Prioritizing Default Logic: Abridged Report

17 Pages·2011·0.11 MB·English

Prioritizing Default Logic: Abridged Report
GerhardBrewka andThomasEiter
(cid:0) (cid:2)
Institutfu¨rInformatik,AbteilungIntelligenteSysteme,Universita¨tLeipzig
(cid:0)
Augustusplatz9/10,D-04109Leipzig,Germany
Email:[email protected]
Institutfu¨rInformatik,Justus-Liebig-Universita¨tGießen
(cid:2)
Arndtstraße2,D-35392Gießen,Germany
Email:[email protected]
Abstract. Anumber ofprioritizedvariants ofReiter’sdefault logichave been
describedintheliterature.Inthispaper,weintroducetwonaturalprinciplesfor
preference handling and show that all existing approaches fail to satisfy them.
Wedevelopanewapproachwhichdoesnotsufferfromtheseshortcomings.We
startwiththesimplestcase,supernormaldefaulttheories,wherepreferencesare
handledinastraightforwardmanner.Thegeneralizationtoprerequisite-freede-
faulttheoriesisbasedonanadditionalfixedpointconditionforextensions.The
full generalization to arbitrary default theories uses a reduction of default the-
ories to prerequisite-free theories. The reduction can be viewed as dual to the
Gelfond/Lifschitzreductionusedinlogicprogrammingforthedefinitionofan-
swersets.Wefinallyshowhowpreferenceinformationcanberepresentedinthe
logicallanguage.
1 Introduction
Innonmonotonicreasoningconflictsamongdefaultsareubiquitous.Forinstance,more
specific rules may be in conflict with more general ones, a problem which has been
studied intensively in the context of inheritancenetworks [21,26,27]. When defaults
are used for representingdesign goals in configurationtasks conflicts naturallyarise.
The same is true in model based diagnosis where defaults are used to represent the
assumptionthatcomponentstypicallyareok.Inlegalreasoningconflictsamongrules
areverycommon[22]andkeepmanylawyersbusy(andrich).
Thestandardnonmontonicformalismshandlesuchconflictsbygeneratingmultiple
beliefsets. Indefaultlogic [23]andautoepistemiclogic [18]thesesets arecalledex-
tensionsorexpansions,respectively.Incircumscription[17]thebeliefsetscorrespond
todifferentclassesofpreferredmodels.
Usually,notallofthebeliefsetsareplausible.Weoftentendtoprefersomeofthe
conflicting rules and are interested in the belief sets generated by the preferred rules
only.Onewaytoachievethisis torerepresentthedefaultsinsuchawaythattheun-
wantedbeliefsetsarenotgenerated,forinstancebyaddingnewconsistencyconditions
toadefault.Thisapproachhastheadvantagethatthelogicalmachineryoftheunderly-
ingnonmonotoniclogicdoesnothavetobechanged.Ontheotherhand,rerepresenting
thedefaultsthatwayisaveryclumsybusiness.Theresultingnewdefaultstendtobe
rathercomplex.Moreover,theadditionofnewinformationtotheknowledgebasemay
leadtofurtherrerepresentations.Inotherwords,elaborationtoleranceisviolated.
For this reason we prefer an approach where preferences among defaults can be
representedviaanexplicit preferencerelationandwherethelogicalmachineryis ex-
tended accordingly. Indeed, for all major nonmonotonicformalisms, such prioritized
versionshavebeenproposedin thepast. Amongthemare prioritizedcircumscription
[12],hierarchicautoepistemiclogic[15],prioritizedtheoryrevision[2,19],prioritized
logicprogramming[25,28],orprioritizedabduction[10].
AlsoseveralprioritizedversionsofReiter’sdefaultlogic,thelogicwearedealing
withinthispaper,havebeendescribedintheliterature[16,4,1,9],aswellasofdefea-
siblelogicsbeyonddefaultlogic[20,13].However,aswewillshowinSection3,these
approaches are not fully satisfactory. It turns out that some of them implicitly recast
Reiter’s default logic to a logic of graded beliefs, while others do overly enforcethe
applicationofruleswithhighpriority,whichleadstocounterintuitivebehavior.
Our approach takes a different perspective, which is dominated by the following
twomainideas.Thefirstisthattheapplicationofadefaultrulemeanstojumptoacon-
clusion,andthisconclusionisyetanotherassumptionwhichhastobeusedgloballyin
the programforthe issue of decidingwhethera ruleis applicableor not.Thesecond
is thattherulesmustbe appliedin anordercompatiblewiththepriorityinformation.
Wetakethistomeanthata ruleisappliedunlessitis defeatedviaitsassumptionsby
rulesofhigherpriorities.Thisviewisnewandavoids theunpleasantbehaviorwhich
is presentwiththeotherapproaches.Ourformalizationoftheseideas involvesadual
ofthestandardGelfond-Lifschitzreductionandacertainoperatorusedtochecksatis-
factionofpriorities.Inordertobaseourapproachonfirmergroundwesetforthsome
abstract principles that, as we believe, any formalization of prioritized default logic
shouldsatisfy.Wedemonstratethatourapproachsatisfiestheseprinciples,whileother
approachesviolatethem.
