Two New Definitions of Stable Models of Logic Programs with Generalized Quantifiers

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by  Joohyung Lee

Two New Definitions of Stable Models of Logic
Programs with Generalized Quantifiers
JoohyungLeeandYunsongMeng
SchoolofComputing,InformaticsandDecisionSystemsEngineering
ArizonaStateUniversity,Tempe,USA
3
1
0 Abstract. We present alternative definitions of the first-order stable model se-
2 manticsanditsextensiontoincorporategeneralizedquantifiersbyreferringtothe
n familiarnotionofareductinsteadofreferringtotheSMoperatorintheoriginal
a definitions.Also,weextendtheFLPstablemodelsemanticstoallowgeneralized
J quantifiers by referring to an operator that is similar to the SM operator. For a

8 reasonablesyntacticclassoflogicprograms,weshowthatthetwostablemodel
semanticsofgeneralizedquantifiersareinterchangeable.

]
O
L 1 Introduction
.
s
c
Mostversionsofthestablemodelsemanticsinvolvegrounding.Forinstance,according
[
totheFLPsemanticsfrom[1;2],assumingthatthedomainis{−1,1,2},program

1
v p(2) ← ınotSUM(cid:104)x:p(x)(cid:105)<2
3
p(−1) ← SUM(cid:104)x:p(x)(cid:105)>−1 (1)
9
p(1) ← p(−1)
3
1
. isidentifiedwithitsgroundinstancew.r.tthedomain:
1
0
p(2) ← ınotSUM(cid:104){−1:p(−1),1:p(1),2:p(2)}(cid:105)<2
3
1 p(−1) ← SUM(cid:104){−1:p(−1),1:p(1),2:p(2)}(cid:105)>−1 (2)
: p(1) ← p(−1).
v
i
X Asdescribedin[1],itisstraightforwardtoextendthedefinitionofsatisfactiontoground
r aggregateexpressions.Forinstance,set{p(−1),p(1)}doesnotsatisfythebodyofthe
a
firstruleof(2),butsatisfiesthebodiesoftheotherrules.TheFLPreductofprogram(2)
relativeto{p(−1),p(1)}consistsofthelasttworules,and{p(−1),p(1)}isitsminimal
model.Indeed,{p(−1),p(1)}istheonlyFLPanswersetofprogram(2).
On the other hand, according to the semantics from [3], program (2) is identified
withsomecomplexpropositionalformulacontainingnestedimplications:
(cid:16) (cid:0) (cid:1) (cid:17)
¬ (p(2)→p(−1)∨p(1))∧(p(1)∧p(2)→p(−1))∧(p(−1)∧p(1)∧p(2)→⊥) →p(2)
(cid:16)(cid:0) (cid:1) (cid:17)
∧ p(−1)→p(1)∨p(2) →p(−1)
(cid:16) (cid:17)
∧ p(−1)→p(1) .
132 J.LeeandY.Meng
Under the stable model semantics of propositional formulas [3], this formula has two
answersets:{p(−1),p(1)}and{p(−1),p(1),p(2)}.TherelationshipbetweentheFLP
andtheFerrarissemanticswasstudiedin[4;5].
Unlike the FLP semantics, the definition from [3] is not applicable when the do-
mainisinfinitebecauseitwouldrequiretherepresentationofanaggregateexpression
to involve “infinite” conjunctions and disjunctions. This limitation was overcome in
thesemanticspresentedin[4;6],whichextendsthefirst-orderstablemodelsemantics
from [7; 8] to incorporate aggregate expressions. Recently, it was further extended to
formulas involving generalized quantifiers [9], which provides a unifying framework
of various extensions of the stable model semantics, including programs with aggre-
gates,programswithabstractconstraintatoms[10],andprogramswithnonmonotonic
dl-atoms[11].
