A Generic Approach for Large-Scale Ontological Reasoning in the Presence of Access Restrictions to the Ontology's Axioms

Reasoning in the Presence of Access Restrictions

to the Ontology's Axioms

Franz Baader

1

,MartinKnechtel

2

,andRafael Peñaloza

1

1

TheoreticalComputerScienceTU Dresden,Germany

{baader,penaloza}@tcs.inf.tu- dresd en.d e

2

SAPAG,SAPResearchCECDresden,Germany

martin.knechtel@sap.com

Abstract. The frameworkdevelopedin this paper candeal with sce-

narios where selected sub-ontologies of a large ontology are oered as

viewsto users, basedon criteria like the user's access right, the trust

levelrequiredbytheapplication, orthelevelofdetailrequested bythe

user.Insteadofmaterializingalargenumberofdierentsub-ontologies,

we propose tokeepjustone ontology,butequipeachaxiomwithala-

belfromanappropriatelabelinglattice.Theaccessright,requiredtrust

level,etc.isthenalsorepresentedbyalabel(calleduserlabel)fromthis

lattice, and the corresponding sub-ontology is determinedby compar-

ing this label with theaxiomlabels. For large-scale ontologies, certain

consequence(liketheconcepthierarchy)areoftenprecomputed.Instead

ofprecomputingtheseconsequencesforeverypossiblesub-ontology,our

approachcomputesjustonelabelforeachconsequencesuchthatacom-

parisonoftheuserlabelwiththeconsequencelabeldetermineswhether

theconsequencefollowsfromthecorrespondingsub-ontologyornot.

In this paper we determine under which restrictions on the user and

axiomlabels suchconsequence labels (called boundaries) alwaysexist,

describe dierentblack-box approaches for computingboundaries, and

present rst experimental results that compare the eciency of these

approachesonlargereal-worldontologies.Black-boxmeansthat,rather

thanrequiringmodicationsofexistingreasoningprocedures, theseap-

proaches canusesuchproceduresdirectlyassub-procedures, whichal-

lowsustoemployexistinghighly-optimizedreasoners.

1 Introduction

Assume thatyouhavealargeontology

T

^{,but youwanttooerdierentusers}

dierentviewsonthisontology,i.e.,eachusercanseeonlyasubsetoftheactual

ontology,whichisselectedbyanappropriatecriterion.Thiscriterioncouldbethe

accessrightthatthisuserhas,theleveloftrust(intheaxiomsoftheontology)

that theuserrequires,thelevelofdetails that isdeemedto beappropriatefor

this user, etc. In principle, you could explicitly create asub-ontologyfor each

(type of) user, but then you might end up with exponentially many dierent

ontologies, where each is a subset of

T

^{. Instead, we propose to keep just the}

bigontology

T

^{, butlabeltheaxiomsin}

T

^{suchthat acomparisonoftheaxiom}

label withtheusercriteriondetermineswhethertheaxiombelongstothesub-

ontologyforthisuserornot.Tobemoreprecise,weusealabelinglattice

(L, ≤)

^,

i.e.,aset oflabels

L

^{togetherwithapartialorder}

≤

^{ontheselabelssuchthata}

nitesetoflabelsalwayshasajoin(supremum,leastupperbound)andameet

(inmum,greatestlowerbound)w.r.t.

≤

^{.3. Allaxioms}

t ∈ T

^{arenowassumed}

tohavealabel

lab(t) ∈ L

^{,andtheuseralsoreceivesalabel}

` ∈ L

^(whichcanbe

read asaccess right,requiredleveloftrust,etc.).The sub-ontologythatauser

withlabel

`

^{canseeisthendened tobe4}

T ` := {t ∈ T | lab(t) ≥ `}.

Ofcourse,theuserofanontologyshouldnotonlybeabletoseeitsaxioms,but

alsotheconsequencesofthese axioms.Thus,auserwithlabel

`

^{should beable}

to see all the consequences of

T `

^{. For large} ontologies, certain relevant conse- quencesareoftenpre-computed.Thegoalofthepre-computationisthatcertain

user queriescan be answered by asimple look-up in the pre-computed conse-

quences, and thus do not require expensive reasoning during the deployment

phaseoftheontology.Forexample,in theversionofthelargemedicalontology

SNOMEDCT 5

thatisdistributedtohospitals,allthesubsumptionrelationships

betweentheconceptnamesoccurringintheontologyarepre-computed.Forala-

beledontologyasintroducedabove,itisnotenoughtopre-computetherelevant

consequencesof

T

^{.In fact,if therelevant}consequence

α

^followsfrom

T

^{, then}

wealsoneedtoknowforwhichuserlabels

`

^{itstillfollowsfrom}

T _`

^{.Otherwise,if}

auserwithlabel

`

^askswhether

α

^{holds,thesystemcouldnotsimplylook this}

up in the pre-computedconsequences,but would need to compute theanswer

on-the-yby reasoningoverthesub-ontology

T `

^{.Our solutionto this problem}

istocomputeaso-calledboundary fortheconsequence

α

^{,i.e.,anelement}

µ α

^of

L

^suchthat

α

^followsfrom

T `

ⁱ

` ≤ µ α

^.

There are basically two approaches for computing a boundary. The glass-

box approach takesaspecicreasoner(orreasoningtechnique) foranontology

language(e.g.,atableau-basedreasonerforOWLDL[20])andmodiesitsuch

that it can compute a boundary. Examples for the application of the glass-

boxapproachtospecicinstancesoftheproblemofcomputingaboundaryare

tableau-based approaches for reasoning in possibilistic Description Logics [15,

13] (where the latticeis the interval

[0, 1]

^{with theusual order)and glass-box}

approaches to axiom pinpointing in Description Logics [19,14,12,3,4] (where

thelatticeconsistsof (equivalenceclassesof)monotoneBooleanformulaewith

implication as order [4]). The problem with glass-box approaches is that they

3

Figure1inSection3shows asmalllattice. Adetailedintroductionto latticesand

orderscan,e.g.,befoundin[9].

4

Todenethissub-ontology,anarbitrarypartialorderwouldbesucient.However,

the existence of suprema and inma will be important for the computation of a

boundaryofaconsequence(seebelow).

5

http://www.ihtsdo.org/snomed-ct/

andthat optimizationsof theoriginalreasoningtechniquedonotalwaysapply

tothemodiedreasoners.Incontrast,theblack-boxapproach canre-useexisting

optimized reasoners withoutmodications,and it can be applied to arbitrary

ontologylanguages:onejust needstoplugin areasonerforthislanguage.