The remainder of this paper is organizedas follows. The next section recalls the
basicdefinitionsofdefaultlogicandintroducesthenotionofaprioritizeddefaultthe-
ory. Section 3 introduces two basic principles for preference handling,reviews some
approaches to prioritized default logic and demonstrates that they fail to satisfy the
principles.InSection 4, we thenpresent ourapproach,byintroducingthe conceptof
preferredextensions.Wewillintroducethisnotioninastepwisemanner,startingwith
the simplest case, namelyprerequisite-freenormaldefaults, which are also calledsu-
pernormaldefaults.Wethenextendourdefinitiontoprerequisite-freedefaulttheories,
showingthatanadditionalfixedpointconditionisneededtogetthedefinitionofpre-
ferredextensionsright.Finally,wehandlearbitrarydefaulttheoriesbyreducingthem
tothe prerequisite-freecase.Thereductioncanbe viewedas dualtothefamousGel-
fond/Lifschitzreductionforextendedlogicprograms.InSection5weshowhowpref-
erenceinformationcanbeexpressedinthelogicallanguage.Thismakesitpossibleto
reasonnotonlywith, butalso aboutpreferencesamongrules.Section6 discusses re-
latedwork,andconcludesthe paperbyconsideringpossibleextensionsandoutlining
furtherwork.
2
Theworkreportedheregeneralizestheapproachpresentedin[7],whichcoversthe
fragmentofdefaultlogicequivalenttoextendedlogicprograms,andextendstheresults.
2 Prioritized DefaultTheories
We firstrecallthebasicdefinitionsunderlyingReiter’s defaultlogic.A defaulttheory
is apair ofatheory containingfirst-ordersentencesandasetofde-
faults .(cid:0)Ea(cid:0)ch(cid:2)dDe(cid:2)faWul(cid:3)t is ofthe forWm , , where , , and are
first-orDderformulas.Theintuitivemeanain(cid:4)gbo(cid:0)f(cid:2)t(cid:3)h(cid:3)e(cid:3)(cid:2)dbenfa(cid:4)ucltnis:(cid:0)if (cid:5)isderiveadabnidthe c are
separatelyconsistentwithwhatisderived,theninfer .Formuala iscalledtheprbeireq-
uisite,each ajustification,and theconsequentoftchedefault.Faoradefault weuse
, bi , and tocdenote the prerequisite,the set of justificatidons, and
pthree(cid:2)cdo(cid:3)nsejquuset(cid:2)ndt(cid:3)of ,recsopnesc(cid:2)tidv(cid:3)ely; denotes .Asusual,we
assumethat andd areinskole(cid:2)mjizuesdt(cid:2)fdo(cid:3)rmandthaft(cid:2)oapejnad(cid:3)efjauulstts(cid:2),di.(cid:3)eg.,defaultswith
freevariableWs,repreDsentthe sets oftheirgroundinstancesovertheHerbranduniverse
[23];adefaulttheorywithopendefaultsisclosedbyreplacingopendefaultswiththeir
groundinstances.Inwhatfollows,weimplicitlyassumethatdefaulttheoriesareclosed
beforeextensionsetcareconsidered.
A (closed) default theory generates extensions which represent acceptable sets of
beliefswhichareasonermightadoptbasedonthegivendefaulttheory .Exten-
sionsaredefinedin[23]asfixedpointsofanoperator . mapsan(cid:2)Dar(cid:2)bWitra(cid:3)rysetof
formulas tothesmallestdeductivelyclosedset th(cid:5)a(cid:2)tc(cid:5)on(cid:2)tains andsatisfiesthe
(cid:0)
condition:Sif , and S then W . Intuitively,an
(cid:0) (cid:0)
extensionisaase(cid:4)tbo(cid:0)f(cid:2)b(cid:3)e(cid:3)l(cid:3)i(cid:2)ebfsn(cid:4)ccon(cid:3)taiDninga (cid:3) Ssuchtha(cid:2)tbi (cid:4)(cid:3) S c (cid:3) S
W
1. asmanydefaultsasconsistentlypossiblehavebeenapplied,and
2. onlyformulaspossessinganoncircularderivationfrom usingdefaultsin are
contained. W D
Adefaulttheory mayhavezero,oneormultipleextensions.Defaulttheoriespos-
sessingatleastonee(cid:0)xtensionwillbecalledcoherent.Wesayadefault
isdefeatedbyasetofformulas ,iff forsome a. (cid:4)b(cid:0)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)bn(cid:4)c
Adefault iscalSledno(cid:2)rmbia(cid:3)l,iSf islogicaill(cid:3)yfeq(cid:5)u(cid:2)(cid:3)iv(cid:3)a(cid:3)l(cid:2)enngtto ;itiscalled
prerequisite-frdee(cid:0), ifa (cid:4) bis(cid:4)ca logical truth, whichbis denoted by . Defaulcts which are
both prerequisite-freae and normal are called supernormal. A (cid:5)default theory is called
normal(prerequisite-free,supernormal),ifallofitsdefaultsarenormal(prerequisite-
free,supernormal),respectively.