Inthispaper,werevisitthefirst-orderstablemodelsemanticsanditsextensionto
incorporategeneralizedquantifiers.Weprovideanalternative,equivalentdefinitionofa
stablemodelbyreferringtogroundingandreductinsteadoftheSMoperator.Ourwork
isinspiredbytheworkofTruszczynski[12],whointroducesinfiniteconjunctionsand
disjunctions to account for grounding quantified sentences. Our definition of a stable
model can be viewed as a reformulation and a further generalization of his definition
toincorporategeneralizedquantifiers.Wedefinegroundinginthesamewayasdonein
theFLPsemantics,butdefineareductdifferentlysothatthesemanticsagreeswiththe
onebyFerraris[3].AsweexplaininSection3.3,ourreductofprogram(2)relativeto
{p(−1),p(1)}is
⊥ ← ⊥
p(−1) ← SUM(cid:104){−1:p(−1),1:p(1),2:⊥)(cid:105)}>−1 (3)
p(1) ← p(−1),
which is the program obtained from (2) by replacing each maximal subformula that
is not satisfied by {p(−1),p(1)} with ⊥. Set {p(−1),p(1)} is an answer set of pro-
gram(1)asitisaminimalmodelofthereduct.Likewisethereductrelativeto{p(−1),p(1),p(2)}
is
p(2) ← (cid:62)
p(−1) ← SUM{(cid:104)−1:p(−1),1:p(1),2:p(2)(cid:105)}>−1
p(1) ← p(−1)
and {p(−1),p(1),p(2)} is a minimal model of the program. The semantics is more
directthantheonefrom[3]asitdoesnotinvolvethecomplextranslationintoapropo-
sitionalformula.
WhiletheFLPsemanticsin[1]wasdefinedinthecontextoflogicprogramswithag-
gregates,itcanbestraightforwardlyextendedtoallowother“complexatoms.”Indeed,
the FLP reduct is the basis of the semantics of HEX programs [13]. In [14], the FLP
reduct was applied to provide a semantics of nonmonotonic dl-programs [11]. In [5],
theFLPsemanticsoflogicprogramswithaggregateswasgeneralizedtothefirst-order
level.ThatsemanticsisdefinedintermsoftheFLPoperator,whichissimilartotheSM
operator.Thispaperfurtherextendsthedefinitiontoallowgeneralizedquantifiers.
By providing an alternative definition in the way that the other semantics was de-
fined, this paper provides a useful insight into the relationship between the first-order
TwoNewDefinitionsofStableModelsofLPswithGeneralizedQuantifiers 133
stablemodelsemanticsandtheFLPstablemodelsemanticsforprogramswithgener-
alized quantifiers. While the two semantics behave differently in the general case, we
show that they coincide on some reasonable syntactic class of logic programs. This
impliesthatanimplementationofoneofthesemanticscanbeviewedasanimplemen-
tationoftheothersemanticsifwelimitattentiontothatclassoflogicprograms.
The paper is organized as follows. Section 2 reviews the first-order stable model
semanticsanditsequivalentdefinitionintermsofgroundingandreduct,andSection3
extendsthatdefinitiontoincorporategeneralizedquantifiers.Section4providesanal-
ternativedefinitionoftheFLPsemanticswithgeneralizedquantifiersviaatranslation
intosecond-orderformulas.Section5comparestheFLPsemanticsandthefirst-order
stablemodelsemanticsinthegeneralcontextofprogramswithgeneralizedquantifiers.
2 First-OrderStableModelSemantics
2.1 ReviewofFirst-OrderStableModelSemantics
This review follows [8], a journal version of [7], which distinguishes between inten-
sionalandnon-intensionalpredicates.
Aformulaisdefinedthesameasinfirst-orderlogic.Asignatureconsistsoffunc-
tion constants and predicate constants. Function constants of arity 0 are also called
object constants. We assume the following set of primitive propositional connectives
andquantifiers:
⊥,(cid:62), ∧, ∨, →, ∀, ∃.
¬F isanabbreviationofF → ⊥,andF ↔ Gstandsfor(F → G)∧(G → F).We
distinguish between atoms and atomic formulas as follows: an atom of a signature σ
is an n-ary predicate constant followed by a list of n terms that can be formed from
function constants in σ and object variables; atomic formulas of σ are atoms of σ,
equalitiesbetweentermsofσ,andthe0-placeconnectives⊥and(cid:62).