In this paper, we introduce three dierent black-box approaches for com-

puting a boundary, and compare their performance on real-world ontologies.

Therstapproachusesanaxiompinpointingalgorithm asblack-boxreasoner,

whereasthesecondonemodiestheHitting-Set-Tree-basedblack-boxapproach

toaxiompinpointing[11,21]. Thethird usesbinarysearchandcanonlybeap-

pliedifthelabelinglatticeisalinearorder.Itcanbeseenasageneralizationof

theblack-boxapproachtoreasoninginpossibilisticDescriptionLogicsdescribed

in [16].Alltheproofsomittedinthispapercanbefoundin[2].

2 Basic Denitions and Results

Tostay asgeneral aspossible,wedo not x aspecic ontologylanguage.We

just assumethat ontologies are nite sets of axioms such that everysubset of

an ontology is again an ontology. If

T ⁰

^{is asubset of theontology}

T

^{, then}

T ⁰

is called a sub-ontology of

T

^{. The ontologylanguagedetermines which sets of}

axiomsareadmissibleasontologies.Foraxedontologylanguage,amonotone

consequencerelation

| =

^{isabinaryrelationbetweenontologies}

T

^{ofthislanguage}

andconsequences

α

^{such that,for everyontology}

T

^{, wehavethat}

T ⁰ ⊆ T

^and

T ⁰ | = α

^imply

T | = α

^{. If}

T | = α

^{, thenwesay that}

α

^{follows from}

T

^{and that}

T

^entails

α

^{.Forinstance,givena}DescriptionLogic

L

^(e.g.,theDL

SHIN (D)

underlying OWL DL), an ontology is an

L

^{-TBox, i.e., a nite set of general}

conceptinclusion axioms(GCIs) oftheform

C v D

^for

L

^-concept descriptions

C, D

^{. As}consequenceswecan, e.g.,consider subsumption relationships

A v B

forconceptnames

A, B

^.

Weconsider alattice

(L, ≤)

^and respectivelydenote by

L

`∈S `

^and

N

`∈S `

thejoin (least upperbound) andmeet (greatestlowerbound) ofthe nite set

S ⊆ L

^{. Alabeled ontology withlabeling lattice}

(L, ≤)

^isanontology

T

^together

withalabelingfunction

lab

^{thatassignsalabel}

lab(t) ∈ L

^{toeveryelement}

t

^of

T

^{6Wedenotewith}

L ^lab

^{thesetofalllabelsoccurringinthelabeledontology}

T

^,

i.e.,

L ^lab := {lab(t) | t ∈ T }

^{.Everyelementofthelabelinglattice}

` ∈ L

^denes

asub-ontology

T `

^{that containstheaxiomsof}

T

^{that arelabeledwithelements}

greaterthanorequalto

`

^:

T ` := {t ∈ T | lab(t) ≥ `}.

Conversely,everysub-ontology

S ⊆ T

^{denesanelement}

λ S ∈ L

^{,calledthelabel}

of

S

^:

λ S := N

t∈S lab(t)

^{. Thefollowinglemma statessomesimple}relationships betweenthesetwonotions.

Lemma1. Forall

` ∈ L

^,

S ⊆ T

^,itholdsthat

` ≤ λ T `

^,

S ⊆ T λ S

^and

T ` = T λ T `

.

6

AnexampleofalabeledontologyisgiveninExample2inSection3.

Noticethat,ifaconsequence

α

^followsfrom

T `

^forsome

` ∈ L

^{,it mustalso}

followfrom

T ` ⁰

^{for every}

` <sup>0</sup> ≤ `

^{, since then}

T ` ⊆ T ` ⁰

^{. A maximal element of}

L

that stillentailstheconsequencewillbecalled amarginforthisconsequence.

Denition1 (Margin).Let

α

^beaconsequencethatfollowsfromtheontology

T

^{. The label}

µ ∈ L

^{is called a}

(T , α)

^{-margin if}

T _µ | = α

^{, and for every}

`

^with

µ < `

^wehave

T ` 6| = α

^.

If

T

^and

α

^{are clear from the context, we usually ignore the prex}

(T , α)

and call

µ

^{simplyamargin. Thefollowinglemma shows threebasic properties}

ofthesetofmarginsthatwillbeusefulthroughout thispaper.

Lemma2. Let

α

^beaconsequencethatfollows fromthe ontology

T

^.Wehave:

1. If

µ

^{isamargin, then}

µ = λ T µ

^;

2. if

T ` | = α

^{,thenthere isamargin}

µ

^suchthat

` ≤ µ

^;

3. thereareatmost

2 ^{|T |}

^marginsfor

α

^.

Ifweknowthat

µ

^{isamarginforthe}consequence

α

^{,thenweknowwhether}

α

followsfrom

T `

^forall

` ∈ L

^{thatarecomparablewith}

µ

^:if

` ≤ µ

^,then

α

^follows

from

T `

^,andif

` > µ

^,then

α

^{doesnotfollowfrom}

T `

^{.However,thefactthat}

µ

^is

amargingivesusnoinformationregardingelementsthatareincomparablewith

µ

^{.Inordertoobtainafullpictureofwhenthe}consequence

α

^followsfrom

T _`

^for

anarbitraryelementof

l

^{,wecantrytostrengthenthenotionofmargintothat}

ofanelement

ν

^of

L

^{thataccuratelydividesthelatticeintothoseelementswhose}

associatedsub-ontologyentails

α

^{andthoseforwhichthisisnotthecase,i.e.,}

ν

shouldsatisfythefollowing:forevery

` ∈ L

^,

T ` | = α

ⁱ

` ≤ ν

^.Unfortunately,such anelementneednotalwaysexist,asdemonstratedbythefollowingexample.

Example 1. Consider thedistributive lattice

(S 4 , ≤ 4 )

^{havingthe four elements}

S 4 = {0, a 1 , a 2 , 1}

^,where

0

^and

1

^{aretheleastandgreatestelements,}respectively,

and

a 1 , a 2

^areincomparablew.r.t.

≤ 4

^.Let

T

^{bethesetformedbytheaxiomsax}

1

andax

2

^{,whicharelabeledby}

a 1

^and

a 2

^,respectively,andlet

α

^beaconsequence such that, for every

S ⊆ T

^{, we have}

S | = α

ⁱ

|S| ≥ 1

^{. It is easy to see}

that there is no element

ν ∈ S 4

^{that satises the condition described above.}

Indeed,ifwechoose

ν = 0

^or

ν = a 1

^,then

a 2

^{violatesthecondition,as}

a 2 6≤ ν

^,

but

T a 2 = {

^ax

2 } | = α

^. Accordingly, if we choose

ν = a 2

^{, then}

a 1

^violatesthe

condition.Finally,if

ν = 1

^{ischosen,then}

1

^{itself violatesthecondition:}

1 ≤ ν

^,

but

T 1 = ∅ 6| = α

^.