Adefault iscalledgeneratinginasetofformulas ,if and
; denodte by the set of all defaultsSfrompre(cid:2)dw(cid:3)(cid:3)hicSh are(cid:2)gjeunsetra(cid:2)dti(cid:3)n(cid:6)g
Sin (cid:0). I(cid:7)t is well-knowGnD[(cid:2)2E3](cid:2)Dth(cid:3)at every extension of a defaultDtheory is
chaEracterizedthrough ,i.e., (cid:0) (cid:0) (cid:2)D(cid:2)W(cid:3)
GD(cid:2)D(cid:2)E(cid:3)
(1)
E (cid:0)Th(cid:2)W (cid:8)cons(cid:2)GD(cid:2)D(cid:2)E(cid:3)(cid:3)(cid:3)
where foranyset .Moreover,if isprerequisite-
(cid:0) (cid:0) (cid:0)
free,thceonnesv(cid:2)eDry(cid:3)(cid:0)wfhciocnhss(cid:2)adti(cid:3)sfijeds(cid:3)(1D)isganextensionD(cf.[16]). (cid:0)
E
3
We now introducethe notion of a prioritizeddefault theory. Basically, we extend
defaulttheorieswitha strict partialorder (i.e., and , implies
(cid:0) (cid:0) (cid:0)(cid:0)
)onthedefaultrules.Adefault w(cid:6)ill becdon(cid:4)(cid:6)sidderedpdre(cid:6)ferdreddov(cid:6)erddefault ,
(cid:0)(cid:0) (cid:0)
dwh(cid:6)endever holds. d d
(cid:0)
d(cid:6)d
Definition1. Aprioritizeddefaulttheoryisatriple where is
adefaulttheoryand isastrictpartialorderon .(cid:0) (cid:0) (cid:2)D(cid:2)W(cid:2)(cid:6)(cid:3) (cid:2)D(cid:2)W(cid:3)
(cid:6) D
Partially ordered default theories have the advantage that the preference ordering
amongcertaindefaultscanbeleftunspecified.Thisisimportantbecauseinmanycases
thereisnonaturalwayofassigningpreferences.However,thecaseofarbitrarypartial
orderscanbereducedtoparticularrefinements,namelywell-orderings,inacanonical
way.Recallthatapartialorderisawell-ordering,iffeverysubsetoftheelementshas
theleastelement;observethatanywell-orderingisatotalordering.
Definition2. A fully prioritized default theory is a prioritized default theory
where isawell-ordering. (cid:0) (cid:0)
(cid:2)D(cid:2)W(cid:2)(cid:6)(cid:3) (cid:6)
Conclusionsofprioritizeddefaulttheoriesaredefinedintermsofpreferredexten-
sions,whichareasubsetoftheclassicalextensionsof ,i.e.,theextensionsof
accordingto[23].Thedefinitionofpreferredextension(cid:0)forfullyprioritizeddefa(cid:2)uDlt(cid:2)tWhe-(cid:3)
orieswillbegiveninthenextsection.Thegeneralcaseofarbitraryprioritizeddefault
theoriescanthenbereducedtothiscaseasfollows.
Definition3. Let beaclosedprioritizeddefaulttheory. isapriori-
tizedextensionof(cid:0)i(cid:0)ff (cid:2)Di(cid:2)sWa(cid:2)p(cid:6)ri(cid:3)oritizedextensionofafullyprioritizeddEefaulttheory
s(cid:0)uchtEhat implies .
(cid:0) (cid:0) (cid:0) (cid:0) (cid:0)
(cid:0) (cid:0)(cid:2)D(cid:2)W(cid:2)(cid:6)(cid:3) d(cid:6)d d(cid:6) d
Thepreferredextensionsofanopenprioritizeddefaulttheory arethepreferred
extensions of obtained by closing . The partial order is i(cid:0)nherited from to
(cid:2)
the ground set(cid:0)of instances in the(cid:0)obvious way. We assu(cid:6)me here that no conDflict
(cid:2)
arises,i.e., doesnotrDesultforany ;otherwise,nopreferredextensions
(cid:2)
aredefined.d (cid:6) d d (cid:3) D
In the remainder of this paper, we will restrict our discussion to fully prioritized
defaulttheories.Unlessstatedotherwise,all defaulttheoriesaretacitlyassumedtobe
closed.
3 Problems with ExistingApproaches
Differentprioritizedversionsofdefaultlogichavebeenproposedintheliterature,e.g.
[16,4,1,24,9].Wewillshowthatallofthemfailtosatisfynaturalprinciplesforpref-
erencehandlingindefaultlogic.
3.1 Principlesforpriorities
Thefirstprinciplecanbeviewedasameaningpostulatefortheterm“preference”and
states what we consider a minimal requirementfor preferencehandlingin rule based
systems:
4
PrincipleI. Let and betwoextensionsofaprioritizeddefaulttheory gen-
eratedbythedeBfa(cid:0)ults B(cid:2) and ,where ,respecti(cid:0)vely.If
ispreferredover R,th(cid:8)enfd(cid:0)gisnoRta(cid:8)prfedfe(cid:2)rgredextendsi(cid:0)o(cid:2)nd(cid:2)of(cid:3)(cid:4) R.
d(cid:0) d(cid:2) B(cid:2) T
We find it hard to see how the use of the term “preference among rules” could be
justifiedincaseswherePrincipleIisviolated.
