Desription Logis (extended abstrat)
EspenH.Lian andArildWaaler
DepartmentofInformatis
UniversityofOslo,Norway,
{elian,arild}ifi.uio.no
1 Introdution
Weproposetheonly-knowinglogiO R
ALC,basedonthedesriptionlogiALC.
Only-knowinglogiswereoriginallydesignedasmodallogisoveralassialon-
sequenerelation fortherepresentationofautoepistemireasoning[17,18℄ and
defaultlogi[15℄.Only-knowinglogisareworthonsideringforseveralreasons:
They are monotoni logis with a lear separation between objet level and
metalevelonepts,andtheyallowfaithfulandmodularenodingsofautoepis-
temianddefaulttheorieswhihdonotinreasethesizeoftherepresentations.
Theenodingsthusprovideautoepistemianddefaultlogiswithformalseman-
tis andoneptuallarity.Besidesbeingompletelyaxiomatized,propositional
only-knowing logis support enodings of propositional autoepistemi and de-
faultlogiswithnieomputationalproperties.
{ There is a simple rewrite proedure that determines extensions. Some of
the rewrite rules are preonditioned by SAT tests, and these are the only
refereneto meta-logialoneptsin theproedure.
{ Theextensionproblem forpropositionalonly-knowinglogishavethesame
omplexityasforpropositionalautoepistemianddefaultlogis.
Toourknowledge,noextensionofonly-knowinglogistodesriptionlogishave
yet beengiven. In thispaperwetakethe desriptionlogi ALC astheunder-
lying logiinsteadof lassialpropositionallogiand generalizethe mahinery
originallydesignedforpropositionallogisofonly-knowing.Thestrengthofthe
proposedonly-knowinglogiis thesimplerewrite proedure that itadmits for
omputing extensions, and thefat that reasoning isnot harderthan in ALC.
ThelogiO R
ALC proposed inthispapersubsumesthepropositionallogithat
weproposedin[20℄and extendsthatworkinanon-trivialway.
The logi ALCK
NF
introdued by Donini, Nardi and Rosati [9℄ is losely
related toO R
ALC.It is onstrutedbyombiningMKNF withALC. It isnot
obvioushowthis should bedone, and in ouradaptation of only-knowinglogi
toALC,wehavebeenguidedbyALCK
NF
.Unlike[9℄,wedonottreatso-alled
subjetivelyquantiedexpressionsinthispaperbutthelogiwepresentisstrong
enoughtorepresent,e.g.,defaulttheoriesforALC.
In[9℄,atableauproedure forALCK
NF
isintrodued. Althoughtheproe-
to guess extensions and then usethe proof proedure to hekwhether ornot
theguesswasorret.Inontrast,theproedurethatweproposedeterminesall
extensionsdiretly.UnliketheproofsystemforALCK
NF
,weformalizeinferene
stepsthataresuÆientfordeterminingextensionsbutnotforharaterizingthe
wholesemantis.
2 O
R
ALC
Thelanguageof O R
ALC is dened intwosteps:rstaonept language,then
amodalformula language.
Conept Language. ThebasisfortheoneptlanguageisALC [1,29℄:
C ;D !>j?jC
at
j:CjCuDjCtDj9R
at :Cj8R
at :C
whereC
at
isanatomioneptandR
at
anatomirole.WewillallALConepts
objetive onepts.TheoneptlanguageforO R
ALCextendsobjetiveonepts
withmodaloneptformationoperators 1
B(belief) andA(assumption):
C ;D !C
ALC
j:CjCuD jCtDjBC jAC
whereC
ALC
isanobjetiveonept.Amodalonept isoftheformBCorAC;
if C is objetive, the modal onept is prime. A onept is subjetive ifevery
objetivesuboneptiswithinthesopeofamodaloneptformationoperator.