ThestablemodelsofF relativetoalistofpredicatesp=(p ,...,p )aredefined
1 n
viathestablemodeloperatorwiththeintensionalpredicatesp,denotedbySM[F;p].1
Let u be a list of distinct predicate variables u ,...,u . By u = p we denote the
1 n
conjunction of the formulas ∀x(u (x) ↔ p (x)), where x is a list of distinct object
i i
variablesofthesamelengthasthearityofp ,foralli=1,...,n.Byu≤pwedenote
i
the conjunction of the formulas ∀x(u (x) → p (x)) for all i = 1,...,n, and u < p
i i
standsfor(u ≤ p)∧¬(u = p).Foranyfirst-ordersentenceF,expressionSM[F;p]
standsforthesecond-ordersentence
F ∧¬∃u((u<p)∧F∗(u)),
whereF∗(u)isdefinedrecursively:
– p (t)∗ =u (t)foranylisttofterms;
i i
– F∗ =F foranyatomicformulaF thatdoesnotcontainmembersofp;
– (F ∧G)∗ =F∗∧G∗;
1Theintensionalpredicatesparethepredicatesthatwe“intendtocharacterize”byF.
134 J.LeeandY.Meng
– (F ∨G)∗ =F∗∨G∗;
– (F →G)∗ =(F∗ →G∗)∧(F →G);
– (∀xF)∗ =∀xF∗;
– (∃xF)∗ =∃xF∗.
A model of a sentence F (in the sense of first-order logic) is called p-stable if it
satisfiesSM[F;p].
Example1 LetF besentence∀x(¬p(x)→q(x)),andletIbeaninterpretationwhose
universeisthesetofallnonnegativeintegersN,andpI(n) = FALSE,qI(n) = TRUE
foralln∈N.Section2.4of[8]tellsusthatI satisfiesSM[F;pq].
2.2 AlternativeDefinitionofFirst-OrderStableModelsviaReduct
For any signature σ and its interpretation I, by σI we mean the signature obtained
fromσ byaddingnewobjectconstantsξ(cid:5),calledobjectnames,foreveryelementξ in
theuniverseofI.WeidentifyaninterpretationI ofσ withitsextensiontoσI defined
byI(ξ(cid:5))=ξ.
In order to facilitate defining a reduct, we provide a reformulation of the standard
semanticsoffirst-orderlogicvia“agroundformulaw.r.t.aninterpretation.”
Definition1. For any interpretation I of a signature σ, a ground formula w.r.t. I is
definedrecursivelyasfollows.
– p(ξ(cid:5),...,ξ(cid:5)),wherepisapredicateconstantofσandξ(cid:5) areobjectnamesofσI,
1 n i
isagroundformulaw.r.t.I;
– (cid:62)and⊥aregroundformulasw.r.t.I;
– IfF andGaregroundformulasw.r.t.I,thenF ∧G,F ∨G,F → Gareground
formulasw.r.t.I;
– IfS isasetofpairsoftheformξ(cid:5):F whereξ(cid:5) isanobjectnameinσI andF isa
groundformulaw.r.t.I,then∀(S)and∃(S)aregroundformulasw.r.t.I.
Thefollowingdefinitiondescribesaprocessthatturnsanyfirst-ordersentenceinto
agroundformulaw.r.t.aninterpretation:
Definition2. LetF beanyfirst-ordersentenceofasignatureσ,andletI beaninter-
pretation of σ whose universe is U. By gr [F] we denote the ground formula w.r.t. I,
I
whichisobtainedbythefollowingprocess:
– gr [p(t ,...,t )]=p((tI)(cid:5),...,(tI)(cid:5));
I 1 n(cid:40) 1 n
(cid:62) iftI =tI,and
– gr [t =t ]= 1 2
I 1 2
⊥ otherwise;
– gr [(cid:62)]=(cid:62); gr [⊥]=⊥;
I I
– gr [F (cid:12)G]=gr [F](cid:12)gr [G] ((cid:12)∈{∧,∨,→});
I I I
– gr [QxF(x)]=Q({ξ(cid:5):gr [F(ξ(cid:5))]|ξ ∈U}) (Q∈{∀,∃}).