The second principle is related to relevance. It tries to capture the idea that the
decisionwhethertobelieveaformula ornotshoulddependontheprioritiesofdefaults
contributingtothederivationof onlpy,notontheprioritiesofdefaultswhichbecome
applicablewhen isbelieved: p
p
PrincipleII. Let be a preferred extension of a prioritized default theory
, a(cElosed)defaultsuchthattheprerequisiteof isnotin .Th(cid:0)en(cid:0)
(cid:2)isDa(cid:2)pWre(cid:2)f(cid:6)er(cid:3)reddextensionof wheneverd agreesEwith oEn
(cid:0) (cid:0) (cid:0)
prioritiesamongdefaultsin(cid:0) .(cid:0)(cid:2)D(cid:8)fdg(cid:2)W(cid:2)(cid:6)(cid:3) (cid:6) (cid:6)
D
Thus,addingarulewhichisnotapplicableinapreferredbeliefsetcanneverrenderthis
beliefsetnon-preferredunlessnewpreferenceinformationchangespreferencesamong
someoftheoldrules(e.g.viatransitivity).Inotherwords,abeliefsetisnotblamedfor
notapplyingruleswhicharenotapplicable.
We will see that each of the existing treatments of preferences for default logic,
describedin[16,4,1,24],violatesoneoftheseprinciples.
3.2 ControlofReiter’squasi-inductivedefinition
The first group of proposals [16,4,1] uses preferences to control the quasi-inductive
definitionofextensions[23]:ineachstep ofthegenerationofextensionsthedefaults
withhighest prioritywhoseprerequisiteshavealreadybeenderivedareapplied.Now
whatis wrongwiththis idea?The answeris: thepreferredextensionsdonottakese-
riously what they believe. It may be the case that a less preferred default is applied
althoughthe prerequisiteof a conflicting,morepreferreddefaultis believedin a pre-
ferredextension.Aswewillsee,thiscanleadtosituationswherePrincipleIisviolated.
Thementionedapproachesdifferintechnicaldetail.Wedonotwanttopresentthe
exact definitions here. Instead, we will illustrate the difficulties using an examplefor
whichalltheseapproachesobtainthesameresult.
Example1. Assumewearegiventhefollowingdefaulttheory:
(1)
(2) a(cid:4)b(cid:4)b
(3) (cid:5)(cid:4)(cid:2)b(cid:4)(cid:2)b
(cid:5)(cid:4)a(cid:4)a
Assume furtherthat (1) is preferredover(2) and(2)over(3).This default theory
hastwoclassicalextensions,namely ,whichisgeneratedbyrules(1)
and (3), and , whEic(cid:0)h(cid:0)isTgehn(cid:2)efraa(cid:2)tbegd(cid:3)by rules (2) and (3). The single
preferred exteEn(cid:2)sio(cid:0)nTinht(cid:2)hfeaa(cid:2)p(cid:2)pbrgo(cid:3)aches mentioned above is . The reason is that the
prerequisite of (2) is derived before the prerequisite of (1)Ein(cid:2)the construction of the
extension.TheapproachesthusviolatePrincipleI.
5
The selection of in the previous example was already observedin [4]. In that
paper,thefirstauthorEtr(cid:2)iedtodefendhisapproacharguingthatthereisonlyweakevi-
dencefortheliteral inourexample.Wereviseourview,however,anddonotsupport
thisargumentanyloanger.Afterall,defaultlogicis notalogicofgradedbeliefwhere
degreesofevidenceshouldplayarole.Defaultlogicmodelsacceptanceofbeliefbased
on defeasiblearguments.Since is an acceptedbelief, we believe rule (1) should be
appliedand shouldbethepreaferredextensionintheexample.
E(cid:0)
3.3 Rintanen’sapproach
An entirely different approach was proposed in [24]. Rintanen uses a total order on
(normal)defaultstoinducealexicographicorderonextensions.
Call anormaldefaultrule : appliedin asetofformulas (denoted
),if and arein .rA(cid:0)neaxtbe(cid:4)nbsion is thenpreferredoverEextension r,(cid:3)if
(cid:0)
aanpdplo(cid:2)nEly(cid:3)ifthaereisabdefaultE E satisfyingthefollowingcondEition:
(cid:0)
if ispreferredover and r (cid:3)appl(cid:2)E(cid:3)n,athpepnl(cid:2)E (cid:3) .
(cid:0) (cid:0) (cid:0) (cid:0)
rUnfortunately,alsorthisarpp(cid:3)roaapcphl(cid:2)leEad(cid:3)stocournt(cid:3)erainptpuli(cid:2)tiEve(cid:3)resultsandtoaviolation
ofourprinciples.
Example2. Considerthefollowingdefaulttheory:
(1)
(2) a(cid:4)b(cid:4)b
(3) (cid:5)(cid:4)(cid:2)a(cid:4)(cid:2)a
(cid:5)(cid:4)a(cid:4)a
Again(1)ispreferredover(2),and(2)over(3).Thedefaulttheoryhastwoclassical
extensions,namely and .Intuitively,sincethedeci-
sionwhethertobeliEev(cid:0)e(cid:0)oTrhn(cid:2)oft(cid:2)daegp(cid:3)endsoEn(cid:2)(2(cid:0))aTnhd(cid:2)(f3a)(cid:2)obngl(cid:3)y,andsince(2)ispreferred
over(3),wewouldexpeacttoconclude ,inotherwords,toprefer .
However,theapproachofRintanen(cid:2)praefers .ThereasonisthatEin(cid:0) default(1)is
applied.Beliefin isthusacceptedonthegrouEn(cid:2)dsthatthisallowsustoEa(cid:2)pplyadefault
ofhighpriority.Tahisisfarfrombeingplausibleandamountstowishfulthinking.Itis
alsoeasytoseethatPrincipleIIisviolated: clearlyisthesinglepreferredextension
ofrules(2)and(3)inRintanen’sapproach.EA(cid:0)ddingrule(1)whichisnotapplicablein
makes anon-preferredextension.