Itisnonethelesspossibletondanelementthatsatisesarestrictedversionof

the condition, where we do not impose that the property must hold for every

elementofthe labelinglattice,but onlyfor thoseelementsthat arejoin prime

relativeto thelabelsoftheaxiomsin theontology.

Denition2 (Join prime).Let

(L, ≤)

^{bealattice. Given aniteset}

K ⊆ L

^,

let

K ⊗ := { N

`∈M ` | M ⊆ K}

^{denote theclosureof}

K

^{underthe meetoperator.}

An element

` ∈ L

^{is called join prime relative to}

K

^{if, for every}

K ⁰ ⊆ K ⊗

^,

` ≤ L

k∈K ⁰ k

^{impliesthat thereisan}

k 0 ∈ K ⁰

^suchthat

` ≤ k 0

^.

In Example 1, all lattice elements with the exception of

1

^{are join prime}

relativeto

{a 1 , a 2 }

^.

Denition3 (Boundary). Let

T

^{be an ontology and}

α

^a consequence. An element

ν ∈ L

^iscalleda

(T , α)

^{-boundaryifforeveryelement}

` ∈ L

^thatisjoin

prime relative to

L ^lab

^{itholds that}

` ≤ ν

ⁱ

T ` | = α

^.

Aswith margins,if

T

^and

α

^{are clearfrom thecontext,wewillsimplycall}

sucha

ν

^{aboundary.InExample1,theelement}

1

^{isaboundary.Indeed,every}

join prime element

`

^relativeto

{a 1 , a 2 }

^{(i.e., everyelementof}

L

^{except for}

1

⁾

issuchthat

` < 1

^and

T ` | = α

^{.Fromapracticalpointofview,ourdenition of}

aboundaryhasthefollowingimplication:wemustenforce thatuserlabelsare

alwaysjoinprime relativetotheset

L ^lab

^{ofalllabelsoccurringintheontology.}

3 Computing a Boundary

Inthissection,wedescribethree black-boxapproachesforcomputingabound-

ary.ThersttwoapproachesarebasedonLemma3below,andthethirdone,a

modicationofbinarysearch,canbeusedifthelabelinglatticeisalinearorder.

Lemma3. Let

µ 1 , . . . , µ n

^beall

(T , α)

^{-margins. Then}

L n

i =1 µ i

^isaboundary.

By Lemma 2, a consequence always has nitely many margins, and thus

Lemma 3 shows that a boundary always exists. Note, however, that a conse-

quencemayhaveboundariesdierentfromtheoneofLemma3.Toidentifythe

particularboundaryofLemma 3,wewill callitthemargin-basedboundary.

3.1 Using Full Axiom Pinpointing

FromLemma3weknowthatthesetofallmarginsyieldssucientinformation

forcomputingaboundary.Thequestionisnowhowtocomputethisset.Inthis

subsection,weshowthatallmargins(andthusthemargin-basedboundary)can

becomputedthroughaxiompinpointing.Axiom-pinpointingreferstothetaskof

computingMinAs [6]:minimal (w.r.t.set inclusion) sub-ontologiesfrom which

aconsequence

α

^{stillfollows.Moreformally,}

S ⊆ T

^{iscalled aMinA for}

T

^and

α

^if

S | = α

^,and

S ⁰ 6| = α

^forevery

S ⁰ ⊂ S

^{.Thefollowinglemmashowsthatevery}

margincanbeobtainedfromsomeMinA.

Lemma4. Foreverymargin

µ

^for

α

^thereisaMinA

S

^suchthat

µ = λ S

^.

Notice that this lemma doesnot imply that the label of any MinA

S

^cor-

responds to amargin. However, as the consequencefollowsfrom everyMinA,

point2ofLemma2showsthat

λ S ≤ µ

^{forsomemargin}

µ

^{.Thefollowingtheorem}

isanimmediateconsequenceofthisfacttogetherwithLemma3andLemma4.

Theorem1. If

S 1 , . . . , S n

^{are all MinAs for}

T

^and

α

^{, then}

L n

i =1 λ S i

^{is the}

margin-basedboundaryfor

α

^.

methods exist for computing the set of all MinAs, either directly [19,11,7] or

through aso-called pinpointingformula [6,4,5],which is a monotoneBoolean

formulaencodingalltheMinAs. Themain advantageofusingthepinpointing-

based approach for computinga boundary is that onecansimply useexisting

implementations forcomputing allMinAs, such asthe ones oeredby theon-

tologyeditor Protégé4 7

andtheCELsystem.

8

3.2 Label-OptimizedAxiom Pinpointing

FromLemma 4we know that everymargin is of the form

λ S

^{for someMinA}

S

^{.Intheprevioussubsectionwehaveusedthisfact tocomputeaboundaryby}

rstobtainingtheMinAsandthencomputingtheirlabels.Thisprocesscanbe

optimized if we directlycompute the labels of theMinAs, withoutnecessarily

computingtheactualMinAs.Additionally,notallthelabelsofMinAsareneces-

sary,butonlythemaximalones.Wepresenthereablack-boxalgorithmthatuses

thelabelsoftheaxiomstondtheboundaryinanoptimizedway.Ouralgorithm

isavariantoftheHitting-Set-Tree-based[17]method(HSTapproach)foraxiom

pinpointing[11,21].First,webrieydescribetheHSTapproachforcomputing

allMinAs,whichwillserveasastartingpointforourmodiedversion.

TheHSTalgorithmcomputesoneMinAatatimewhilebuildingatreethat

expressesthedistinctpossibilitiestobeexploredinthesearchoffurtherMinAs.

ItrstcomputesanarbitraryMinA

S 0

^for

T

^{,whichisusedtolabeltherootof}

the tree.Then, for everyaxiom

t

ⁱⁿ

S 0

^{, asuccessornodeis created.If}

T \ {t}

doesnotentailtheconsequence,thenthisnodeisadeadend.Otherwise,

T \ {t}

still entailsthe consequence. In this case, a MinA

S 1

^for

T \ {t}

^{is computed}

and usedto labelthenode.TheMinA

S 1

^for

T \ {t}

^{obtainedthis way isalso}

a MinA of

T

^{, and it is guaranteed to bedistinct from}

S 0

^since

t / ∈ S 1

^{. Then,}

foreachaxiom

s

ⁱⁿ

S 1

^{,anewsuccessoriscreated,andtreatedin thesameway}

asthesuccessorsoftherootnode,i.e.,itis checkedwhether

T \ {t, s}

^stillhas

theconsequence,etc.Thisprocessobviouslyterminates,andtheendresultisa

tree,whereeachnodethatisnotadeadendislabeledwithaMinA,andevery

MinAappearsasthelabelofatleastonenodeofthetree(see[11,21]).