I I
Definition3. For any interpretation I and any ground formula F w.r.t. I, the truth
valueofF underI,denotedbyFI,isdefinedrecursivelyasfollows.
TwoNewDefinitionsofStableModelsofLPswithGeneralizedQuantifiers 135
– p(ξ(cid:5),...,ξ(cid:5))I =pI(ξ ,...,ξ );
1 n 1 n
– (cid:62)I = TRUE; ⊥I = FALSE;
– (F ∧G)I = TRUEiffFI = TRUEandGI = TRUE;
– (F ∨G)I = TRUEiffFI = TRUEorGI = TRUE;
– (F →G)I = TRUEiffGI = TRUEwheneverFI = TRUE;
– ∀(S)I = TRUEifftheset{ξ | ξ(cid:5):F(ξ(cid:5)) ∈ S andF(ξ(cid:5))I = TRUE}isthesameas
theuniverseofI;
– ∃(S)I = TRUEifftheset{ξ |ξ(cid:5):F(ξ(cid:5))∈S andF(ξ(cid:5))I = TRUE}isnotempty.
WesaythatI satisfiesF,denotedI |=F,ifFI = TRUE.
Example1continued(I). gr [F]is∀({n(cid:5):(¬p(n(cid:5))→q(n(cid:5)))|n∈N}).Clearly,I
I
satisfiesgr [F].
I
An interpretation I of a signature σ can be represented as a pair (cid:104)Ifunc,Ipred(cid:105),
whereIfuncistherestrictionofI tothefunctionconstantsofσ,andIpredisthesetof
atoms,formedusingpredicateconstantsfromσ andtheobjectnamesfromσI,which
are satisfied by I. For example, interpretation I in Example 1 can be represented as
(cid:104)Ifunc, {q(n(cid:5))|n∈N}(cid:105),whereIfuncmapseachintegertoitself.
Thefollowingpropositionisimmediatefromthedefinitions:
Proposition1. Letσbeasignaturethatcontainsfinitelymanypredicateconstants,let
σpredbethesetofpredicateconstantsinσ,letI =(cid:104)Ifunc,Ipred(cid:105)beaninterpretation
ofσ,andletF beafirst-ordersentenceofσ.ThenI |=F iffIpred |=gr [F].
I
Theintroductionoftheintermediateformofagroundformulaw.r.t.aninterpreta-
tionhelpsusdefineareduct.
Definition4. ForanygroundformulaF w.r.t.I,thereductofF relativetoI,denoted
by FI, is obtained by replacing each maximal subformula that is not satisfied by I
with⊥.Itcanalsobedefinedrecursivelyasfollows.
(cid:40)
p(ξ(cid:5),...,ξ(cid:5)) ifI |=p(ξ(cid:5),...,ξ(cid:5)),
– (p(ξ(cid:5),...,ξ(cid:5)))I = 1 n 1 n
1 n ⊥ otherwise;
– (cid:62)I =(cid:62); ⊥I =⊥;
(cid:40)
FI (cid:12)GI ifI |=F (cid:12)G ((cid:12)∈{∧,∨,→}),
– (F (cid:12)G)I =
⊥ otherwise;
(cid:40)
Q({ξ(cid:5):(F(ξ(cid:5)))I |ξ(cid:5):F(ξ(cid:5))∈S}) ifI |=Q(S) (Q∈{∀,∃}),
– Q(S)I =
⊥ otherwise.
The following theorem tells us how first-order stable models can be characterized
intermsofgroundingandreduct.