E(cid:0) E(cid:0)
Sincealltheseapproachessufferfromdrawbacks,wedevelopournewapproachin
thefollowingsection.
4 Preferred Extensions
In this section we introduce our new notion of preferred extensions for fully priori-
tizeddefaulttheories.Asmentionedbefore,arbitraryprioritizeddefaulttheoriescanbe
reducedtothatcaseinacanonicalmanner.
We will considerthe simplest case first, namelysupernormaldefaulttheories.We
thenproceedtoprerequisite-freedefaulttheoriesand,finally,toarbitrarydefaulttheo-
ries.
6
4.1 Supernormaldefaulttheories
Preferencehandlinginprioritizedsupernormaldefaulttheoriesisrathereasy.Theob-
viousideais tocheckthe applicabilityofdefaultsin theorderofpreference.We first
introduceanoperator which,givenafullyprioritizedprerequisite-freedefaulttheory
(whichisnotnecesCsarilysupernormal),producestentativeconclusionsof .
(cid:0) Calladefault activeinasetofformulas ,if , (cid:0) and
allhdold.Intuitively,adefaultisSactivperien(cid:2)d(cid:3)i(cid:3)fiSt is(cid:2)ajpupslitc(cid:2)adb(cid:3)l(cid:6)ewSr(cid:0)t. (cid:7)and
choanssn(cid:2)odt(cid:3)ye(cid:3)(cid:4)tbSeenapplied. S S
Definition4. Let beafullyprioritizedprerequisite-freedefaulttheory.
Theoperator is(cid:0)de(cid:0)fin(cid:2)eDd(cid:2)aWs(cid:2)fo(cid:6)ll(cid:3)ows: , where ,and
foreveryordinCal , C(cid:2)(cid:0)(cid:3) (cid:0) (cid:3)(cid:3)(cid:3)E(cid:3) E(cid:3) (cid:0) Th(cid:2)W(cid:3)
(cid:7)(cid:8)(cid:6) S
ifnodefaultfrom isactivein
E(cid:3)(cid:2) otherwise,where D E(cid:3)(cid:7)
E(cid:3) (cid:0)(cid:2)Th(cid:2)E(cid:3)(cid:8)fcons(cid:2)d(cid:3)g(cid:3) isactivein
(cid:3) (cid:0) (cid:0)
d(cid:0)min(cid:4)fd (cid:3)D jd E(cid:3)g(cid:2)
where .(Notethatforeachsuccessorordinal , .)
(cid:4)
E(cid:3) (cid:0) (cid:5)(cid:4)(cid:3)E(cid:5) (cid:7)(cid:0)(cid:9)(cid:8)(cid:5) E(cid:3) (cid:0)E(cid:5)
In the case of supernormal default theories, the operator always produces an
extensionintShesenseofReiterandthuscandirectlybeusedtoCdefinepreferredexten-
sions:
Definition5. Let beafullyprioritizedsupernormaldefaulttheory.
isthepreferredext(cid:0)en(cid:0)sio(cid:2)nDo(cid:2)fW(cid:2)i(cid:6)fa(cid:3)ndonlyif . E
(cid:0) E (cid:0)C(cid:2)(cid:0)(cid:3)
Itisobviousthatthereisalwaysexactlyonepreferredextension.Notethatthedefinition
ofthisextensionisfullyconstructive.Itextendsthenotionofpreferredsubtheoriesas
developedin[3]totheinfinitecase.
4.2 Prerequisite-freedefaulttheories
Canwesimplyextendthedefinitionforsupernormaldefaultstothiscase?Theanswer
is obviously no. It may be the case that defaults are applied during the construction
whicharedefeatedlaterthroughtheapplicationofdefaultsoflowerpriority.
So,canwesimplysay:iftheconstructiongivesusanextension,thenthatextension
ispreferred?Unfortunately,theanswerisagainno.
Example3. Considerthefollowingdefaulttheory:
(1) (3)
(2)(cid:5)(cid:4)(cid:2)b(cid:4)a (4) (cid:5)(cid:4)a(cid:4)a
(cid:5)(cid:4)(cid:2)a(cid:4)(cid:2)a (cid:5)(cid:4)b(cid:4)b
Assume .Applyingoperator tothisdefaulttheoryyields
(cid:2)(cid:5)(cid:3) (cid:6).A(cid:2)s(cid:9)(cid:3)ea(cid:6)sil(cid:2)y(cid:10)s(cid:3)e(cid:6)en,(cid:2)(cid:11)th(cid:3)isisaclassicalextenCsion.Nonetheless,onewould
Ecer(cid:0)tainTlhy(cid:2)nfoat(cid:2)bsgay(cid:3) that this extension preserves priorities. What went wrong? Default
(2) is defeated in by applying a default which is less preferred than (2), namely
default(3).InthecEonstructionof thisremainsunnoticed,sincerule(1),although
defeatedin ,blockstheapplicabCili(cid:2)t(cid:0)yo(cid:3)f(2).Inotherwords,withoutaspecialtreatment
ofsuchcaseEs,arule(e.g.(3))mayinheritahighpreferencefromarulewiththesame
consequent(namely(1)),evenifthatlatterruleisnotapplicableintheextension.