AnimportantingredientoftheHSTalgorithmisaprocedurethatcomputes

asingleMinAfromanontology.Suchaprocedurecan,forexample,beobtained

bygoingthroughtheaxiomsoftheontologyinanarbitraryorder,andremoving

redundant axioms,i.e., ones such that the ontologyobtainedbyremoving this

axiomfrom thecurrent sub-ontologystill entailsthe consequence(see [6]fora

descriptionofthis andof amoresophisticated logarithmicprocedure). Assaid

before, in our modied HST algorithm,weare now not interestedin actually

computingaMinA,butonlyitslabel.Thisallowsustoremoveallaxiomshaving

aredundant labelratherthan asingleaxiom.Algorithm1describesablack-

boxmethod forcomputing

λ S

^{forsome MinA}

S

^{that isbased onthis idea.In}

7

http://protege.stanford.edu/

8

http://code.google.com/p/cel/

Proceduremin-lab

(T , α)

Input:

T

^:ontology;

α

^:consequence

Output:

M L ⊆ L

^{:minimallabelsetforaMinA}

1: if

T 6| = α

^then

2: return noMinA

3:

S := T

4:

M ^L := ∅

5: forevery

k ∈ L lab

^do

6: if

N

l ∈ M _L l 6≤ k

^then

7: if

S − k | = α

^then

8:

S := S − k

9: else

10:

M L := (M ^L \ {l | k < l}) ∪ {k}

11: return

M ^L

fact,thealgorithmcomputesaminimallabelset ofaMinA

S

^{,anotionthatwill}

alsobeusefulwhendescribingourvariantoftheHSTalgorithm.

Denition4 (Minimallabelset).Let

S

^beaMinAfor

α

^.Aset

K ⊆ {lab(t) | t ∈ S}

^{iscalleda minimallabelset of}

S

^{ifdistinct elementsof}

K

^areincompa-

rableand

λ S = N

`∈K `

^.

Algorithm1removesallthelabelsthatdonotcontributetoaminimallabelset.

If

T

^{is anontology and}

` ∈ L

^{, then the expression}

T − `

^{appearing at Line 7}

denotesthesub-ontology

T − ` := {t ∈ T | lab(t) 6= `}

^{.If,afterremovingallthe}

axiomslabeledwith

k

^,theconsequencestillfollows,thenthereisaMinAnone ofwhoseaxiomsislabeledwith

k

^.Inparticular,thisMinAhasaminimallabel set notcontaining

k

^{;thus alltheaxiomslabeled with}

k

^{canberemovedin our}

searchforaminimallabelset.Iftheaxiomslabeledwith

k

^{cannotberemoved,}

then allMinAs ofthecurrentsub-ontologyneed anaxiomlabeledwith

k

^{, and}

hence

k

^{is storedin theset}

M L

^{. This set is used to avoiduseless} consequence tests:ifalabelisgreaterthanorequalto

N

`∈M L `

^{,thenthepresenceorabsence}

of axiomswith thislabel willnotinuence thenalresult, which willbegiven

bytheinmumof

M L

^{;hence,there isnoneedtoapplythe(possiblycomplex)}

decisionprocedurefortheconsequencerelation.

Theorem2. Let

T

^and

α

^besuchthat

T | = α

^{.ThereisaMinA}

S 0

^for

α

^such

that Algorithm1outputsaminimal labelsetof

S 0

^.

OncethelabelofaMinAhasbeenfound,wecancomputenewMinAlabels

by asuccessivedeletion of axiomsfrom theontologyusing the HSTapproach.

Supposethatwehavecomputedaminimallabelset

M 0

^,andthat

` ∈ M 0

^.Ifwe

removealltheaxiomsintheontologylabeledwith

`

^{,andcomputeanewminimal}

label set

M 1

^{ofaMinA ofthis} sub-ontology,then

M 1

^{doesnotcontain}

`

^{, and}

thus

M 0 6= M 1

^{.Byiteratingthisprocedure,wecouldcomputeallminimallabel}

sets, and hencethelabelsof allMinAs. However,sinceour goalisto compute

Procedurehst-boundary

(T , α)

Input:

T

^:ontology;

α

^:consequence Output:boundary

ν

^for

α

1:

Global : C , H := ∅; ν

2:

M :=

^min-lab

(T , α)

3:

C := {M}

4:

ν := N

` ∈M `

5: foreachlabel

` ∈ M

^do

6: expand-hst

(T 6≤ ` , α, {`})

7: return

ν

Procedureexpand-hst

(T , α, H)

Input:

T

^:ontology;

α

^:consequence;

H

^{:list oflatticeelements}

Side eects:modicationsto

C

,

H

and

ν

1: if thereexists some

H ⁰ ∈ H

suchthat

{h ∈ H ⁰ | h 6≤ ν} ⊆ H

^or

H ⁰

^{containsaprex-path}

P

^with

{h ∈ P | h 6≤ ν} = H

^then

2: return (earlypathtermination

3

⁾

3: if thereexistssome

M ∈ C

suchthatforall

` ∈ M, h ∈ H

^,

` 6≤ h

^and

` 6≤ ν

^then

4:

M ⁰ := M

^{(MinLabreuse)}

5: else

6:

M ⁰ :=

^min-lab

(T 6≤ ν , α)

7: if

T 6≤ ν | = α

^then

8:

C := C ∪ {M ⁰ }

9:

ν := L {ν, N

` ∈M <sup>0</sup> `}

10: foreachlabel

` ∈ M ⁰

^do

11: expand-hst

(T 6≤ ` , α, H ∪ { ` })

12: else

13:

H := H ∪ { H }

^(normaltermination

⁾

thesupremumofthese labels,the algorithmcan beoptimized byavoidingthe

computationofMinAswhoselabelswillhavenoimpactonthenalresult.Based

on this wecan actually dobetterthan just removingthe axiomswith label

`

^:

instead,all axiomswithlabels

≤ `

^{canbe removed. Foran element}

` ∈ L

^and

anontology

T

^,

T 6≤`

^denotesthesub-ontologyobtainedfrom

T

^{byremovingall}

axiomswhoselabelsare

≤ `

^{.Now,assumethat wehavecomputedtheminimal}

labelset

M 0

^,andthat

M 1 6= M 0

^{istheminimallabelsetoftheMinA}

S 1

^.For

all

` ∈ M 0

^,if

S 1

^{isnotcontainedin}

T 6≤`

^,then

S 1

^{containsanaxiomwithlabel}

≤ `

^. Consequently,

N

m∈M 1 m = λ S 1 ≤ N

m∈M 0 m

^{, andthus}

M 1

^{neednot be}

computed.Algorithm2describesourmethodforcomputingtheboundaryusing

avariantoftheHSTalgorithmthat isbasedonthisidea.