Theorem1. Let σ be a signature that contains finitely many predicate constants, let
σpredbethesetofpredicateconstantsinσ,letI =(cid:104)Ifunc,Ipred(cid:105)beaninterpretation
of σ, and let F be a first-order sentence of σ. I satisfies SM[F;σpred] iff Ipred is a
minimalsetofatomsthatsatisfies(ıgr [F])I.
I
136 J.LeeandY.Meng
Example1continued(II). Thereductofgr [F]relativetoI,(gr [F])I,is
I I
∀({n(cid:5):(¬⊥ → q(n(cid:5))) | n ∈ N}), which is equivalent to ∀({n(cid:5):q(n(cid:5)) | n ∈ N}).
Clearly,Ipred ={q(n(cid:5))|n∈N}isaminimalsetofatomsthatsatisfies(gr [F])I.
I
2.3 RelationtoInfinitaryFormulasbyTruszczynski
The definitions of grounding and a reduct in the previous section are inspired by the
workofTruszczynski[12],whereheintroducesinfiniteconjunctionsanddisjunctions
toaccountfortheresultofgrounding∀and∃w.r.t.agiveninterpretation.Differences
betweenthetwoapproachesareillustratedinthefollowingexample:
Example2 Consider the formula F = ∀x p(x) and the interpretation I whose uni-
verseisthesetofallnonnegativeintegersN.Accordingto[12],groundingofF w.r.t.
I resultsintheinfinitarypropositionalformula
{p(n(cid:5))|n∈N}∧.
Ontheotherhand,formulagr [F]is
I
∀({n(cid:5):p(n(cid:5))|n∈N}).
Ourdefinitionofareductisessentiallyequivalenttotheonedefinedin[12].Inthe
nextsection,weextendourdefinitiontoincorporategeneralizedquantifiers.
3 StableModelsofFormulaswithGeneralizedQuantifiers
3.1 Review:FormulaswithGeneralizedQuantifiers
Wefollowthedefinitionofaformulawithgeneralizedquantifiersfrom[15,Section5]
(thatistosay,withLindstro¨mquantifiers[16]withouttheisomorphismclosurecondi-
tion).
WeassumeasetQofsymbolsforgeneralizedquantifiers.EachsymbolinQisas-
sociatedwithatupleofnonnegativeintegers(cid:104)n ,...,n (cid:105)(k ≥0,andeachn is≥0),
1 k i
calledthetype.A(GQ-)formula(withthesetQofgeneralizedquantifiers)isdefinedin
arecursiveway:
– anatomicformula(inthesenseoffirst-orderlogic)isaGQ-formula;
– if F ,...,F (k ≥ 0) are GQ-formulas and Q is a generalized quantifier of type
1 k
(cid:104)n ,...,n (cid:105)inQ,then
1 k
Q[x ]...[x ](F (x ),...,F (x )) (4)
1 k 1 1 k k
is a GQ-formula, where each x (1 ≤ i ≤ k) is a list of distinct object variables
i
whoselengthisn .
i
TwoNewDefinitionsofStableModelsofLPswithGeneralizedQuantifiers 137
WesaythatanoccurrenceofavariablexinaGQ-formulaF isboundifitbelongs
toasubformulaofF thathastheformQ[x ]...[x ](F (x ),...,F (x ))suchthatx
1 k 1 1 k k
isinsomex .Otherwisetheoccurrenceisfree.WesaythatxisfreeinF ifF contains
i
afreeoccurrenceofx.A(GQ-)sentenceisaGQ-formulawithnofreevariables.
We assume that Q contains type (cid:104)(cid:105) quantifiers Q and Q , type (cid:104)0,0(cid:105) quanti-
⊥ (cid:62)
fiers Q ,Q ,Q , and type (cid:104)1(cid:105) quantifiers Q ,Q . Each of them corresponds to the
∧ ∨ → ∀ ∃
standardlogicalconnectivesandquantifiers—⊥,(cid:62),∧,∨,→,∀,∃.Thesegeneralized
quantifierswilloftenbewritteninthefamiliarform.Forexample,wewriteF ∧Gin
placeofQ [][](F,G),andwrite∀xF(x)inplaceofQ [x](F(x)).
