Guarded Hybrid Knowledge Bases

Under conside ratio n for public ation in Theory and Practice of Logic Pro grammi ng 1 Guar ded Hybrid Knowledge Bases ∗ STIJN HEYMANS 1 , JOS DE BR UIJN 2 , LIVIA PREDOIU 3 , CRISTIN A FEIER 1 , D A VY V AN NIEUWENBORGH 4 † 1 Digital Enterprise Resear ch Insti tute , Univer sity of Innsbruck, T ech nik erstr asse 21a, Innsbruc k, Austria ( e-mail: { stijn.heymans ,cristina.feie r } @deri.at ) 2 F aculty of Computer Science , F r ee Uni versit y of Bozen-Bolzano, Ital y ( e-mail: debruijn@in f.unibz.it ) 3 Institut e of Compute r Science , Univer sity of Mannheim, A5, 6 68159 Mannheim, Germany ( e-mail: livia@infor matik.uni-mannh eim.de ) 4 Dept. of Computer Scienc e, Vrije Universitei t Brussel, VUB, Pleinlaa n 2, B1050 Brussels, Belgiu m ( e-mail: dvnieuwe@vu b.ac.be ) submitte d 20 J une 2006; r e vised 3 J anuary 2007; acce pted 18 October 2007 Abstract Recently , there has been a lot of interest in the integration of Description Logics and rules on the Semantic W eb . W e deﬁne gua r ded hy brid knowledg e bases (or g-hybrid knowledg e bases ) as kno wl- edge bases that c onsist of a Desc ription Logic kno wled ge base and a guar ded logic program , similar to the D L + l o g knowledge bases from (Rosati 2006). G- hybrid knowled ge bases enable an integ ra- tion of Description Logics and Logic Programming where, unlike in other approach es, v ariables in the rules of a guarded program do not need to appear i n positiv e non-DL atoms of the body , i.e. DL atoms can act as guar ds as well. Decidability of satisﬁabili ty checking of g-hybrid kno wledge bases is sho wn for t he particular DL D LRO −{≤} , which is close to O WL DL, by a reduction to guarded programs under the open answer set semantics. Moreov er , we sho w 2- E X P T I M E -com pleteness for satisﬁability checking of such g-hybrid kno wledge bases. Fi nally , we discuss advantage s and disad- v antages of our approach compared with DL + lo g kno wledge bases. KEYWORDS : g-hybrid kno wledge bases, open answer set programming, guarded logic programs, description logics 1 Introduction The integration o f Description Logics with rule s has r ecently received a lot of atten- tion in the co ntext of the Sema ntic W eb ( Rosati 2005a; Rosati 2 006; Eiter et al. 2004; Motik et al. 2004; Horrocks and Patel-Schneider 2004b ; Motik and Rosati 2007; de Bruijn et al. 2007). R-hybrid knowledge bases (Rosati 2005a), and its extension D L + lo g (Rosati 2006), is ∗ A preliminary version of this paper appe ared in the proceedin gs of the ICLP’06 W orkshop on Applica tions of Logic P r ogr amming in the Semantic W eb and Semantic W eb Service s (ALPSWS2006) pages 39-54, Seattle, W ashington, USA, August 16 2006. † The w ork is funded by the European Commission under the pr oject s ASG, DIP , enIRaF , In fraW ebs, Kn o wledge W eb, Musing, Saler o, SEKT , SEEMP , SemanticGO V , Super , SW ING and TripCom; by Scienc e Foundatio n Ireland under the DERI-Lion Grant No.SFI/02/CE1/I13 ; by the F FG ( ¨ Osterreic hische Forschun gsfrderungs- geselle schaft mbH) under the project s Grisino, R W 2 , SemNetMan, SEnSE , TSC and OnT ourism. Da vy V an Nieuwenbo rgh was supported by the Flemish Fund for Scientiﬁc Research (FWO-Vlaan deren). 2 S. Heymans e t al. an elegant form alism based o n co mbined mod els for Descr iption Log ic kn owledge bases and no nmon otonic logic p rogra ms. W e pro pose a variant of r-hyb rid knowledge bases, called g -hybrid kno wledge bases , which d o n ot req uire standar d nam es or a spe cial saf e- ness restriction o n rules, b ut instead req uire the pro gram to be guar ded . W e show se v- eral co mputatio nal proper ties by a red uction to guarded o pen an swer set progr amming (Heymans et al. 2005a; Heymans et al. 2006b). Open answer set p rogr amming (OASP) (Heymans et al. 2005a; Heymans et al. 2006b) combines the logic pro grammin g a nd ﬁrst-ord er logic paradigms. From the logic p ro- grammin g parad igm it inherits a ru le-based p resentation and a non monoto nic seman tics by means of negation as failure. In contrast with usual logic programming semantics, such as the answer set semantics ( Gelfond and Lifschitz 1988), OASP allows for doma ins con- sisting of other