In the procedure hst-boundary, three global variables are declared:

C

and

H

, initialized with

∅

^{, and}

ν

^{. The variable}

C

stores all the minimal label sets

computedsofar, whileeach element of

H

isaset oflabelssuch that, whenall theaxiomswithalabellessthanorequaltoanylabelfromthesetareremoved

fromtheontology,theconsequencedoesnotfollowanymore;thevariable

ν

^stores

thesupremumofthelabelsofalltheelementsin

C

andultimatelycorresponds

arstminimal labelset

M

^{, which isused to labelthe root of atree. Foreach}

elementof

M

^{,abranchiscreatedbycalling theprocedure}expand-hst.

Theprocedureexpand-hstimplementstheideasofHSTconstructionforpin-

pointing[11,21]withadditionaloptimizationsthathelpreducethesearchspace

aswellasthenumberofcallstomin-lab.Firstnoticethateach

M ∈ C

isamin- imallabelset,andhencetheinmumofitselementscorrespondstothelabelof

someMinA for

α

^.Thus,

ν

^{isthesupremumof thelabelsofaset ofMinAs for}

α

^{.Ifthisisnotyettheboundary,thentheremustexistanotherMinA}

S

^whose

labelisnotlessthanorequalto

ν

^{.Thisinparticularmeansthatnoelementof}

S

^{may havealabellessthanorequalto}

ν

^{, asthe labelof}

S

^{is theinmumof}

thelabelsof theaxiomsin it. Whensearching forthis newMinA wecanthen

exclude allaxiomshavinga label

≤ ν

^{, as donein Line 6of} expand-hst. Every timeweexpandanode,weextendtheset

H

^{,whichstoresthelabelsthathave}

beenremovedonthepathinthetreetoreachthecurrentnode.Ifwereachnor-

mal termination,it meansthat theconsequencedoesnot followanymorefrom

the reducedontology. Thus, any

H

^{stored in}

H

is such that, ifall the axioms havingalabellessthanorequaltoanelementin

H

^{areremovedfrom}

T

^,then

α

^{does notfollow anymore. Lines 1 to 4 of expand-hst are used to reduce the}

numberofcallstothesubroutinemin-labandthetotalsearchspace.Wedescribe

themnowinmoredetail.Therstoptimization,earlypathtermination,prunes

the tree once we know that nonew information canbe obtainedfrom further

expansion.Therearetwoconditionsthattriggerthisoptimization.Therstone

tries to decidewhether

T 6≤ν | = α

^{withoutexecutingthe decisionprocedure.As}

saidbefore,weknowthatforeach

H ⁰ ∈ H

,ifalllabelslessthanorequaltoany in

H ⁰

^{areremoved,thenthe}consequencedoesnotfollow.Hence,ifthecurrent listofremovallabels

H

^containsaset

H ⁰ ∈ H

weknowthatenoughlabelshave beenremovedto makesurethat theconsequencedoesnotfollow.Itisactually

enough to test whether

{h ∈ H ⁰ | h 6≤ ν } ⊆ H

^{since the}consequence test we need to perform iswhether

T 6≤ν | = α

^{. Thesecond conditionfor earlypath ter-}

mination asks for a prex-path

P

^of

H ⁰

^{such that}

P = H

^{. If we consider}

H ⁰

asalistof elements,thenaprex-pathisobtainedbyremovinganalportion

of this list. The idea is that, if at some point we have noticed that we have

removedthesameaxiomsasin apreviousportion ofthesearch, weknowthat

allpossibilitiesthatarisefromthatsearchhavealreadybeentestedbefore,and

henceitisunnecessaryto repeatthework.Hence wecanprunethetreehere.

The second optimization avoids a call to min-lab by reusing a previously

computed minimallabelset.Notice that ouronlyrequirementonmin-lab that

it producesaminimal label set. Hence,anyminimal labelset for theontology

obtained after removing all labels less than or equal to any

h ∈ H

^{or to}

ν

would work. The MinLab reuse optimization checks whether there is such a

previouslycomputedminimallabelset.Ifthisisthecase,itusesthissetinstead

ofcomputinganewonebycalling min-lab.

Theorem3. Let

T

^and

α

^besuchthat

T | = α

^{.Then Algorithm2computesthe}

margin-basedboundaryof

α

^.

` 0

` 5 ` 4

` 3 ` 2

` 1

Fig.1.Alattice

n 0 : {` 4 , ` 5 }

n 1 : {` 2 , ` 3 } n 4 : {` 2 , ` 3 }

n 2 : n 3 : n 5 : 3 n 6 : 3

` 4 ` 5

` 2 ` 3 ` 2 ` 3

Fig.2.AnexpansionoftheHSTmethod

Aproofof this theoremcanbefound in [2].Here, wejust illustrate howit

worksbyasmallexample.

Example 2. ConsiderthelatticeinFigure1,andlet

T

^bethe(DescriptionLogic) ontologyconsistingofthefollowingveaxioms:

t 1 : A v P 1 u Q 1 , t 2 : P 1 v P 2 u Q 2 , t 3 : P 2 v B, t 4 : Q 1 v P 2 u Q 2 , t 5 : Q 2 v B,

where eachaxiom

t i

^{is labeled with}

lab(t i ) = ` i

^{. Thereare four MinAs forthe}

subsumption relation

A v B

^w.r.t.