7
To avoid this, we have to impose an additional condition on an extension: in the
construction of the tentative conclusions, we have to discard each rule whose conse-
quentisin ,butwhichisdefeatedin .Sincewehavetotake astheresultofthe
constructionEintoaccount,thisamountsEtoaddingafixedpointconEdition.Whatwewill
doischeckwhetherwearriveatthesamesetofformulasaftereliminatingruleswhich
aredefeatedin andwhoseheadisin .
E E
Definition6. Let beafullyprioritizedprerequisite-freedefaulttheory.
Then,aset offo(cid:0)rm(cid:0)ula(cid:2)Dsi(cid:2)sWa(cid:2)p(cid:6)ri(cid:3)oritizedextensionof ,ifandonlyif
E (cid:0)
,
(cid:6)E
E (cid:0)C(cid:2)(cid:0) (cid:3)
where isobtainedfrom bydeletingalldefaultswhoseconsequentsarein and
(cid:6)E
whicha(cid:0)redefeatedin . (cid:0) E
E
This definition is coherent with the intuition that preferred extensions are distin-
guishedclassicalextensions.Moreover,asinthecaseofsupernormaltheories,thepre-
ferredextension(ifitexists)isunique.
Proposition1. Let beafullyprioritizedprerequisite-freedefaultthe-
ory.Then,everypref(cid:0)err(cid:0)ed(cid:2)Dex(cid:2)teWns(cid:2)i(cid:6)on(cid:3) of isaclassicalextension,and hasatmost
onepreferredextension. E (cid:0) (cid:0)
Proof. Toshowthefirstpart,assumethat isapreferredextensionof .Weshow
that (*)holEds;sincealldefaultsareprere(cid:0)quisite-free,
thisiEmp(cid:0)lieTsht(cid:2)hWat (cid:8)cisonasc(cid:2)lGasDsic(cid:2)Dal(cid:2)eExt(cid:3)e(cid:3)n(cid:3)sionof (cf.paragraphafterEquation(1)).
Since E ,nodefaultfrom (cid:0) isappliedintheconstruction
(cid:6)E
of ;hencEe,(cid:0)C(cid:2)(cid:0) (cid:3) DnGDfo(cid:2)lDlow(cid:2)Es.(cid:3)Ontheotherhand,sinceeach
Eis includeEd(cid:9)inTh(cid:2)aWnd(cid:8)cdooness(cid:2)GnoDt(cid:2)dDef(cid:2)eEat(cid:3)(cid:3)(cid:3), for every , it follows
E(cid:3) E; hencEe, d d (cid:3) GfDoll(cid:2)oDw(cid:2)sE.T(cid:3)his implies
(cid:6)E
c(*o)n,sa(cid:2)ndd(cid:3)p(cid:3)rovCe(cid:2)s(cid:0)that(cid:3) isaclaTsshic(cid:2)aWlex(cid:8)tecnosniosn(cid:2)GofD(cid:2)D. (cid:2)E(cid:3)(cid:3)(cid:3) (cid:9) E
ForthesecondpEart,assumethat differentpr(cid:0)eferredextensions exist.We
(cid:0)
deriveacontradiction.Let betheleastdefaultin suchthateitherE (cid:4)(cid:0) E
and ,or d and D .Sinc(cid:2)ei(cid:3)d(cid:3)GD,(cid:2)Dm(cid:2)Eus(cid:3)t
(cid:0) (cid:0) (cid:0)
existc.oCnosn(cid:2)dsi(cid:3)de(cid:3)(cid:4)r Efirst the(cid:2)ici(cid:3)asde(cid:3) G.DIt(cid:2)fDoll(cid:2)oEws(cid:3)thatcons(cid:2)d(cid:3) (cid:3)(cid:4)(cid:0);Efor,otherEwis(cid:4)(cid:0)e E d
(cid:6)E
holds, which is a contrad(cid:2)ici(cid:3)tion. The defaultd (cid:3)muDst be defeated by c;ofnrso(cid:2)md(cid:3)th(cid:3)e
(cid:0) (cid:0)
Edefinition of , it follows that is defeatded by E for
(cid:6)E
C(cid:2)(cid:0) (cid:3) .Fromthedminimalityof ,itTfho(cid:2)llWow(cid:8)sthcoatnfso(cid:2)rKev(cid:3)e(cid:3)ry K (cid:0)
(cid:0) (cid:0) (cid:0) (cid:0)
fitdho(cid:3)ldGstDha(cid:2)tD(cid:2)E (cid:3)jd (cid:6)d.gHence, d ,whichmeans dedfea(cid:3)tsK.
(cid:0)
Thisisaconctorandsi(cid:2)cdtio(cid:3)n(cid:3)toE Th(cid:2)W,h(cid:8)ocwoenvse(cid:2)rK.T(cid:3)h(cid:3)e(cid:9)caEse ,i.e., E d
(cid:0)
and isanalodgo(cid:3)usG.DTh(cid:2)iDsp(cid:2)Ero(cid:3)vestheresult. (cid:2)ii(cid:3) d(cid:3)GD(cid:2)D(cid:2)E (cid:3)
cons(cid:2)d(cid:3)(cid:3)(cid:4) E
4.3 Generaldefaulttheories
We will now reduce the general case to the prerequisite-free case. The basic idea is
the following:in orderto checkwhetheran extension of a fully prioritizeddefault
theory ispreferred,weevaluatetheprerequisitesofthEedefaultrulesaccordingtothe
(cid:0)
8
extension . Evaluating prerequisites means (1) eliminating prerequisites which are
containedEin the extension from the correspondingrules, and (2) eliminating rules
whoseprerequisitesarenotEcontainedin .Observethatthisoperationcanbeviewed
asadualofthestandardGelfond/LifschitEzreductionfromlogicprogramming[11],in
whichthejustificationsratherthantheprerequisitesareusedtoeliminateandsimplify
rules.