objects than those present in the logic program at hand. Such open do- mains are inspired by ﬁrst-order logic based lan guage s such as Description Logics (DLs) (Baader et al. 2003) and make O ASP a viable can didate for conceptu al rea soning. Due to its rule-based presentation and its support for nonmon otonic reasoning and open domains, O ASP can be u sed to reason with both rule-based and con ceptual knowledge o n the Se- mantic W eb, as illustrated in (Heymans et al. 2005b). A major challen ge for O ASP is to control undecidability of satisﬁability ch ecking, a challenge it shar es with DL-based lan guage s. In (Heymans et al. 2005a; Heyman s et al. 2006b), we identify a decidab le class o f prog rams, the so-called guar ded p r o grams , f or which decidability o f satisﬁability check ing is ob tained by a translation to guard ed ﬁxed point logic (Gr ¨ adel and W alukie wicz 1999). In (Heymans et al. 2006), we show the expressi ve- ness of such guarde d programs by simulating a DL with n -ary ro les and no minals. In par- ticular , we extend the DL D LR (Calvanese et al. 1997) with both concep t n ominals { o } and r ole nominals { ( o 1 , . . . , o n ) } , resulting in D LRO . W e d enote the DL D LRO with- out number restrictions as DLRO −{≤} . Satisﬁability checking of concept expressions w .r .t. D LRO −{≤} knowledge bases can be reduce d to checking satisﬁability of guarded progr ams (Heymans et al. 2006b). A g-hyb rid knowledge base c onsists of a Description Log ic knowledge base an d a guarde d program. T he D L + lo g knowledge bases fro m (Rosati 200 6) are weakly safe , which means that the intera ction between the prog ram and th e DL knowledge base is restricted by requiring that v ariables which appear in n on-DL atoms, appear in positi ve non-DL atoms in the bod y , where DL atom s are atoms in volving a concept or role symbol from the DL kno wledge base. G-hybrid k nowledge bases do not require such a restriction; instead, variables must appear in a gua r d of the r ule, but this guard can be a DL atom as well. In this paper, we show decidability of g-h ybrid knowledge b ases f or DLRO −{≤} knowledge b ases b y a redu ction to guard ed progra ms, and show that satisﬁability check- ing o f g-hybrid knowledge ba ses is 2- E X P T I M E -co mplete. The DL D LRO −{≤} is close to S HO I N , the Description Logic u nderly ing O WL DL (Horrock s and P atel-Schneider 2004a). Compared with S HO I N , D LRO −{≤} does n ot inclu de transitive r oles and numb er r e- strictions, but does include n -ary roles and complex r ole expressions. T o see wh y a co mbination of r ules and ontologies, as proposed in g- hybrid knowledge bases, is usef ul, and why the safeness cond itions considered so far in the literatur e are not G-Hybrid Knowledge Bases 3 approp riate in all scenarios, consider the Description Logic ontolog y F r aternityMe mb er ⊑ Drinker ⊓ ∃ hasDrinkingBuddy . F r aternityMemb er which says that fraternity members are drinkers who have drinkin g buddies, whic h are also fraternity members. Now c onsider the logic progra m pr oblemDrinker ( X ) ← Drinker ( X ) , not so cialDrinker ( X ) so cialDrinker ( X ) ← Drinker ( X ) , not pr oblemDrinker ( Y ) , hasDrinkingBuddy ( X , Y ) F r aternityMe mb er ( John ) ← which says th at drin kers are by default pro blem drinkers, unless it is known that they are social drinkers; dr inkers with drin king b uddies who are no t problem drinkers are social drinkers; and Joh n is a fra ternity mem ber . From the combin ation of the ontolog y and the logic pro gram, one would expect to deriv e that Joh n is a social d rinker, and not a prob- lem drinker . This logic p rogram cannot b e expressed using r-hybrid knowledge bases, or DL + lo g , b ecause the rules in the pro gram are not weakly safe . Howe ver , the logic p ro- gram is guar ded , and thus part of a v alid g -hybr id knowledge base, which has th e e xpected consequen ces. The remainde r of the paper starts with an introduc tion to open answer set programmin g and De scription Logics in Section 2. Section 3 deﬁnes g-h ybrid knowledge bases, tra nslates them to gu arded progra ms when th e DL D LRO −{≤} is co nsidered, an d p rovides a com- plexity characteriz ation for satisﬁability che cking of these particular g-hybr id knowledge bases. In Section 5, we discuss the relation of g-hybrid kn owledge bases with DL + lo g and other related work. W e conclud e and gi ve directions for further research in Section 6. 