An interpretation I = (;
I
) over onsists of a non-empty domain
andaninterpretationfuntion I
mappingatomioneptsandrolestosubsets
of and resp. Following [9℄ we assume that the domain ontains all
individuals in the language, and that a I
= a for eah individual a. Conept
desriptionsareinterpretedrelativetoapairM=(U;V)andaninterpretation
I; werequirethatU andV areinterpretationsover andthat V U,butwe
donotrequirethatI 2U.ALC oneptsevaluaterelativetoanI asusual,e.g.,
(:C) M;I
=nC M;I
,whilemodaloneptsareinterpretedasfollows:
(BC) M;I
=
\
J2U C
M;J
(AC) M;I
=
\
J2V C
M;J
AnABox Aisanitesetofobjetive membershipassertions,i.e.onept asser-
tions C(a) and role assertions R (a;b) for individualsa andb.O
A
is theset of
individualsexpliitlymentionedinA.Amodalatom isanassertionoftheform
M(a), whereM is amodalonept; M(a) isprime ifM isprime.M satises
C(a) in I if a 2 C M;I
; M satises R (a;b) in I if (a;b) 2 R M;I
. As we will
see,modalatomsplayanessentialroleintherewritesystem.ADBox isanite
setofterminologial axioms,i.e.inlusions CvDforoneptsCandD,while
1
Inthe literature, K is sometimes used instead ofB, and :M: insteadof A. For
intuition about the modalities,onsult the literature about MKNF[9,22,23℄ and
a TBox is an objetiveDBox. M satises C vD in I if C D . We
dene a knowledge base (KB) asa tuple (T;A;D), where T is aTBox,A an
ABox,and DisaDBox.TheDBoxwill typially(but notneessarily)ontain
representation of default rules. Weassume standardnotionsand notations for
default theories.Let ,
k
for 16k6n and beALC-onepts. Below isan
ALC-default anditsrepresentation(followingKonolige[14℄;notethat [9℄hasa
Bin frontof)asaninlusionstatement:
:
1
;:::;
n
= ; Bu:A:
1
uu:A:
n v
Bdenotesthatisbelieved,while:A: denotesthatisonsideredpossible,
orthat isonsistent withonesbeliefs.BisstrongerthanAin thesense that
everyMsatisesBCvAC in everyinterpretationI.
Interpretation of anobjetive oneptis independentof M, hene we may
write C I
forC M;I
andsay thatI satises C(a),writtenI C(a),ifa2C I
;
similarlyforroles,roleassertionsandinlusionstatements.Henewemaywrite
IAandI T ifeveryelementin therespetiveABoxandTBoxissatised
byI.WealsowriteI (T;A)ifI T andIA.Foranobjetiveassertion,
wewriteT;AifI foreveryI suhthatI (T;A).Similarly,T;AA 0
ifIA 0
foreveryI suhthat I(T;A).
FormulaLanguage. Formulaearedenedasfollows.Coneptandroleassertions,
TandFareatomi formulae.:','^ and'_ areformulaeif'and are.
B, A,
B and O are modal operators. B and A orrespond to their onept
formation operator ounterparts, O is an \only knowing"-operator[17,33,20℄,
while
B isa\knowingatmost"-operator,orrespondingtoC: of[20℄.L'isa
formulaifL isamodaloperatorand 'is aformula withoutanyourreneof
. ' is aformula if'is a subjetive formula, i.e. everyatomi formula that
ours asasubformulain 'is eitherasubjetiveassertionorwithin thesope
ofamodaloperator.NoteinpartiularthatifCisanobjetiveonept,BC(a)
isamodalatom,andheneanatomiformula,whileB(C(a)) isnotanatomi
formula. Abbreviations:') and ', aredened asusual;
3
'is::';
O R
'isO'^:O'. LetL beamodaloperator.A formulaisL-freeifitdoes
notontainL(neitherasamodaloperatornorasaoneptformationoperator)
andL-basi ifitis subjetiveandontainsnoothermodalitythanL.
Relativeto the universal set U of allinterpretations over that satisfyT,
a model is a pair (U;V) suh that V U U. The modality quanties
over models by means of a binary relation >. Let M = (U;V). M 0
> M if
M 0
= (U 0
;U)for someU 0
U; in whih ase we say that M 0
is larger than
M. Truth onditionsaregivenrelativetoaninterpretationI,whihneedsnot
bein V.Atomiformulaeandonnetivesareinterpretedasonewouldexpet,
e.g., Mj=
I
C(a) i a2 C M;I
, Mj=
I
T and M 6j=
I
F.The modal operators
areinterpretedasfollows:
{ Mj=
I
B'iMj=
J
'foreahJ 2U;
{ Mj=
I
A'iMj=
J
'foreah J 2V;
{ Mj=
B'iJ 2U foreah J 2U suhthat Mj= ';
{ Mj=
I
O'i(Mj=
J
'ifandonlyifJ 2U)foreahJ 2U;
{ Mj=
I
' iM
0
j=
I
'foreveryM 0
>M.