∧ ∀
Asinfirst-orderlogic,aninterpretationI consistsoftheuniverseU andtheevalua-
tionofpredicateconstantsandfunctionconstants.ForeachgeneralizedquantifierQof
type(cid:104)n1,...,nk(cid:105),QU isafunctionfromP(Un1)×···×P(Unk)to{TRUE,FALSE},
whereP(Uni)denotesthepowersetofUni.
Example3 Besides the standard connectives and quantifiers, the following are some
examplesofgeneralizedquantifiers.
– type(cid:104)1(cid:105)quantifierQ≤2suchthatQU≤2(R)= TRUEiff|R|≤2; 2
– type(cid:104)1(cid:105)quantifierQmajority suchthatQUmajority(R)= TRUEiff|R|>|U \R|;
– type(cid:104)1,1(cid:105)quantifierQ(SUM,<)suchthatQU(SUM,<)(R1,R2)= TRUEiff
• SUM(R1)isdefined,
• R ={b},wherebisaninteger,and
2
• SUM(R1)<b.
GivenasentenceF ofσI,FI isdefinedrecursivelyasfollows:
– p(t ,...,t )I =pI(tI,...,tI),
1 n 1 n
– (t =t )I =(tI =tI),
1 2 1 2
– ForageneralizedquantifierQoftype(cid:104)n ,...,n (cid:105),
1 k
(Q[x ]...[x ](F (x ),...,F (x )))I =QU((x :F (x ))I,...,(x :F (x ))I),
1 k 1 1 k k 1 1 1 k k k
where(xi:Fi(xi))I ={ξ ∈Uni |(Fi(ξ(cid:5)))I = TRUE}.
We assume that, for the standard logical connectives and quantifiers Q, functions
QU havethestandardmeaning:
– QU∀(R)= TRUEiffR=U; – QU→(R1,R2) = TRUE iff R1 is ∅ or
– QU(R)= TRUEiffR∩U (cid:54)=∅; R is{(cid:15)};
∃ 2
– QU∧(R1,R2) = TRUEiffR1 = R2 = – QU⊥()= FALSE;
{(cid:15)};3 – QU()= TRUE.
– QU∨(R1,R2)= TRUEiffR1 ={(cid:15)}or (cid:62)
R ={(cid:15)};
2
2ItisclearfromthetypeofthequantifierthatRisanysubsetofU.Wewillskipsuchexplana-
tion.
3(cid:15)denotestheemptytuple.ForanyinterpretationI,U0 ={(cid:15)}.ForI tosatisfyQ [][](F,G),
∧
both((cid:15):F)I and((cid:15):G)I havetobe{(cid:15)},whichmeansthatFI =GI =TRUE.
138 J.LeeandY.Meng
WesaythataninterpretationI satisfiesaGQ-sentenceF,orisamodelofF,and
writeI |=F,ifFI = TRUE.AGQ-sentenceF islogicallyvalidifeveryinterpretation
satisfiesF.AGQ-formulawithfreevariablesissaidtobelogicallyvalidifitsuniversal
closureislogicallyvalid.
Example4 Program(1)intheintroductionisidentifiedwiththefollowingGQ-formulaF :
1
(¬Q [x][y](p(x), y=2)→p(2))
(SUM,<)
∧(Q [x][y](p(x), y=−1)→p(−1))
(SUM,>)
∧(p(−1)→p(1)).
ConsidertwoHerbrandinterpretationsoftheuniverseU ={−1,1,2}:I ={p(−1),p(1)}
1
and I2 = {p(−1),p(1),p(2)}. We have (Q(SUM,<)[x][y](p(x), y = 2))I1 = TRUE
since
– (x:p(x))I1 ={−1,1}and(y :y=2)I1 ={2};
– QU ({−1,1},{2})= TRUE.
(SUM,<)
Similarly,(Q(SUM,>)[x][y](p(x), y=−1))I2 = TRUEsince
– (x:p(x))I2 ={−1,1,2}and(y :y=−1)I2 ={−1};
– QU ({−1,1,2},{−1})= TRUE.
