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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).

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