We write Mj= ' ifM j=
I
'for eah I 2 U. Relative to amodelM, k'k M
denotes the truth set of ' in M, i.e. fI 2 U j M j=
I
'g. Note that if ' is
objetive,k'k M
is givenindependentlyofM,asitonlydependsonthepoints
in U. Foranyobjetive':
{ U =k'k i(U;V)j=O';
{ V k'ki(U;V)j=A'.
{ U k'k i(U;V)j=B';
{ U k'k i(U;V)j=
B';
Also note that in the lauses that dene truth for the modal operators, the
interpretation I playsno ativerole in thedenition.When'is subjetive,it
is immediatethat Mj=
I
'i Mj= ',i.e. weansafelyskip thereferene to
I. Thisisalsothereasonwhythefollowingobservationholds.
Lemma1. Foranysubjetive onept M,
either Mj=M(a),>(a) orMj=M(a),?(a).
It is also the asethat Mj= ' orMj= :', for any subjetiveformula '. A
formula' isstrongly valid, written j',if Mj= 'foreverymodelM. There
is also a weaker notionof validity, whih is the notion of validity that we are
primarilyinterestedin.Itisdenedrelativetothesetofweak models:(U;V)is
aweakmodelifU =V.'isvalid,writtenj=',ifMj='foreveryweakmodel
M.Clearly,strongvalidityimpliesvalidity,but notonversely.
Lemma2. jB')A'andj=A')B'.
Proof. Follows from the onditions V U (for arbitrarymodels) and V =U
(forweakmodels). ut
We are interested in models M of O R
', thus wewant M to satisfy O' but
not
3
O',i.e.nolargermodelthanMshouldalsosatisfyO'.Thegurebelow
illustratesthetruthonditionsrelativetoM
2
>M
1
,whereM
1
=(U
1
;V
1 )for
anobjetive'andanarbitraryV
1 U
1
k'k,andM
2
=(U
2
;V
2
)=(k'k;U
1 ).
M
2
j=O'^:
3 O' V
2 U
2
=k'k
M
1
j=:O'^
3 O' U
1 k'k
= Examining M
1
, we see
that U
1
k'k, thus
B'doesnothold inM
1 ,
hene neither does O'.
O' is, however, true in
M
2
, and sine M
2
>
M
1 ,
3
O'istruein M
1 .
Examining M
2
, we see
that U
2
=k'k, thus O'
istrue.Butasthereisno
M>M
2
thatmakesO'true,
3
O'isnottrue.Hene O R
'holdsin M
2 .
2
We ould have dened Oin terms ofB and
B (syntatially),as Mj=I O' ,
modelM
1
:Anymodelof
3
O'musthavetheshapeofM
1
,inwhihtheremust
be apoint I 62 U
1
at whih ' is true.Note that B' and :
B' are bothtrue
in M
1
. Conversely, any model ofB'^:
B ' must alsohave theshapeof M
1 ,
satisfying
3 O'.
Lemma3. j
3
O',(B'^:
B') if'isobjetive.
ForformulaeintheA-freefragmentofthelanguage,thetwonotionsofvalidity
oinide.Itiseasytoseethat inthisasetheweakmodelsofO R
'areexatly
themodels(U;U)withthelargestbeliefstateU thatsatisfyO'.
Let[℄denotethefuntionthatreplaesAwithB,and(fortheservieofthe
rewrite rules)puts theresultingformulaonnegation normalform,i.e. [A'℄=
B['℄, [:A'℄=:B['℄,[::'℄=['℄,[:('^ )℄=[:'℄_[: ℄,andsoforth.
Lemma4. j
3
')['℄ if'isA-basi.
Proof. Let M = (U;V). If M j=
3
', then M 0
j= ' for some M 0
> M. By
denition,M 0
=(W;U)for someW U.Sine 'is A-basi,itis interpreted
in M 0
onlyrelativeto U.ButifwesubstituteBfor allourrenesof Ain ',
theresultingformula['℄isB-basiandisheneinterpretedinMrelativetoU
in exatlythesamewayas'isinterpretedinM 0
.Hene Mj=['℄. ut
Weemploytheonventionthat whenanitesetofformulaeX oursin plae
of aformula,this isto beread astheonjuntion ofits elements, i.e.