T

^{, namely}

{t 1 , t 2 , t 3 }, {t 1 , t 2 , t 5 }, {t 1 , t 3 , t 4 }

^,

and

{t 1 , t 4 , t 5 }

^{. All the elements of the labeling lattice except}

` 1

^and

` 3

^are

join prime relative to

L ^lab

^{. Figure 2 showsa possible run of the} hst-boundary algorithm. The algorithm rst calls the routine min-lab

(T , A v B)

^{. Consider}

thattheforloopof min-labisexecutedusingthelabels

` 1 , . . . , ` 5

^{in thatorder.}

Thus, wetryrst to remove

t 1

^labeledwith

` 1

^{. Wesee that}

T − ` 1 6| = A v B

^;

hence

t 1

^{isnotremovedfrom}

T

^,and

M L

^isupdatedto

M L = {` 1 }

^.Wethensee

that

T − ` 2 | = A v B

^{, andthus}

t 2

^{is removedfrom}

T

^.Again,

T − ` 3 | = A v B

^,

so

t 3

^{is removed from}

T

^{. Atthis point,}

T = {t 1 , t 4 , t 5 }

^{.We test then whether}

T − ` 4 | = A v B

^{and receive a negative answer; thus,}

` 4

^{is added to}

M L

^;

additionally, since

` 4 < ` 1

^{, the latter is removed from}

M L

^{. Finally,}

T − ` 5 6| = A v B

^{, andsoweobtain}

M L = {` 4 , ` 5 }

^{asanoutputof min-lab.}

Theminimallabelset

{` 4 , ` 5 }

^{,isusedastherootnode}

n 0

^{,settingthevalue}

of

ν = ` 4 ⊗ ` 5 = ` 0

^{.Wethencreate therstbranchontheleftbyremovingall}

the axiomswitha label

≤ ` 4

^{, which isonly}

t 4

^{, and computinganew minimal}

labelset.Assume,forthesakeoftheexample,thatmin-labreturnstheminimal

labelset

{` 2 , ` 3 }

^,and

ν

^isaccordinglychangedto

` 4

^{.Whenweexpandthetree}

from this node, by removingall theaxioms below

` 2

^{(left branch) or}

` 3

^(right

branch), thesubsumptionrelation

A v B

^{doesnotfollowanymore,andhence}

wehave anormal termination, adding the sets

{` 4 , ` 2 }

^and

{` 4 , ` 3 }

^to

H

. We thencreatethesecondbranchfromtheroot,byremovingtheelementsbelow

` 5

^.

Wesee thatthepreviouslycomputedminimalaxiomset ofnode

n 1

^worksalso

asaminimalaxiomsetinthiscase,andhenceitcanbereused(MinLabreuse),

Input:

T

^:ontology;

α

^:consequence Output:

ν

^:

(T , α)

^-boundary

1: if

T 6| = α

^then

2: return noboundary

3:

` := 0 lab ; h := 1 lab

4: while

l < h

^do

5: set

m, ` < m ≤ h

^suchthat

δ(`, m) − δ(m, h) ≤ 1

^.

6: if

T ^m | = α

^then

7:

` := m

8: else

9:

h := pred(m)

10: return

ν := `

representedasanunderlinedset.Thealgorithmcontinuesnowbycallingexpand-

hst

(T 6≤` 2 , A v B, {` 5 , ` 2 })

^{. Atthis point, wedetect that there is}

H ⁰ = {` 4 , ` 2 }

satisfyingtherstconditionofearlypathtermination (recallthat

ν = ` 4

^),and

hence the expansion of that branch at that point. Analogously, we obtain an

early path termination on the second expansion branch of the node

n 4

^{. The}

algorithmthenoutputs

ν = ` 4

^{,whichcanbeeasilyveriedto beaboundary.}

3.3 Binary Searchfor Linear Ordering

In thissubsection, weassumethat thelabeling lattice

(L, ≤)

^{isalinear order,}

i.e.,foranytwoelements

` 1 , ` 2

^of

L

^wehave

` 1 ≤ ` 2

^or

` 2 ≤ ` 1

^.

Lemma5. Let

T

^and

α

^besuchthat

T | = α

^{.Thenthe uniqueboundaryof}

α

^is

the maximalelement

µ

^of

L ^lab

^with

T µ | = α

^.

Adirectwayforcomputingtheboundaryin thisrestrictedsettingthuscon-

sists of testing, for every element in

` ∈ L ^lab

^{, in order (either increasing or}

decreasing) whether

T ` | = α

^{until the desired maximal elementis found. This}

process requires in the worst case

n := |L ^lab |

iterations.This can be improved using binarysearch, whichrequires alogarithmicnumberof stepsmeasuredin

n

^{.Algorithm 3describesthebinary search algorithm.Inthe}descriptionofthe

algorithm, the following abbreviations havebeen used:

0 lab

^and

1 lab

^represent

theminimal andthemaximal elementsof

L ^lab

^,respectively;for

` 1 ≤ ` 2 ∈ L ^lab

^,

δ(` 1 , ` 2 ) := |{` <sup>0</sup> ∈ L <sup>lab</sup> | ` 1 < ` <sup>0</sup> ≤ ` 2 }|

^{isthedistance function in}

L ^lab

^andfora

given

` ∈ L ^lab

^,

pred (`)

^{isthemaximalelement}

` ⁰ ∈ L ^lab

^suchthat

` <sup>0</sup> < `

^.

Thevariables

`

^and

h

^{are used to keep trackof the relevantsearch space.}

Ateveryiterationof thewhileloop,theboundaryis between

`

^and

h

^.Atthe

beginning these values are set to the minimum and maximum of

L lab

^{and are}

latermodiedasfollows:werstndthemiddleelement

m

^{ofthesearchspace;}

i.e., anelementwhosedistance to

`

^{diersbyatmostonefromthedistance to}

h

^{.Wethentestwhether}

T m | = α

^{.Ifthatisthecase,weknowthattheboundary}

must belarger orequal to

m

^{, and hence thelowerbound}

`

^{is updated to the}

valueof

m

^{.Otherwise,weknowthattheboundaryisstrictlysmallerthan}

m

^as

m

^{itself cannot be one;hence, thehigher bound}

h

^{isupdated to the maximal}

elementof

L lab

^{thatissmallerthan}

m : pred(m)

^{.Thisprocess terminateswhen}

thesearchspacehasbeenreducedtoasinglepoint,whichmustbetheboundary.

4 Empirical Evaluation

4.1 Test data and testenvironment

We test ona PC with 2GBRAM and Intel CoreDuo CPU 3.16GHz. Weim-

plemented all approaches in Java and used Java 1.6, CEL 1.0, Pellet 2.0.0-rc5

and OWL API trunk revision 1150. The boundary computation with full ax-

iom pinpointing (FP in the following) uses log-extract-mina()(Alg. 2 from [7],

whichisidenticaltoAlg.8from[21])andtheHSTbasedhst-extract-all-minas()

(Alg. 9 from [21]). The set of extracted MinAs is then used to calculate the

label ofthe consequence.Webreak after 10found MinAsin order to limitthe

runtime, sothere mightbenon-nal label results.The boundary computation

withlabel-optimizedaxiompinpointing(LPinthefollowing)withmin-lab()and

hst-boundary() are implementations of Alg. 1and Alg. 2 of the present paper.

