordingtoLDP2andLDP3re-
thattheconstituenttriplesofthemulti-tripleaxiomap-
spectively.LetD :=(V,E)representasimpledirected
pearinonesource.Totakeanexample,fortheT-atoms
graph where V ⊂ S is a set of sources (vertices), and
E ⊂S×Sisasetofpairsofvertices(edges).Letting
19http://www.w3.org/TR/2009/ si,sj ∈V betwovertices,then(si,sj)∈Eiffsi (cid:54)=sj
REC-owl2-syntax-20091027/ and there exists some u ∈ U such that redirs(u) = sj
9
and u appears in some triple t ∈ get(s ): i.e., an edge statingatripleshouldincreasetherankofthetriple.20
i
extendsfroms tos ifftheRDFgraphreturnedbys Thus, for calculating the rank of a triple t, we use the
i j i
mentionsaURIwhichredirectstos . summation of the ranks of the sources it appears in as
j
Now, let E(s) denote the set of direct successors of follows:
s (outlinks), let E denote the set of vertices with no
∅
(cid:88)
outlinks(danglingnodes),andletE−(s)denotetheset r(t):= r(s )
t
of direct predecessors of s (inlinks). The PageRank of
st∈{s∈S|t∈get(s)}
a vertex s in the directed graph D := (V,E) is then
i
givenasfollows[8]:
4. Thelogicalframework
r(s ):= 1−d +d (cid:88) r(s∅) +d (cid:88) r(sj) In this section, we look at incorporating the above
i |V| |V| |E(s )| three dimensions of trust and provenance—which we
j
s∅∈E∅ sj∈E−(si) will herein refer to as annotation properties—into an
annotated logic programming framework which tracks
where d is a damping constant (usually set to 0.85) this information during reasoning, and determines the
which helps ensure convergence in the following it- annotations of inferences based on the annotations of
erative calculation, and where the middle component the rule and the relevant instances, where the resultant
splitstheranksofdanglingnodesevenlyacrossallother valuesoftheannotationpropertiescanbeviewedasde-
nodes. Note also that the first and second components notingthestrengthofaderivation.Weproposeandfor-
are independent of i, and constitute the minimum pos- maliseageneralannotationframework,introducesome
sible rank of all nodes (ensures that ranks do not need high-level tasks it enables, and discuss issues relating
tobenormalisedduringiterativecalculation). toscalabilityinourscenario.
Nowletw := 1|V−d| representtheweightofauniversal We begin in Section 4.1 by formalising annotation
(weaklink)givenbyallnon-danglingnodestoallother functions which abstract the mechanisms used for an-
nodes—danglingnodessplittheirvoteevenlyandthus notating facts and rules. In Section 4.2, we propose an
don’trequireaweaklink;wecanuseaweightedadja- annotatedprogramframeworkandassociatedsemantics
cency matrix M as follows to encode the graph D := based on the previous work of Kifer et al. [42] (intro-
(V,E): ducedpreviouslyinSection2.4).InSection4.3,wein-
troducesomehigh-levelreasoningtasksthatthisframe-
 d work enables; in Section 4.4 we look at how each task
|E(sj)| +w, if(sj,si)∈E scales in the general case, and in Section 4.5 we focus
m := 1 on the scalability of the task required for our use-case
i,j , ifs ∈E
w|V,| othejrwise∅ owveerbroiuerflysedleisccteudssansonmoteataioltnerpnraotpiveertiseesm.aInntSicescttihoant4o.n6e,
mightconsiderforourframework.WewrapupinSec-
where this stochastic matrix can be thought of as a tion4.7byintroducingannotatedconstraintruleswhich
Markov chain (dubbed the random-surfer model). The allowfordetectingandlabellinginconsistenciesinour
ranks of all sources can be expressed algebraically as use-case.
theprincipaleigenvectorofM,whichinturncanbees-
timatedusingthepoweriterationmethodupuntilsome
4.1. Abstractingannotationfunctions
terminationcriteria(fixednumberofiterations,conver-
gencemeasures,etc.)isreached.Werefertheinterested
Thefirststeptowardsaformalsemanticsofannotated
readerto[8]formoredetail.
logicprogramsconsistsingeneralisingtheannotations
(suchasblacklisting,authoritativeness,andranking)re-
calledintheprevioussections.Theannotationdomains
3.3.2. Calculatingtripleranks
Based on the rank values for the data sources, we
nowcalculatetheranksforindividualtriples.Weusea 20Note that one could imagine a spamming scheme where a large
numberofspuriouslow-rankeddocumentsrepeatedlymakethesame
simplemodelforrankingtriples,basedontheintuition
assertionstocreateasetofhighly-rankedtriples.Infuture,wemay
that triples appearing in highly ranked sources should
revise this algorithm to take into account some limiting function
benefit from that rank, and that each additional source derivedfromPLD-levelanalysis.
10