V
X; for
X = ; this amounts to T. Hene we will refer to a onjuntion of objetive
assertions as an ABox. Next we show under whih onditions OA, for some
ABoxA,impliesaprime modalatom oritsnegation.Aswewill see,thisgives
usthesideonditionsfortheollapserulesintherewritesystem.
Lemma5. Let C andR beobjetive.
1. jOA)BC(a) ifT;AC(a);
2. jOA):(BC(a))if T;A6C(a).
Proof. Let M = (U;V) be a model suh that M j= OA. Then U = kAk. 1.
Assume that T;AC(a).Then I C(a) for everyI s.t. I j=T and I j=A.
Hene U kC(a)k. It followsthat Mj=BC(a).2. Assume that T;A6C(a).
Then I :C(a) for some I s.t. I j= T and I j= A, hene there is some
interpretationJ 2U \k:C(a)k.HeneMj=:(BC(a)). ut
Corollary 1. LetC andR beobjetive.
1. jOA)AC(a) ifT;AC(a);
2. j=OA):(AC(a)) if T;A6C(a)
Proof. ByLemmata2and5. ut
Lemma6. Let AandA 0
beABoxes.
1. jOA)
BA 0
ifT;A 0
A.
0 0
Generalizing the rewrite system for the underlying propositional language in
[20℄, the system in Fig. 1 onsists of two rewrite relations on formulae. The
rulesoftherelationarebasedonstrongequivalenes,whereasthe^ relation
extends with rules whose underlying equivalenes are merelyweak. We say
that 'reduesto if' , where isthereexivetransitivelosure of.
Redutionanbeperformedonanysubformula.Aformula'isonnormalform
wrt.ifthereis noformula suh that' .Thesamenotationisusedfor
the^ relation.Redutionisperformedmoduloommutativityandassoiativity
of ^ and _,and ' is identied with'^T and '_F;this implies that T and
F behaveasemptyonjuntionanddisjuntionresp.Wedenetobetheset
of rules l r in Fig. 1, while ^ is the union of and theset of rules l^ r.
Applyingtherelationexhaustivelybeforethe^isappliedguaranteesaorret
rewriteproess.
Forformulae and , h=i is asubstitution funtion:'h=i denotes the
resultofsubstitutingeveryourreneofin'with.Substitutionisperformed
stritly on the formula level for the reason that assertions do not onsist of
subassertionsin thesense that formulaeonsistof subformulae.A substitution
of avaluefor anassertionC(a) will notapply to, e.g.,theassertionCtD(a)
butwillapplytotheequivalentformulaC(a)_D(a).Thesubstitutionfuntion
hM(a)=V(a)iforaprimemodalatomM(a)andV 2f>;?gisabinding,whih
binds M(a) toV(a).
Theexpandruleworksbybinding prime modalatoms in O',assumingno
subformulaof'isoftheformB orA foranobjetiveformula .Aformula
with this property is objetive if no prime modal atoms our in it. Observe
that theprimemodalatom BC(a) hasthis property,whereasB(C(a))donot.
Bindings might break this property, e.g.,B(BC(a))hBC(a)=>(a)i =B(>(a)).
ForthisreasonwehavetheM
4
ruleswhihregaintheproperty,shoulditbelost
intheourseofabindingoperation,i.e.aftertheexpandrulehasbeenapplied:
B(>(a))(B>)(a).WhentherearenoprimemodalatomsleftinO',onemay
applytheollapserulestoredue(orollapse) O'^M(a) toeitherO'orF.
The last two rules of C
1
are stritly in .^ These do not preserve strong
equivalene and are hene not sound in all ontexts. Applying the relation
exhaustivelytoaformulaO R
'resultsin formulaeofaformwhihreetsthat
thelasttworulesofC
1
havenotbeenapplied.Toharaterizethiswesaythata
formulaissemi-normal ifitisoftheformOA^foranABoxAandapossibly
emptysetofformulaeoftheformAC(a)and:(AC(a))withCobjetiveand
T;A6:C(a).
Lemma7. For eahsemi-normal OA^,
1. OA^ison normalformwrt.;
2. eitherOA^
^
OAorOA^
^
F.