The boundary computation with binary search for linear ordering (BS in the

following)implementsAlg.3ofthepresentpaper.

Althoughwefocusoncomparingtheeciency ofthepresentedalgorithms,

andnotonpracticalapplicationsofthesealgorithms,wehavetriedtouseinputs

that are closely related to ones encountered in applications. The two labeling

lattices

(L d , ≤ _d )

^and

(L l , ≤ _l )

^{aresimilartoones}encounteredinreal-worldappli- cations.Thelabelinglattice

(L d , ≤ _d )

^{wasalreadyintroducedinFig.1.Lattices}

ofthisstructure(wheretheelementscorrespondtohierarchicallyorganizeduser

roles) can be obtainedfrom a real-world access matrix with the methodology

presentedin [8]. The set of elements of

L d

^{that are allowed to represent user}

roles ifallelementsof thelatticecanbeusedasaxiomlabelsare theelements

that are join prime relativeto thewhole lattice, i.e.,

` 0 , ` 2 , ` 4 , ` 5

^{. The labeling}

lattice

(L l , ≤ l )

^{is a linear order with 6 elements}

L l = L d = {` 0 , . . . , ` 5 }

^with

≤ _l := {(` n , ` n +1 ) | ` n , ` n +1 ∈ L l ∧ 0 ≤ n ≤ 5}

^{,whichcouldrepresentanorderof}

trustvaluesasin[18]ordatesfromarevisionhistory.

Weusedthetwoontologies

O

^{Snomed and}

O

^{Funct withdierent}expressivity andtypesofconsequencesforourexperiments.TheSystematizedNomenclature

ofMedicine,ClinicalTerms(Snomed ct)isacomprehensivemedicalandclin-

ical ontologywhichisbuiltusingtheDescriptionLogic(DL)

EL+

^.Ourversion

O

^{Snomedis the}January/2005releaseoftheDLversion,whichcontains379,691 conceptnames,62objectpropertynames,and379,704axioms.Sincemorethan

vemillionsubsumptionsare consequencesof

O

^{Snomed, testingallof themwas}

not feasible and we used the same sample subset as described in [7], i.e., we

sampled 0.5% of all concepts in each top-level category of

O

^{Snomed. For each}

sampledconcept

A

^,allpositivesubsumptions

A v _O

^Snomed

B

^with

A

^assubsumee

were considered.Overall,this yielded 27,477positive subsumptions. Following

theideasof[7],weprecomputedthereachability-basedmoduleforeachsampled

]

^early

termination

]

^reuse

]

^callsto

extract

MinA

(MinLab)

]

^MinA

(

]

^MinLab)

]

^axioms

(

]

^{lab els)p er}

MinA

(MinLab)

lattice

op erations

time

total

lab eling

time

O

Snomed FP

avg 81.05 9.06 26.43 2.07 5.40 0.25 143.55

max 57,188.00 4,850.00 4,567.00 9.00 28.67 45.00 101,616.00

stddev 874.34 82.00 90.48 1.86 3.80 0.86 1,754.03

LP

avg 0.01 0.00 2.76 1.03 1.73 0.35 4.29

max 2.00 1.00 6.00 3.00 3.00 57.00 70.00

stddev 0.13 0.02 0.59 0.16 0.56 0.98 3.62

O

Funct FP

avg 43.59 29.52 26.56 4.26 3.05 0.49 3,403.56

max 567.00 433.00 126.00 9.00 6.50 41.00 13,431.00

stddev 92.16 64.04 30.90 2.84 1.01 2.38 3,254.25

LP

avg 0.09 0.02 2.80 1.33 1.40 0.76 207.32

max 2.00 1.00 7.00 4.00 3.00 22.00 1,295.00

stddev 0.34 0.13 0.90 0.54 0.48 1.56 87.29

Table 1. Emprical results of FP and LP with lattice

(L d , ≤ ^d )

^{on a sampled set of}

21,001subsumptionsfrom

O

^{Snomedandonasetof307}consequencesfrom

O

^Functwith

lessthan10MinAs(timeinms)

concept

A

^{withCELandstoredthesemodules.Thismodulefor}

A

^wasthenused

asthestartontologywhenconsideringsubsumptionswithsubsumee

A

^.

O

^{Funct is an OWL ontologyfor functional} description of mechanicalengi- neeringsolutionspresentedin[10].Ithas115conceptnames,47objectproperty

names, 16 data property names, 545 individual names, 3,176 axioms, and the

DLexpressivity usedin the ontologyis

SHOIN ( D )

^{. Its716}consequences are 12subsumptionand704instancerelationships(classassertions).

To obtain labeled ontologies, axiomsin both labeled ontologies received a

random label assignment of elements from

L l = L d

^{. As black-box subsump-}

tion and instance reasoner we used the reasonerPellet since it can deal with

theexpressivity ofbothontologies.FortheexpressiveDL

SHOIN ( D )

^ituses

a tableau-based algorithm and for

EL+

^{it uses anoptimized classier for the}

OWL2ELprole,which isbasedonthealgorithmdescribedin[1].

4.2 Results

The results for

O

^{Snomed and}

(L d , ≤ _d )

^{are given in the upper part of Table 1.}

LP computedalllabels,but sincewelimitFPto<10MinAs, only21,001sub-

sumptionshaveanallabel,whichisguaranteed tobeequalto theboundary.

The 6,476 remaining subsumptions (31%) have a non-nal label which might

betoolowinthelatticesincethere mightbefurtherMinAsprovidingahigher

label. Theoveralllabelingtime forall21,001subsumptionswith FPwas50.25

minutes,forLP1.50minuteswhichmeansthatLPisabout34timesfasterthan

FP,butagainthisisonlyforthesubsetofsubsumptionswhichwerenishedby

FP. An estimation for the time neededto label allof the morethan 5million

subsumptionsin

O

^{Snomed withLPwouldbe}approximately6hours.