2 Preliminaries In this sectio n we introduc e Open Answer Set Program ming, guard ed pro grams, and the Description Logic DLRO −{≤} . 2.1 Decidab le Open An swer Set Programming W e intro duce the o pen answer set sem antics from ( Heymans et al. 2005a; Heyman s et al. 2006b), modiﬁed as in (Heymans et al. 20 06) such that it does not assume uniqueness of names by default. Con stants , varia bles , terms , and ato ms ar e deﬁned as u sual. A literal is an atom p ( ~ t ) o r a naf-literal not p ( ~ t ) , with ~ t a tup le of term s. 1 The po sitive part of a set of literals α is α + = { p ( ~ t ) | p ( ~ t ) ∈ α } and th e ne gative part of α is α − = { p ( ~ t ) | not p ( ~ t ) ∈ α } . W e assume th e existence of the (in)equ ality p redicates = and 6 = , u sually written in inﬁx notation; t = s is an atom and t 6 = s is sh ort for not t = s . A r e gular atom is an atom without equality . For a set A of atoms, not A = { not l | l ∈ A } . A pr ogram is a coun table set of rules α ← β , where α and β are ﬁnite sets of literals, | α + | ≤ 1 (b ut α − may b e of arbitrar y size), and every a tom in α + is regu lar , i.e. α contain s 1 W e do not allo w “classical” negatio n ¬ , ho we ve r , programs with ¬ ca n be reduced to programs without it, see e.g. (Lifschitz et al. 2001). 4 S. Heymans e t al. at most one po siti ve atom, which may not con tain the equality pr edicate. 2 The set α is the head o f the rule and repr esents a disjunction of literals, while β is the body and re presents a conjunction of literals. If α = ∅ , the rule is called a constraint . F r ee rules are rules of th e form q ( ~ X ) ∨ n ot q ( ~ X ) ← ; they enable a choice for the inclusion of atoms in a model. W e call a pr edicate p fr ee if there is a free ru le p ( ~ X ) ∨ n ot p ( ~ X ) ← . Atoms, literals, rules, an d progr ams that do not contain v ariables are gr ou nd . For a literal, rule, or program o , let cts ( o ) , vars ( o ) , pr e ds ( o ) b e the con stants, v ariables, and predicate s, respectively , in o . A p r e-interpr etation U fo r a pro gram P is a pair ( D , σ ) where D is a n on-em pty do main and σ : cts ( P ) → D is a f unction which maps all constants in P to elemen ts fro m D . 3 P U is the gro und p rogram obtained fro m P b y sub - stituting e v ery variable in P with e very possible element from D and e very constant c with σ ( c ) . E.g., f or a ru le r : p ( X ) ← f ( X , c ) and U = ( { x, y } , σ ) where σ ( c ) = x , we h av e that the ground ing w .r .t. U is: p ( x ) ← f ( x , x ) p ( y ) ← f ( y , x ) Let B P be the set of re gular atom s o btained f rom the languag e of th e ground prog ram P . An interpr etation I of a gr ound progra m P is a subset of B P . For a groun d regular atom p ( ~ t ) , we write I | = p ( ~ t ) if p ( ~ t ) ∈ I ; for an e quality atom t = s , we wr ite I | = t = s if s a nd t are equal terms. W e write I | = not p ( ~ t ) if I 6| = p ( ~ t ) , for p ( ~ t ) an atom. For a set of groun d literals A , I | = A holds if I | = l for every l ∈ A . A ground rule r : α ← β is satisﬁed w .r .t. I , deno ted I | = r , if I | = l for some l ∈ α when ev er I | = β . A g roun d constraint ← β is satisﬁed w .r .t. I if I 6| = β . For a ground progr am P withou t no t , an interpretation I of P is a mod el of P if I satisﬁes e very ru le in P ; it is an answer set of P if it is a subset minim al model o f P . For g round programs P containing not , the r educt (Inoue and Sakama 1998) w . r .t. I is P I , where P I consists of α + ← β + for e very α ← β in P such that I | = not β − and I | = α − . I is an answer set of P if I is an answer set of P I . Note that allo wing n egation in the head of rules leads to the loss of th e anti-chain pr o perty (Ino ue and Sakama 1998) which states that no answer set can be a strict subset of another answer s et. E.g, a rule a ∨ not a ← h as the answer sets ∅ and { a } . Howe ver , n egation in the he ad is r equired to en sure ﬁrst-order behavior for certain