Theprimary funtion of thesystemis to redue formulaeofthe form O'and
R
Theexpandrule:
(M
1 )
O'(O'hM(a)=>(a)i^M(a))_(O'hM(a)=?(a)i^:(M(a)))
foranyprimemodalatomM(a)ourringin '
Thedominationanddistributionrules:
'^FF (M
2 )
('_)^ ('^ )_(^ ) (M
3 )
Theassertionalrules:
(M
4 )
:(C(a))(:C)(a)
B(C(a))(BC)(a)
A(C(a))(AC)(a)
C(a)^D(a)(CuD)(a)
C(a)_D(a)(CtD)(a)
For
B
=BC(a) and
A
=AC(a):
(C
1 )
OA^
B
OA
OA^:
B F
OA^
A
OA
OA^:
A F
ifT;AC(a)
ifT;AC(a)
ifT;AC(a)
ifT;AC(a)
OA^
B F
OA^:
B
OA
OA^
A
^ F
OA^:
A
^
OA
ifT;A6C(a)
ifT;A6C(a)
ifT;A6C(a)
ifT;A6C(a)
ForanABoxA 0
:
(C
2 )
OA^
BA 0
OA
OA^:BA 0
F
ifT;A 0
A
ifT;AA 0
OA^
BA 0
F
OA^:BA 0
OA
ifT;A 0
6A
ifT;A6A 0
Rulesfor reduing O R
and
O R
'O'^:O' (R
1 )
:('_ ):'^: (R
2 )
:FT
(R
3 )
:(OA^):BA_
BA_[:℄ forsemi-normalOA^ (R
4 )
IfinruleR
4
is empty,weget
:OA:BA_
BA (R
0
4 )
Fig.1:Therules.AisanABox,CandDobjetiveonepts,andRanobjetive
role.TherulesinC
k
arealled ollapserules.R
1
isnotaproperrule,astheleft
hand sideisan abbreviation ofthe right.Weinlude itfor readability reasons.
a knowledge base,eah disjunt represents aReiter extension of ' (when O R
is used) or anautoepistemi extension (when Ois used). To ahieve this, the
rewritesystemneedsrulesfor reduingformulaeprexed withOand :, and
whatevertheyarereduedto.
Note that an objetive formula may or may not be an assertion. C(a)_
:C(a)is,forinstane,notanassertion,butanbetransformedtoanequivalent
assertionifwe\pushtheaoutward"andhange_ tot.Belowweaddressthe
reverse operation of \pushing the a inward." We do this to get prime modal
atomsassubformulae,sothat theexpandrulemaybeapplied. Thisoperation,
JK,isdenedasfollows.Forroleassertions,JR (a;b)K =R (a;b),andforonept
assertions,JC(a)K =C(a) forC2f>;?goratomi,and
JLC(a)K= (
LC(a) ifCisobjetive
^
LJC(a)K otherwise
forL2f:;B;Ag;
JC?D(a)K= (
C?D(a) ifC?D isobjetive
JC(a)K ^?JD(a)K otherwise
for?2fu;tg;
where
^
u=^and
^
t=_,and
^
L=LforL2f:;B;Ag.Inlusionsareinstanti-
ated andtranslated intoformulaeasfollows:Foran individuala, JC vDK
a
=
JC(a)K)JD(a)K.
Example 1. LetusaddressthedefaultC:>=C,whihisrepresentedasBCv
C. Let ' =JBC v CK
a
=BC(a) ) C(a). To redue O', we rst apply the
expandruleandthen rulesfrom theC
1
group.Thesameredutions alsoapply
inaboxedontext.AstheformulaisA-free,the^ relationwillnotbeneeded.
O(:BC(a)_C(a))(OC(a)^BC(a))_(O>(a)^:BC(a)) (M
1 )
OC(a)_O>(a) (C
1 )
:O(:BC(a)_C(a)):(OC(a)_O>(a))
:OC(a)^:O>(a) (R
2 )
(:BC(a)_
BC(a))^(:B>(a)_
B>(a)) (R
4 )
HavingreduedO'and:O',weredueO R
'.