Thenal labelsof FP and LP (i.e.,the computedboundaries) were identi-

cal, the non-nallabelsof FPwereidentical to thenal labels ofLP (i.e., the

boundaries) in 6,376 of the 6,476 cases (98%), i.e., in most cases the missing

MinAs would not have changed the alreadycomputed label. Table 2 provides

resultsfor the subsumptionswith more than10 MinAs: FP took 2.5hours on

]

^early

termination

]

^reuse

]

^callsto

extract

MinA

(MinLab)

]

^MinA

(

]

^MinLab)

]

^axioms

(

]

^{lab els)p er}

MinA

(MinLab)

lattice

op erations

time

total

(non-nal)

lab eling

time

O

Snomed FP

avg 432.11 42.25 126.54 10.20 16.38 0.30 1,378.66

max 42,963.00 5,003.00 4,623.00 16.00 37.80 14.00 148,119.00

stddev 1,125.06 121.15 186.33 0.49 5.00 0.54 3,493.02

LP

avg 0.04 0.00 3.12 1.06 2.05 0.32 8.88

max 3.00 2.00 6.00 3.00 3.00 46.00 86.00

stddev 0.21 0.04 0.50 0.25 0.44 1.04 4.26

O

Funct FP

avg 30.01 16.00 26.44 10.04 4.41 0.56 8,214.91

max 760.00 511.00 411.00 11.00 6.50 3.00 25,148.00

stddev 85.33 47.79 40.61 0.20 1.08 0.55 3,428.97

LP

avg 0.09 0.01 2.76 1.38 1.32 0.77 200.55

max 3.00 2.00 7.00 4.00 2.00 16.00 596.00

stddev 0.33 0.12 0.91 0.64 0.43 1.40 61.11

Table2.EmpricalresultsofFPandLPwithlattice

(L d , ≤ ^d )

^{onasampledsetof6,476}

subsumptionsfrom

O

^{Snomedand onasetof 409classassertions from}

O

^{Funct withat}

least10MinAs(timeinms)

LP BS

]

^early

termina-

tion

]

^reuse

]

^callsto

extract

MinLab

]

^MinLab

]

^{lab els}

p er

MinLab lattice

op era-

tions

time total

lab eling

time

iterations total

lab eling

time

O

^Snomed

avg 0.03 0.00 2.24 1.03 1.23 0.37 4.75 2.41 2.81

max 1.00 0.00 5.00 3.00 2.00 329.00 330.00 3.00 75.00

stddev 0.18 0.00 0.45 0.19 0.42 4.85 6.37 0.49 2.94

O

^Funct

avg 0.09 0.00 2.50 1.27 1.24 0.82 186.98 2.55 95.80

max 1.00 0.00 5.00 3.00 2.00 62.00 1147.00 3.00 877.00

stddev 0.28 0.00 0.72 0.49 0.40 2.74 69.55 0.50 45.44

Table 3.EmpricalresultsofLPandBSonasampledsetof27,477subsumptionsin

O

^{Snomed/all716}consequencesof

O

^{Funct withlattice}

(L ^l , ≤ ^l )

^(timeinms)

thissetwithoutnalresults(sinceitstoppedafter10MinAs),whereasLPtook

0.6%ofthattime andreturnednalresultsafter58seconds.Westartedatest

serieslimitingrunsofFPto<30MinAs,whichdidnotterminateafter90hours,

with1,572labelssuccessfullycomputedand30subsumptionsskippedsincethey

had

≥

^30MinAs. Interestingly,inbothconsequencesets, LP canrarelytakead- vantageoftheoptimizationsearlyterminationandMinAreuse,whichmightbe

dueto thesimplestructureofthelattice.

For

O

^{Funct the comparison between FP and LP is givenin thelower part}

ofTables1and2.Again,thecomputationofFPwasrestrictedto<10MinAs.

This time, only 363outof 409(88%) non-nal labels of FP were equalto the

nallabelsofLP(i.e.,theboundary).Althoughtheontologyisquitesmall,LP

againbehavesmuch betterthan FP. Thereasoncouldbethat in thisontology

consequences frequently have a largeset of MinAs. From Tables 1 and 2,one

can see that LP requires at mostthree MinLabs for

O

^{Snomed, at mostfour for}

O

^{Funct,andusuallyjustoneMinLabwhereasFPusuallyrequiresmoreMinAs.}

Table3providesresultsforLPvs.BSwiththetotalorder

(L l , ≤ l )

^aslabeling

lattice.For

O

^{Snomed,LPtakes130.4andBStakes77.1secondstolabelall27,477}

subsumptions. For

O

^{Funct, LP takes133.9 and BStakes68.6 seconds to label}

all716consequences.SoBSisabouttwice asfastasLP.Interestingly,labeling

allconsequencesof

O

^{Funct and}

O

^{Snomed takesroughlythesametime, perhaps}

dueto atradeobetweenontologysizeandexpressivity.

We haveconsidereda scenariowhere ontologyaxiomsarelabeledand userla-

belsdetermine views onthe ontology,i.e., sub-ontologies that are obtainedby

comparing theuserlabelwith theaxiomlabels.Our approach canbeused for

large-scale ontologies since, on the one hand, it allows to precompute conse-

quences withouthavingto dodothis separatelyforallpossibleviews:once we

have computed aboundary for the consequence, checking whether this conse-

quence entailedby asub-ontologyisreducedto asimplelabelcomparison. On

the other hand, the fact that we employ ablack-box approach for computing

the boundary allowsus to use existing highly-optimzed reasoners,rather than

havingto implementanewreasonerfrom scratch.

Our general framework allows to use any restriction criterion that can be

represented using a lattice, such as user roles, levels of trust, granularity, or

degreesofuncertainty.Inthepresenceofaccessrestrictions,eachuserlabelde-

nes asub-ontologycontainingtheaxiomsvisible tothis user.In thepresence

of trustrestrictions,the userlabel species thetrust levelrequiredfor theon-

tology axiom. This supports scenarioswith axioms from dierent sources,like

company-internalwithhigh trustleveland publicWeb withlowtrust level.In

the presence of uncertainty, e.g. in possibilistic reasoning, each axiom has an

associated certainty degree in the interval

[0, 1]

^{. The user label then species}

the certainty degree required for the axioms and the consequences. Similarly,

granularityrestrictions(i.e., on how much details the ontologyshould provide

fortheuser)canbeexpressedbyatotalorder.

Our experiments have shown that this framework can be applied to large

ontologies.Fromthetwoblack-boxalgorithmsthatcandealwitharbitrarylat-

tices,theFullAxiomPinpointingapproachisclearlyoutperformedbytheLabel-

OptimizedAxiomPinpointingapproach.Forthespecialcasewherethelabeling

lattice is atotal order,the latter is again outperformed by the Binary Search

approach.

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dresden.de/research/reports.html.

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