(SUM,>)
Consequently,bothI andI satisfyF .
1 2 1
3.2 Review:SM-BasedDefinitionofStableModelsofGQ-Formulas
ForanyGQ-formulaF andanylistofpredicatesp = (p ,...,p ),formulaSM[F;p]
1 n
isdefinedas
F ∧¬∃u((u<p)∧F∗(u)),
whereF∗(u)isdefinedrecursively:
– p (t)∗ =u (t)foranylisttofterms;
i i
– F∗ =F foranyatomicformulaF thatdoesnotcontainmembersofp;
–
(Q[x ]...[x ](F (x ),...,F (x )))∗ =
1 k 1 1 k k
Q[x ]...[x ](F∗(x ),...,F∗(x ))∧ Q[x ]...[x ](F (x ),...,F (x )).
1 k 1 1 k k 1 k 1 1 k k
WhenF isasentence,themodelsofSM[F;p]arecalledthep-stablemodelsofF:
theyarethemodelsofF thatare“stable”onp.WeoftensimplywriteSM[F]inplaceof
SM[F;p]whenpisthelistofallpredicateconstantsoccurringinF,andcallp-stable
modelssimplystablemodels.
Asexplainedin[17],thisdefinitionofastablemodelisapropergeneralizationof
thefirst-orderstablemodelsemantics.
Example4continued(I). ForGQ-sentenceF consideredearlier,SM[F ]is
1 1
F ∧¬∃u(u<p∧F∗(u)), (5)
1 1
TwoNewDefinitionsofStableModelsofLPswithGeneralizedQuantifiers 139
whereF∗(u)isequivalenttotheconjunctionofF and
1 1
(¬Q [x][y](p(x),y=2)→u(2))
(SUM,<)
∧((Q [x][y](u(x),y=−1)∧ Q [x][y](p(x),y=−1))→u(−1))
(SUM,>) (SUM,>)
∧(u(−1)→u(1)).
TheequivalencecanbeexplainedbyProposition1from[9],whichsimplifiesthetrans-
formationformonotoneandantimonotoneGQs.I andI consideredearliersatisfy(5)
1 2
andthusarestablemodelsofF .
1
3.3 Reduct-BasedDefinitionofStableModelsofGQ-Formulas
Thereduct-baseddefinitionofstablemodelspresentedinSection2.2canbeextended
toGQ-formulasasfollows.
LetIbeaninterpretationofasignatureσ.Asbefore,weassumeasetQofgeneral-
izedquantifiers,whichcontainsallpropositionalconnectivesandstandardquantifiers.
Definition5. AgroundGQ-formulaw.r.t.I isdefinedrecursivelyasfollows:
– p(ξ(cid:5),...,ξ(cid:5)),wherepisapredicateconstantofσandξ(cid:5) areobjectnamesofσI,
1 n i
isagroundGQ-formulaw.r.t.I;
– foranyQ ∈ Qoftype(cid:104)n ,...,n (cid:105),ifeachS isasetofpairsoftheformξ(cid:5):F
1 k i
where ξ(cid:5) is a list of object names from σI whose length is n and F is a ground
i
GQ-formulaw.r.t.I,then
Q(S ,...,S )
1 k
isagroundGQ-formulaw.r.t.I.
The following definition of grounding turns any GQ-sentence into a ground GQ-
formulaw.r.t.aninterpretation:
Definition6. LetF beaGQ-sentenceofasignatureσ,andletI beaninterpretation
ofσ.Bygr [F]wedenotethegroundGQ-formulaw.r.t.Ithatisobtainedbytheprocess
I
similar to the one in Definition 2 except that the last two clauses are replaced by the
followingsingleclause:
– gr [Q[x ]...[x ](F (x ),...,F (x ))]=Q(S ,...,S )
I 1 k 1 1 k k 1 k
whereS ={ξ(cid:5):gr [F (ξ(cid:5))]|ξ(cid:5)isalistofobjectnamesfromσI whoselengthisn }.
i I i i
For any interpretation I and any ground GQ-formula F w.r.t. I, the satisfaction
relationI |=F isdefinedrecursivelyasfollows.