O R
'O'^:O'
(OC(a)_O>(a))^(:BC(a)_
BC(a))^(:B>(a)_
B>(a))
(OC(a)^(:BC(a)_
BC(a))^(:B>(a)_
B>(a)))_
(O>(a)^(:BC(a)_
BC(a))^(:B>(a)_
B>(a))) (M
3 )
Distributingonjuntions overdisjuntions, usingM
3
,weobtaina formulaon
DNF,whihreduestoO>(a).ThisorrespondstotheuniqueReiterextension.
u t
TheModalRedutionTheoremforO R
ALCstatesthatwheneveraformulaO R
'
enodesaknowledgebase,itislogiallyequivalenttoadisjuntion,whereeah
disjuntisoftheformOAforsomeABoxA.Infat,eahofthesedisjuntshas
anessentiallyuniqueweakmodel.Itishenepossible,withinthelogiitself, to
deomposeaformulaO R
'intoaformwhihdiretlyexhibitsitsmodels.
Wetranslateanentireknowledgebase =(T;A;D) into aformulaasfol-
lows: JK
J
=A^JDK
J
, where JDK
J
=fJC vDK
a
j a2J &C vD 2Dgfor
somenon-emptyset ofindividualsJ .ObservethattheTBoxT seemingly
disappears.Itdoes,however,reappearinthesideonditionsoftheollapserules.
The Modal Redution Theorem. Foreah=(T;A;D), thereareABoxes
A
1
;:::;A
n
for somen>0, suhthatAA
k
for 16k6n,and
j=O R
JK
O
A
,(OA
1
__OA
n ):
Proof. Byompleteness(Theorem1)andsoundness(Theorems2and3). ut
We dene the extensions 3
of to be exatly the ABoxes A
1
;:::;A
n
in The
ModalRedutionTheorem. Hene thenotionof extensionmakesnoappeal to
theformulalanguage,onlytheoneptlanguage.
The rewrite system is just strong enough for establishing the Modal Re-
dutionTheorem: It issound andomplete forredutions ofO R
-formulaeinto
disjuntions of theappropriatetype. It is, however, notomplete forthe logi
of O R
itself. From the point of view of omputing default extensions this is
enough, beause only a subset of the logi of O R
is atually needed for the
ModalRedutionTheorem.
Example 2. The ALC-defaultEmployee ::Manager =EngineertMathematiian
(adapted from [9℄) is represented as BEmployeeu:AManager v (Engineert
Mathematiian).LetDonsistofthisinlusion,andletJ =fBobg:
O R
(A^JDK
J )
=O R
(A^JBEmployeeu:AManagervEngtMatK
Bob )
=O R
(A^(JBEmployeeu:AManager(Bob)K)JEngtMat(Bob)K)
=O R
(A^(JBEmployee(Bob)K^J:AManager (Bob)K)JEngtMat(Bob)K)
=O R
(A^(BEmployee(Bob)^:AManager (Bob))EngtMat(Bob)))
This formulaanbereduedto asimplerform,dependingontheABoxAand
the TBox. Let T = fManager v Employeeg, and let
1
= BEmployee(Bob),
2
=AManager (Bob)and =EngineertMathematiian(Bob).ForanyAsuh
that AEmployee(Bob),OA^
1
OAandOA^:
1
F,hene
O(A^(
1
^:
2
)))(OA^
1
^
2
)_(O(A^)^
1
^:
2 )_
3
Extensions are here taken inthe sense of defaultlogi, i.e. Reiter extensions; au-
toepistemiextensions(stableexpansions)anbedenedfromtheModalRedution
1 2 1 2
(OA^
2
)_(O(A^)^:
2 ):
FortheABoxA
1
=fEmployee(Bob)g, theredutisonnormalformwrt.but
notfortheABoxA
2
=fManager(Bob)g:
(OA
1
^
2
)_(O(A
1
^)^:
2 )
^
O(A
1
^)
(OA
2
^
2
)_(O(A
2
^)^:
2
)OA
2
Intheformerase,Bobisanengineer oramathematiian,in thelatterBob is
onlyamanager.ReduingO R
(A^JDK
J
)produesthesameextensions. ut
4.1 Soundness and Completeness
Lemma8. Foreah=(T;A;D),thereisaset
1
=fO(A^A
k )^
k g
16k 6n
of semi-normalformulae for somen>0suhthat for someset
2
1
(a) OJK
J
_
1
and (b) O R
JK
J
_
2 :
Theorem1 (Completeness).Foreah =(T;A;D), forsome n>0,there
are ABoxesA
1
;:::;A
n
suhthat for someset ofsemi-normal formulae,
O R
JK
O
A
_
^
(O(A^A
1
)__O(A^A
n )):
Proof. ByLemmata7and8. ut
Lemma9. If Mj=
I
(,)and does notourwithin thesopeof in',
thenMj=
I
(','h=i):
Thepreviousresultstatetheonditionunderwhihsubstitution ofequivalents
isvalid.Forsubstitutionwithintheontextofweneedthestrongernotionof
validity. From thepointofviewof formularewriting,thesigniane ofstrong
validityisthatitisrequiredforgeneralsubstitutionofequivalents.