predicates, e.g., when simulating Description Logic reasoning. In the following, a progra m is assumed to be a ﬁnite set of rules; inﬁnite p rogram s only appear as b yprod ucts of ground ing a ﬁnite program u sing an in ﬁnite pre-interpretation . An open interpretation of a p rogram P is a pair ( U, M ) where U is a pre-interp retation for P and M is an interpretation of P U . An open answer set of P is an open interp retation ( U, M ) of P with M an answer set of P U . An n -ary predicate p in P is satisﬁable if there is an open an swer set (( D , σ ) , M ) o f P and a ~ x ∈ D n such that p ( ~ x ) ∈ M . A pro gram P is satisﬁable i ff it has an open an swer set. Note that satisﬁability checking o f programs can be easily reduce d to satisﬁability checking of predicates: P is satisﬁable iff p is satisﬁable 2 The condit ion | α + | ≤ 1 makes the GL-reduct non-disjunct i ve, ensuring that the immediate consequence operat or is well-de ﬁned, s ee (He ymans et al. 2006b). 3 In (Heyman s et al. 2006b), we only use the domain D which is there deﬁned as a non-empty superset of the constant s in P . This corre sponds to a pre-interp retat ion ( D, σ ) wher e σ is the identi ty functi on on D . G-Hybrid Knowledge Bases 5 w .r .t. P ∪ { p ( ~ X ) ∨ not p ( ~ X ) ←} , where p is a pred icate symbo l no t used in P and ~ X is a tuple of v ariables. In the follo wing, when we s peak of satisﬁability checking, we refer to satisﬁability checking of predicates, unless speciﬁed otherwise. Satisﬁability checking w .r .t. the open answer set seman tics is und ecidable in gen eral. In (He ymans et al. 2006b), we ide ntify a syn tactically restricted fragment o f programs, so- called g uar ded pr ogr ams , for which satisﬁability checking is decidable, wh ich is shown throug h a reduction to guard ed ﬁxed po int lo gic ( Gr ¨ adel and W alukiewicz 1999). Th e de - cidability of g uarded prog rams relies on the presence of a g uar d in each rule, whe re a guard is an atom that contains all variables of the rule. F ormally , a rule r : α ← β is guarded if there is an atom γ b ∈ β + such that vars ( r ) ⊆ vars ( γ b ) ; γ b is the gua r d of r . A pro gram P is a gu ar ded pr ogram (GP) if every non-fr ee rule in P is g uarded . E.g ., a rule a ( X , Y ) ← not f ( X , Y ) is not guar ded, but a ( X, Y ) ← g ( X , Y ) , not f ( X , Y ) is guarde d with guard g ( X , Y ) . Satisﬁability checking of pr edicates w .r .t. guarded program s is 2- E X P T I M E -com plete ( Heymans et al. 2006b) – a result that stems f rom the correspond- ing complexity in guarded ﬁx ed point logic. 2.2 T he Description Logic D LRO −{≤} DLR (Calv anese et al. 1997; Baader et al. 2003) is a DL wh ich supports roles of arbitrary arity , whereas most DLs only supp ort b inary roles. W e in troduce an extension of DLR with nominals, called D LRO (Heyman s e t al. 200 6). The basic building blocks of D LRO are concept names A and r elation names P where P de notes an ar bitrary n -ary relation for 2 ≤ n ≤ n max and n max is a g iv en ﬁnite non- negativ e integer . Role expre ssions R and concept expressions C are deﬁned as: R → ⊤ n | P | ($ i/n : C ) | ¬ R | R 1 ⊓ R 2 | { ( o 1 , . . . , o n ) } C → ⊤ 1 | A | ¬ C | C 1 ⊓ C 2 | ∃ [$ i ] R | ≤ k [$ i ] R | { o } where i is between 1 and n in ($ i/n : C ) ; similarly in ∃ [$ i ] R and ≤ k [$ i ] R f or R an n -ary relation. Moreover, we a ssume that the above constructs are well-typed , e.g. , R 1 ⊓ R 2 is deﬁned only for relations of the same arity . The semantics of D LRO is given by interpre- tations I = (∆ I , · I ) where ∆ I is a no n-empty set, the domain , and · I is an interpr etation function such that C I ⊆ ∆ I , R I ⊆ (∆ I ) n for an n -ary relatio n R , and the following condition s ar e satisﬁed ( P , R , R 1 , and R 2 have arity n ): ⊤ I n ⊆ (∆ I ) n P I ⊆ ⊤ I n ( ¬ R ) I = ⊤ I n \ R I ( R 1 ⊓ R 2 ) I = R I 1 ∩ R I 2 ($ i/n : C ) I = { ( d 1 , . . . , d n ) ∈ ⊤ I n | d i ∈ C I } 6 S. Heymans e t al. ⊤ I 1 = ∆ I A I ⊆ ∆ I ( ¬ C ) I = ∆ I \ C I ( C 1 ⊓ C 2 ) I = C I 1 ∩ C I 2 ( ∃ [$ i ] R ) I = { d ∈ ∆ I | ∃ ( d 1 , . . . , d n ) ∈ R I . d i = d } ( ≤ k [$ i ] R ) I = { d ∈ ∆ I | |{ ( d 1 , . . . , d n ) ∈ R I | d i = d }| ≤ k } { o } I = { o I } ⊆ ∆ I { ( o 1 , . . . , o n ) } I = { ( o I 1 , . . . , o I n ) } Note that in D LRO the negation of role expressions is deﬁned w .r .t. ⊤ I n and not w .r .t. (∆ I ) n . A DLRO knowledge base Σ is a set of termino logical axioms and role ax ioms, which de- note subset relations be tween concept and ro le expressions ( of the same a rity), respectively . A te rminolog ical axiom C 1 ⊑ C 2 is satisﬁed by I iff C I 1 ⊆ C I 2 . A role axiom R 1 ⊑ R 2 is satisﬁed by I iff R I 1 ⊆ R I 2 . An interpretatio n I is a model of a knowledge base Σ (i.e. Σ is satisﬁed by I ) if all ax ioms in Σ ar e satisﬁed by I ; if Σ has a model, then Σ is satisﬁable . A con cept expression C is satisﬁable w .r .t. a knowledge base Σ if ther e is a model I of Σ such that C I 6 = ∅ . Note that for ev ery interpretation I , ( { ( o 1 , . . . , o n ) } ) I = (($1 /n : { o 1 } ) ⊓ . . . ⊓ ($ n/n : { o n } )) I . Therefo re, in the remainder of th e paper , we will restrict ourselves to nom inals of the f orm { o } . W e denote the fragme nt of DLRO with out the numb er restriction ≤ k [$ i ] R with DLRO −{≤} . 3 G-hybrid Knowledge Bases G-hybr id knowledge bases are com binations o f Description Log ic (DL ) knowledge bases and guarded lo gic p rogram s (GP). They are a v ariant o f the r -hybrid kn owledge bases introdu ced in (Rosati 2005a). Deﬁnition 1 Giv en a Descriptio n Logic DL , a g -hybrid kn owledge base is a pair (Σ , P ) wh ere Σ is a DL knowledge base and P is a guar ded program. Note that in the above deﬁnition there are n o restrictions on th e use of predica te symbols. W e call the atom s and literals in P that h ave un derlying pred icate symbols which cor- respond to concept o r role names in the DL k nowledge base DL atoms an d DL literals , respectively . V a riables i n rules are not required to appear in positi ve non-DL atoms, which is the case in, e.g., th e DL + lo g knowledge bases in (Rosati 2006), the r-hyb rid knowledge bases in (Rosati 2005a), and th e DL-saf e rules in (Motik et al. 2004). DL-atoms can appear in the hea d of rules, thereb y enabling a bi-dir ectional ﬂow of information between the DL knowledge base and t he logic pro gram. Example 1 Consider the D LRO −{≤} knowledge b ase Σ wher e so cialDrinker is a co ncept, dri nks is G-Hybrid Knowledge Bases 7 a ternary role such th at, intuitively , ( x, y , z ) is in the interpr etation of drinks if a person x drinks some drink z with a perso n y . Σ consists of the single axiom so cialDrinker ⊑ ∃ [$ 1 ]( drinks ⊓ ($ 3 / 3 : { wine } )) which ind icates that social drinkers dr ink wine with someon e. Consider a GP P that indicates that someone has an increased risk of alc oholism if the person is a social drinker and knows someo ne f rom the association of Alcoh olics Anonymous (AA). Fur thermor e, we state that john is a social drinker and knows michael from AA: pr oblematic ( X ) ← so cialDrinker ( X ) , knowsF r o mAA ( X , Y ) knowsF r omA A ( john , michael ) ← so cialDrinker ( john ) ← T o gether, Σ and P for m a g-hy brid knowledge base. The literals so cialDrinker ( X ) and so cialDrinker ( john ) are DL ato ms wher e the latter appears in the hea d of a ru le in P . The literal knowsF r omAA(X,Y) appears only in the program P (and is thus not a DL atom). Giv en a DL interpretation I = (∆ I , · I ) and a grou nd program P , we deﬁne Π( P, I ) as the pr ojection of P with respect to I , which is obta ined as follo ws: for e very rule r in P , • if there exists a DL literal in the head of the form — A ( ~ t ) with ~ t ∈ A I , or — not A ( ~ t ) with ~ t 6∈ A I , then delete r , • if there exists a DL literal in the body of the form — A ( ~ t ) with ~ t 6∈ A I , or — not A ( ~ t ) with ~ t ∈ A I , then delete r , • otherwise, delete all DL literals from r . Intuitively , th e pro jection “ev aluates” the progr am with respect to I by removin g ( ev alu- ating) rules and DL literals consistently with I ; concep tually this is similar to the r educt, which removes r ules and negative literals consistently with an in terpretation of the pro- gram. Deﬁnition 2 Let (Σ , P ) b e a g-hybrid kno wledge base. An interpretation o f (Σ , P ) is a tuple ( U, I , M ) such that • U = ( D, σ ) is a pr e-interpr etation for P , • I = ( D, · I ) is an interpr etation of Σ , • M is an interpr etation of Π( P U , I ) , and • b I = σ ( b ) f or ev ery constant symbol b appearin g both in Σ and in P . Then, ( U = ( D , σ ) , I , M ) is a model of a g-hybrid knowledge base (Σ , P ) if I is a model of Σ and M is an answer set of Π( P U , I ) . For p a con cept expression from Σ o r a pred icate from P , we say th at p is satisﬁable w .r .t. (Σ , P ) if there is a model ( U, I , M ) su ch that p I 6 = ∅ or p ( ~ x ) ∈ M fo r some ~ x from D , respectively . 