Definition7. ForanyinterpretationIandanygroundGQ-formulaF w.r.t.I,thesatis-
factionrelationI |=F isdefinedsimilartoDefinition3exceptthatthelastfiveclauses
arereplacedbythefollowingsingleclause:
– Q(S ,...,S )I = QU(SI,...,SI)whereSI = {ξ | ξ(cid:5):F(ξ(cid:5)) ∈ S , F(ξ(cid:5))I =
1 k 1 k i i
TRUE}.
140 J.LeeandY.Meng
Example4continued(II). ForHerbrandinterpretationI = {p(−1),p(1)},formula
1
gr [F ]is4
I1 1
(¬Q ({−1:p(−1),1:p(1),2:p(2)},{−1:⊥,1:⊥,2:(cid:62)})→p(2))
(SUM,<)
∧(Q ({−1:p(−1),1:p(1),2:p(2)},{−1:(cid:62),1:⊥,2:⊥})→p(−1)) (6)
(SUM,>)
∧(p(−1)→p(1)).
I satisfies Q ({−1:p(−1),1:p(1),2:p(2)},{−1:⊥,1:⊥,2:(cid:62)}) because
1 (SUM,<)
I |=p(−1),I |=p(1),I (cid:54)|=p(2),and
1 1 1
QU ({−1,1},{2})= TRUE.
(SUM,<)
I satisfiesQ ({−1:p(−1),1:p(1),2:p(2)},{−1:(cid:62),1:⊥,2:⊥})because
1 (SUM,>)
QU ({−1,1},{−1})= TRUE.
(SUM,>)
Consequently,I satisfies(6).
1
Proposition2. Letσbeasignaturethatcontainsfinitelymanypredicateconstants,let
σpredbethesetofpredicateconstantsinσ,letI =(cid:104)Ifunc,Ipred(cid:105)beaninterpretation
ofσ,andletF beaGQ-sentenceofσ.ThenI |=F iffIpred |=gr [F].
I
Definition8. ForanyGQ-formulaF w.r.t.I,thereductofF relativetoI,denotedby
FI,isdefinedinthesamewayasinDefinition4byreplacingthelasttwoclauseswith
thefollowingsingleclause:
(cid:40)
Q(SI,...,SI) ifI |=Q(S ,...,S ),
– (Q(S ,...,S ))I = 1 k 1 k
1 k
⊥ otherwise;
whereSI ={ξ(cid:5):(F(ξ(cid:5)))I |ξ(cid:5):F(ξ(cid:5))∈S }.
i i
Theorem2. Let σ be a signature that contains finitely many predicate constants, let
σpredbethesetofpredicateconstantsinσ,letI =(cid:104)Ifunc,Ipred(cid:105)beaninterpretation
ofσ,andletF beaGQ-sentenceofσ.I |= SM[F;σpred]iffIpred isaminimalsetof
atomsthatsatisfies(ıgr [F])I .
I
Example4continued(III). InterpretationI consideredearliercanbeidentifiedwith
1
the tuple (cid:104)Ifunc,{p(−1),p(1)}(cid:105) where Ifunc maps every term to itself. The reduct
(grI1[F1])I1 is
(⊥→⊥)
∧(Q ({−1:p(−1),1:p(1),2:⊥},{−1:(cid:62),1:⊥,2:⊥})→p(−1))
(SUM,>)
∧(p(−1)→p(1)),
which is the GQ-formula representation of (3). We can check that {p(−1),p(1)} is a
minimalmodelofthereduct.
ExtendingTheorem2toallowanarbitrarylistofintensionalpredicates,ratherthan
σpred,isstraightforwardinviewofProposition1from[18].
4Forsimplicity,wewrite−1,1,2insteadoftheirobjectnames(−1)(cid:5),1(cid:5),2(cid:5).