Lemma10. j(','h=i)if j(,).
Theorem2 (Soundnessof ). If' thenj', .
Proof. ByLemma10,itissuÆienttoshowthatjl,rforeahrulelr,in
whihasewesaythat theruleis stronglyvalid. RuleR
1
istrivial.R
2 follows
from the fat that j('^ ),('^ ) andDe Morgan'slaw,while R
3
followsfrom the fat that j T. M
2
and M
3
arepropositionally valid, M
4 is
left to thereader. C
1
followsfrom Lemma 5and Corollary1,while C
2
follows
fromLemma 6.Thetworemainingasesaretreatedin detail.
R
4
:We showthat j:(O'^),((B'^:
B'))[:℄)for any semi-
normal O'^. By Lemma 3,j
3
O', (B'^:
B'), henebyLemma 10,
we have to show that j :(O'^) , (
3
O' ) [:℄). ()) Assume that
Mj=(O'):)andMj=
O'.ThenMj=
(O'^:),thusMj=
:,
:(O'^) for every M. Now there are two ases. If M j= :O', then
Mj=:(O'^).IfMj=[:℄,then Mj=:[℄, thusMj=:byLemma
4,thus Mj=:(O'^).
M
1
:WeshowthatjO',(O'hM(a)=>(a)i^M(a))_(O'hM(a)=?(a)i^
:(M(a))) for any prime modal atom M(a). Let M be an arbitrary model.
Then either M j=(M(a) , >(a)) orMj=(M(a) ,?(a)) by Lemma 1.By
Lemma 9,as'is -free, eitherMj=O',O'hM(a)=>(a)iorMj=O',
O'hM(a)=?(a)i,resp.IneitheraseMsatiseseither(M(a),>(a))^(O',
O'hM(a)=>(a)i) or(M(a) , ?(a))^(O' , O'hM(a)=?(a)i); from whih
theequivalenefollows. ut
Theorem3 (Soundnessof).^ Foranydisjuntion'ofsemi-normalformu-
lae,if '
^
thenj=', .
Proof. Assemi-normal formulaedo notontain, byLemma9,it issuÆient
to showthat j=l,rfor eah rulel^ r,in whihasewesaythatthe ruleis
valid.Thattherulesofarevalidisobvious,giventhattheyarestronglyvalid.
Theremainingrulesarein theC
1
group;validityfollowsfromCorollary1. ut
4.2 Complexity
The extension problem is determining whether aKB has an extension. A re-
deemingfeatureofO R
ALC isthattheextensionproblemisnotharderthanthe
ALC problemofinstane heking.
Theorem 4. The extensionproblem ofO R
ALC is PSpae-omplete.
Proof (Sketh). The translation from to JK
O
A
an be done in polynomial
time: jO
A
jjDj. The orresponding extension problem when the underlying
logiis propositionalisin p
2
=NP NP
asitanbesolvednondeterministially
withapolynomialnumberofallstoapropositionalSATorale[33℄.Themain
dierene here is that instead of a SAT orale, we need an orale that an
do instaneheksin ALC, whihis PSpae-omplete [1℄. Thusthe extension
problemisin NP PSpae
,whihisequaltoPSpae[25℄. ut
5 Future Work
TheunderlyingoneptlanguagedoesnothavetobeALC.Otheroneptlan-
guages with lower omplexity like DL -Lite [4℄ might prove more useful in an
atualimplementation.
Annaturalextensionistointrodueapartialorder(I;4),intuitivelyrepre-
sentingondenelevels,andforeahindexk2I,addingmodaloperatorsB
k ,
A
k ,
B
k ,O
k
, and
k
to thesignatureofthelogi,inordertorepresentordered
defaulttheories.Anotherextensionwouldbeallowingsubjetivelyquantiedex-
pressions,i.e.oneptsof theform 8XR :YC and9XR :YC forX;Y2fB;Ag,
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