Finally, we check whether the resulting prerequisite-free theory has as its
preferredextension. (cid:0)E E
Definition7. Let beafullyprioritizeddefaulttheory and aset of
formulas.Thedefa(cid:0)ult(cid:0)the(cid:2)oDr(cid:2)yW(cid:2)(cid:6)(cid:3) isobtainedfrom asfolloEws:
resultsfrom by (cid:0)E (cid:0)(cid:2)DE(cid:2)W(cid:2)(cid:6)E(cid:3) (cid:0)
DE D
1. eliminatingeverydefault suchthat ,and
2. replacing by indal(cid:3)lreDmainingdefparuel(cid:2)tsd;(cid:3)(cid:3)(cid:4) E
pre(cid:2)d(cid:3) (cid:5)
isinheritedfrom asfollows:foranyrules and in , holdsifand
(cid:0) (cid:0)
(cid:6)onElyif holds(cid:6)forthe -leastrules andd idn wDhEichdg(cid:6)ivEerdiseto and
(cid:0) (cid:0) (cid:0)
(i.e., d(cid:5)(cid:6)d(cid:5)and ),(cid:6)respectively.d(cid:5) d(cid:5) D d d
(cid:0) (cid:0)
d(cid:5)E (cid:0)d d(cid:5)E (cid:0)d
The resulting default theory is clearly prerequisite-free.We thus can define preferred
extensionsforgeneraldefaulttheoriesasfollows:
Definition8. Let beafullyprioritizeddefaulttheory.Then, isapri-
oritizedextension(cid:0)of(cid:0),(cid:2)iDf (cid:2)W(cid:2)(cid:6)is(cid:3)aclassicalextensionof ,and isapErioritized
extensionof . (cid:0) (cid:2)i(cid:3)E (cid:0) (cid:2)ii(cid:3)E
(cid:0)E
LetusshowthattheproblematicexamplesdiscussedinSection3arehandledcorrectly
inourapproach:
Example4. Considerthedefaulttheory :
(cid:0)
(1)
(2) a(cid:4)b(cid:4)b
(3) (cid:5)(cid:4)(cid:2)b(cid:4)(cid:2)b
(cid:5)(cid:4)a(cid:4)a
Againweassumethat(1)ispreferredover(2)and(2)over(3).Thisdefaulttheory
hastwoclassical extensions,namely and .
consistsoftherules E(cid:0) (cid:0) Th(cid:2)fa(cid:2)bg(cid:3) E(cid:2) (cid:0) Th(cid:2)fa(cid:2)(cid:2)bg(cid:3) (cid:0)E(cid:0)
(1)
(2) (cid:5)(cid:4)b(cid:4)b
(3) (cid:5)(cid:4)(cid:2)b(cid:4)(cid:2)b
(cid:5)(cid:4)a(cid:4)a
Itisnotdifficulttoseethat .Sincetherearenoruleswhoseheadis
in butwhicharedefeatedinC(cid:2)(cid:0), E(cid:0)(cid:3)is(cid:0)aEpr(cid:0)eferredextension.Onthecontrary, is
notEp(cid:0)referred.NotethatthedualE(cid:0)-Ere(cid:0)ductof isthesameasthedual -reductE,a(cid:2)nd
that appliedtothereductdoEes(cid:2)notreprod(cid:0)uce . E(cid:0)
C(cid:2)(cid:10)(cid:3) E(cid:2)
9
Example5. Considerthefollowingdefaulttheory :
(cid:0)
(cid:0)
(1)
(2) a(cid:4)b(cid:4)b
(3) (cid:5)(cid:4)(cid:2)a(cid:4)(cid:2)a
(cid:5)(cid:4)a(cid:4)a
Again(1)ispreferredover(2),and(2)over(3).ThedefaulttheoryhastwoReiter
extensions,namely and .We arguedin Section 3
that shouldbeprEef(cid:0)er(cid:0)redT.Cho(cid:2)fn(cid:2)siadger(cid:3) :E(cid:2) (cid:0) Th(cid:2)fa(cid:2)bg(cid:3)
(cid:0)
E(cid:0) (cid:0)E(cid:0)
(2)
(3) (cid:5)(cid:4)(cid:2)a(cid:4)(cid:2)a
(cid:5)(cid:4)a(cid:4)a
Clearly, , and since again thereare no rules defeatedin whose
(cid:0)
headisin C,(cid:2)w(cid:0)eEh(cid:0)a(cid:3)ve(cid:0)thEat(cid:0) ispreferred. E(cid:0)
isnEo(cid:0)tpreferredsinceE(cid:0) whichdiffersfrom .
(cid:0)
E(cid:2) C(cid:2)(cid:0)E(cid:2)(cid:3)(cid:0)Th(cid:2)fb(cid:2)(cid:2)ag(cid:3) E(cid:2)
Thefollowingpropositiontells usthatall ruleswhicharenotgeneratinginapre-
ferred extension must be defeated by some appropriate generating rules, which must
havehigherpriority.
Proposition2. Let beafullyprioritizedgrounddefaulttheory,andlet
beaclassicalexte(cid:0)ns(cid:0)ion(cid:2)Wof(cid:2)D.(cid:2)T(cid:6)h(cid:3)en, isapreferredextensionof ,ifandonlyiffor
Eeachdefault suchthat(cid:0) E and ,thereex(cid:0)istsasetofdefaults
d(cid:3)D pre(cid:2)d(cid:3)su(cid:3)chEthat ciosndse(cid:2)fde(cid:3)at(cid:3)(cid:4)edEin .
(cid:0) (cid:0)
Kd (cid:9)fd (cid:3)GD(cid:2)D(cid:2)E(cid:3)jd (cid:6)dg d Th(cid:2)W (cid:8)cons(cid:2)Kd(cid:3)(cid:3)
Proof. ( )Supposethatforeverydefault suchthat and
aset (cid:11) existsdsuchthat pre(cid:2)d(cid:3)(cid:3)E consd(cid:2)edfe(cid:3)a(cid:3)(cid:4)tsE.
(cid:0) (cid:0)
By(trKands(cid:9)finfitde)i(cid:3)ndGuDcti(cid:2)oDn(cid:2)oEn(cid:3)thjedset(cid:6)s dg, ,weshowTthh(cid:2)aWtth(cid:8)elceoanstsa(cid:2)cKtidv(cid:3)e(cid:3)default d
from in ,providedoneexistsE,(cid:3)ste(cid:7)m(cid:0)sfr(cid:5)omsome and dE
(cid:6)E
hoDldEs. E(cid:3) d(cid:3)GD(cid:2)D(cid:2)E(cid:3) cons(cid:2)d(cid:3)(cid:3)
E(cid:3)For , the statement holds. Indeed. the least rule of is active. Let
(cid:6)E
be the(cid:7)le(cid:0)ast(cid:5)parent of in , i.e., dE DE. Assuming that
(cid:0) (cid:0)
d , wedoEbtainD d,(cid:0)andmhienn(cid:4)cfed ijsddEefe(cid:0)ateddEbgy .
dTh(cid:3)isDconntrGadDic(cid:2)tDst(cid:2)hEat(cid:3) isactiveKind (cid:0) (cid:7) ,howevder.Thus, E(cid:3) (cid:0) Thh(cid:2)oWlds(cid:3),
and fodllEows. E(cid:0) (cid:0) E(cid:3) d (cid:3) GD(cid:2)D(cid:2)E(cid:3)
Lceotntsh(cid:2)edn(cid:3)(cid:3)E(cid:3) andassume the statement holds forall . Supposethe
least default(cid:7) (cid:8)fr(cid:5)om active in exists, and that its(cid:5)l(cid:12)eas(cid:9)t p(cid:6)are(cid:7)nt in is not
(cid:6)E
in d.EThe indDucEtion hypotheEsis(cid:3)implies that for each D such
(cid:0)
thaGt D(cid:2)D(cid:2)Ei(cid:3)t holdsthat . Hence, d (cid:3) GD(cid:2)D(cid:2)E,(cid:3)which
(cid:0) (cid:0)
impldies(cid:6)thadt isdefeatedcboyns(cid:2)d.(cid:3)Th(cid:3)isEc(cid:3)ontradictstThhat(cid:2)W (cid:8)iscaocntisv(cid:2)eK.Tdh(cid:3)(cid:3)us(cid:9),ifE(cid:3) exists,
then d holds;clEea(cid:3)rly, .ThidsEconcludestheinductdiEon,from
whichd(cid:3)GD(cid:2)D(cid:2)E(cid:3) cons(cid:2)d(cid:3)(cid:3)E(cid:3)follows. By Equation (1), it follows
(cid:6)E
C(cid:2)(cid:0)E,(cid:3)w(cid:0)hicThhm(cid:2)eWans(cid:8)thcaotns(cid:2)GisDa(cid:2)pDre(cid:2)feErr(cid:3)e(cid:3)d(cid:3)extension.
(cid:6)E
E (cid:0)(C)(cid:2)(cid:0)SuEpp(cid:3)ose is a preferredEextension,but some such that
and (cid:13) Eis not defeated by any d (cid:3) D , where pre(cid:2)d(cid:3) (cid:3) E
(cid:0)
cons(cid:2)d(cid:3) (cid:3)(cid:4) E .Let betheleastsuchTruhle(cid:2)Win (cid:8).cSoinncse(cid:2)K(cid:3)i(cid:3)sapreferrKede(cid:9)xtefndsion(cid:3),
(cid:0)
GfoDr e(cid:2)vDe(cid:2)ryE(cid:3)jd (cid:6)dg d we have D .EBy the minimality of ,
(cid:0) (cid:0) (cid:6)E
it followsdtha(cid:3)t GDb(cid:2)eDco(cid:2)mEe(cid:3)s the leastcaocntisv(cid:2)edr(cid:3)ule(cid:3)inC(cid:2)(cid:0)E (cid:3)at some step , and up tdo
(cid:6)E
dE DE (cid:7)
10