8 S. Heymans e t al. Example 2 Consider the g- hybrid knowledge base in Examp le 1. T ake U = ( D , σ ) with D = { j ohn, michael , w ine, x } and σ the identity fun ction on the constant symbo ls in (Σ , P ) . Further- more, deﬁne · I as follows: • so cialDrinker I = { john } , • drinks I = { ( john , x , wine ) } , • wine I = wine . If M = { knowsfr omAA ( john , michael ) , pr oblematic ( john ) } , t hen ( U, I , M ) is a model of th is g -hybr id knowledge base. No te that the projection Π( P , I ) doe s n ot co ntain th e ru le so cialDrinker ( john ) ← . 4 T ranslation to Guarded Logic Programs In th is section we introd uce a translation of g -hyb rid knowledge bases to g uarded logic progr ams (GP) under th e o pen answer set semantics, show tha t th is tran slation p reserves satisﬁability , and use this translation to obtain co mplexity r esults for reasoning i n g-hybrid knowledge bases. Before intr oducin g the tran slation to gu arded pro grams f ormally , we introdu ce the translation t hroug h an example. Consider the knowledge b ase in Example 1. The axiom so cialDrinker ⊑ ∃ [$ 1 ]( drinks ⊓ ($ 3 / 3 : { wine } )) translates to the constraint ← so cialDrinker ( X ) , not ( ∃ [$ 1 ]( drinks ⊓ ($ 3 / 3 : { wine } )))( X ) Thus, th e concept expressions on either side of the ⊑ symbol ar e associated with a new unary p redicate name. F or con venience, we name the new p redicates according to the orig- inal co ncept e xpressions. Th e constraint simulates the beh avior of the DLRO −{≤} axiom. If the left- hand side of th e axiom hold s and the r ight-han d side d oes not h old, there is a contradictio n. It remains to ensure that those newly introduced predic ates behav e according to th e DL semantics. First, all the concept and role names occu rring in the axiom above need to b e deﬁned as fr ee predicates, in order to simulate the ﬁrst-order semantics of co ncept and r ole names in DLs. I n DLs, a tuple is eit her true or fals e in a g iv en interpretation (cf. the law of the excluded middle); this beha vior can be captured e xactly by the free predicates: so cialDrinker ( X ) ∨ not so cia lDrinker ( X ) ← drinks ( X , Y , Z ) ∨ not dri nks ( X , Y , Z ) ← Note that concep t names are translated to u nary free p redicates, while n - ary role names are translated to n -ary free predicates. The d eﬁnition of the truth symbols ⊤ 1 and ⊤ 3 which a re imp licit in ou r D LRO −{≤} axiom (since the axio m co ntains a conc ept n ame and a ter nary role) are tr anslated to free predicates as we ll. Note that we do not need a pr edicate for ⊤ 2 since the axiom do es not contain binary predicates. ⊤ 1 ( X ) ∨ not ⊤ 1 ( X ) ← ⊤ 3 ( X , Y , Z ) ∨ n ot ⊤ 3 ( X , Y , Z ) ← G-Hybrid Knowledge Bases 9 W e en sure that, for the ternary DLRO −{≤} role drinks , drin k s I ⊆ ⊤ I 3 holds by adding the constraint: ← drinks ( X , Y , Z ) , not ⊤ 3 ( X , Y , Z ) T o ensure that ⊤ I 1 = ∆ I , we add the constraint: ← not ⊤ 1 ( X ) For rules contain ing only one variable, we ca n alw ays assum e that X = X is in the body and a cts as the gu ard of the rule, so th at th e latter r ule is gua rded; cf. the equiv alent rule ← not ⊤ 1 ( X ) , X = X . W e tran slate the nominal { wine } to the rule { wine } ( wine ) ← Intuitively , since this r ule will be the only rule with the predicate { wine } in the head, every open answer set of the tr anslated pr ogram will contain { wine } ( x ) with σ ( wine ) = x if and only if the correspond ing interpretation { wine } I = { x } for wine I = x . The D LRO −{≤} role expression ($3 / 3 : { wine } ) indicates the tern ary tuples f or which the thir d argument belongs to the e xtension of { win e } , w hich is translated to the follo wing rule: ($ 3 / 3 : { wine } )( X , Y , Z ) ← ⊤ 3 ( X , Y , Z ) , { wine } ( Z ) Note that the above rule is guarded by the ⊤ 3 literal. Finally , the concept expression ( drinks ⊓ ($ 3 / 3 : { wine } )) can be rep resented by the following rule: ( drinks ⊓ ($ 3 / 3 : { wine } ))( X , Y , Z ) ← drinks ( X , Y , Z ) , ($3 / 3 : { wine } )( X , Y , Z ) As we can see, the DL construct ⊓ is translated to con junction in the body of a rule. The D LRO −{≤} role ∃ [$1]( dr ink s ⊓ ($ 3 / 3 : { wine } )) can be rep resented using the following rule: ( ∃ [$1]( dri nk s ⊓ ($ 3 / 3 : { wine } )))( X ) ← ( dri n k s ⊓ ($3 / 3 : { wine } ))( X , Y , Z ) Indeed , the elemen ts which be long to the extension of ∃ [$1 ]( drink s ⊓ ($3 / 3 : { wine } )) are exactly those that are connected to the role ($3 / 3 : { wine } ) , as speciﬁed in the rule. This conclu des the translation of the DL kn owledge base in the g -hybr id k nowledge base of Exam ple 1 . The pr ogram can be co nsidered as is, since, by deﬁnitio n o f g-hy brid knowledge bases, it is already a guarded program. W e no w proceed with the f ormal tran slation. The closure clos (Σ) of a D LRO −{≤} knowl- edge base Σ is deﬁned as the smallest set satisfying the following conditions: • ⊤ 1 ∈ clos (Σ) , • for each C ⊑ D an axiom in Σ (role or terminolo gical), { C, D } ⊆ clos ( Σ) , • for every D in clos (Σ) , clos (Σ) contains every subform ula which is a concept ex- pression or a role expression, • if clos (Σ) contains an n -ary relation name, it contains ⊤ n . W e de ﬁne Φ(Σ) as the smallest logic pro gram satisfying t he following conditions: 10 S. Heymans e t al. • For each terminological axiom C ⊑ D ∈ Σ , Φ(Σ) contains the constraint: ← C ( X ) , not D ( X ) (1) • For each role axiom R ⊑ S ∈ Σ where R and S are n -ary , Φ(Σ) con tains: ← R ( X 1 , . . . , X n ) , not S ( X 1 , . . . , X n ) (2) • For each ⊤ n ∈ clos (Σ) , Φ(Σ) c ontains the free rule: ⊤ n ( X 1 , . . . , X n ) ∨ not ⊤ n ( X 1 , . . . , X n ) ← (3) Furthermo re, for each n -ary relation name P ∈ clos (Σ) , Φ(Σ) con tains: ← P ( X 1 , . . . , X n ) , not ⊤ n ( X 1 , . . . , X n ) (4) Intuitively , the latter rule ensures that P I ⊆ ⊤ I n . Add itionally , Φ(Σ) has to contain the constraint: ← not ⊤ 1 ( X ) (5) which ensures that, for every element x in the pre- interpretatio n, ⊤ 1 ( x ) is true in the open an swer set. The latter rule ensures that ⊤ I 1 = D fo r the corresp onding interpretatio n. The rule is implicitly guarded with X = X . • Next, we disting uish b etween the ty pes of concept an d r ole expr essions that appea r in clos (Σ) . For each D ∈ clos (Σ) : — if D is a concep t nominal { o } , Φ(Σ) contain s t he fact: D ( o ) ← (6) This fact ensures that { o } ( x ) ho lds in any open answer set iff x = σ ( o ) = o I for an interpretation of (Σ , P ) . — if D is a concep t name, Φ( Σ) con tains: D ( X ) ∨ n ot D ( X ) ← (7) — if D is an n -ary relation name, Φ(Σ) contain s: D ( X 1 , . . . , X n ) ∨ not D ( X 1 , . . . , X n ) ← (8) — if D = ¬ E for a concept expression E , Φ(Σ) contains the rule: D ( X ) ← not E ( X ) (9) Note that we can again assume that such a rule is guarded by X = X . — if D = ¬ R for an n -ary role expression R , Φ(Σ) contains: D ( X 1 , . . . , X n ) ← ⊤ n ( X 1 , . . . , X n ) , not R ( X 1 , . . . , X n ) (10) Note that if n egation wou ld have been deﬁn ed w .r .t. D n instead of ⊤ I n , we would not be able to write the above as a guar ded rule. — if D = E ⊓ F for concep t e xpressions E and F , Φ(Σ) co ntains: D ( X ) ← E ( X ) , F ( X ) (11) G-Hybrid Knowledge Bases 11 — if D = E ⊓ F for n -ary role e xpressions E and F , Φ(Σ) co ntains: D ( X 1 , . . . , X n ) ← E ( X 1 , . . . , X n ) , F ( X 1 , . . . , X n ) (12) — if D = ($ i/n : C ) , Φ(Σ) conta ins: D ( X 1 , . . . , X i , . . . , X n ) ← ⊤ n ( X 1 , . . . , X i , . . . , X n ) , C ( X i ) (13) — if D = ∃ [$ i ] R , Φ(Σ) co ntains: D ( X ) ← R ( X 1 , . . . , X i − 1 , X , X i +1 , . . . , X n ) (14) The following theorem sho ws that this translation preserves satisﬁability . Theor em 1 Let (Σ , P ) be a g-hyb rid kno wledge base with Σ a D LRO −{≤} knowledge base. Then, a predicate or conce pt expre ssion p is satisﬁ able w .r . t. (Σ , P ) iff p is satisﬁable w .r .t. Φ(Σ) ∪ P . Pr oof ( ⇒ ) Assume p is satisﬁ able w .r .t. (Σ , P ) , i.e., there exists a mod el ( U, I , M ) o f (Σ , P ) , with U = ( D , σ ) , in which p has a n on-em pty e xtension. Now , we c onstruct the open interpretatio n ( V , N ) of Φ(Σ , P ) as follows. V = ( D , σ ′ ) with σ ′ : cts (Φ(Σ) ∪ P ) → D , and σ ′ ( x ) = σ ( x ) for every c onstant symbol x from P and σ ′ ( x ) = x I for every c onstant symbol x from Σ . Note th at σ ′ is well-deﬁned, since, for a constant symbo l x which o ccurs in both Σ and P , we hav e that σ ( x ) = x I . W e deﬁne the set N as f ollows: N = M ∪ { C ( x ) | x ∈ C I , C ∈ clos (Σ) } ∪ { R ( x 1 , . . . , x n ) | ( x 1 , . . . , x n ) ∈ R I , R ∈ clos (Σ) } with C and R con cept expressions and role e xpressions respectively . It is easy to verify that ( V , N ) is an open a nswer set o f Φ(Σ) ∪ P and ( V , N ) satisﬁes p . ( ⇐ ) Assume ( V , N ) is an open answer set of Φ(Σ) ∪ P with V = ( D , σ ′ ) such that p is satisﬁed. W e d eﬁne the interpretation ( U, I , N ) of (Σ , P ) as fo llows. • U = ( D , σ ) wher e σ : cts ( P ) → D with σ ( x ) = σ ′ ( x ) ( note that this is po ssible since cts ( P ) ⊆ cts (Φ(Σ) ∪ P ) ). U is then a pre-in terpretation for P . • I = ( D, · I ) is deﬁned such that A I = { x | A ( x ) ∈ N } fo r conce pt names A , P I = { ( x 1 , . . . , x n ) | P ( x 1 , . . . , x n ) ∈ N } for n -ary role names P and o I = σ ′ ( o ) , for o a constant symbol in Σ ( note that σ ′ is indeed deﬁned on o ). I is then an interp retation of Σ . • M = N \ { p ( ~ x ) | p ∈ clos (Σ) } , such that M is an interp retation of Π( P U , I ) . Moreover , for every constant symb ol b which ap pears in both Σ and P , b I = σ ( b ) . As a consequen ce, ( U , I , M ) is an interp retation of (Σ , P ) . It is easy to verify that ( U, I , M ) is a model of (Σ , P ) which satisﬁes p . Theor em 2 Let (Σ , P ) be a g -hybr id knowledge base where Σ is a D LRO −{≤} knowledge base. Then, Φ(Σ) ∪ P is a guard ed program with a size polynomial in the size of (Σ , P ) . 12 S. Heymans e t al. Pr oof The r ules in Φ(Σ) are obviou sly g uarded . Since P is a guarded p rogra m, Φ(Σ) ∪ P is a guarde d program as well. The size of clos (Σ) is of the o rder n log n where n is the size o f Σ . Intu itiv ely , g iv en that th e size of an expression is n , we have that the size of the set of its subexpressions is at most the size of a tree with depth log n where the size of the subexpr essions at a certain lev el of the tree is at most n . The size of Φ(Σ) is clearly polyno mial in the size of clos (Σ) , assumin g th at the arity n of an added role e xpression is p olyno mial in the size o f the maximal arity of role e xpressions in Σ . If we were to add a relation n ame R with arity 2 n , where n is the maximal arity of relation names in C and Σ , the size of Σ would increase li nearly , but the s ize of Φ(Σ) ∪ P would increase exponentially: o ne needs to add, e.g., rules ⊤ 2 n ( X 1 , . . . , X 2 n ) ∨ not ⊤ 2 n ( X 1 , . . . , X 2 n ) ← which introduce an expo nential number o f arguments while the size of the r ole R does not depend on its arity . Note th at in g-hybr id knowledge bases, we consider D LRO −{≤} , wh ich is D LRO without expressions of the form ≤ k [$ i ] R , since such exp ressions cannot be simulated with guarded pro grams. E .g., consider the concep t expression ≤ 1[$1] R where R is a binary role. One can simulate the ≤ by negation as f ailure: ≤ 1[$1] R ( X ) ← not q ( X ) for some new q , with q deﬁned such that there are at least 2 different R -successors: q ( X ) ← R ( X , Y 1 ) , R ( X , Y 2 ) , Y 1 6 = Y 2 Howe ver , the latter rule is not gu arded – the re is n o atom th at contain s X , Y 1 , and Y 2 . So, in gener al, expressing number restrictions such as ≤ k [$ i ] R is out of r each fo r GPs. From Theorem s 1 and 2 we obtain the following corollary . Cor o llary 1 Satisﬁability check ing w .r .t. g-h ybrid knowledge bases (Σ , P ) , with Σ a DLRO −{≤} knowledge base, can be polynomially reduced to satis ﬁability check ing w .r .t. GPs. Since satisﬁability checking w .r .t. GPs is 2- E X P T I M E -c omplete (Heymans et al. 20 06b), we obtain the same 2- E X P T I M E ch aracterization for g-h ybrid knowledge b ases. W e ﬁrst make e xplicit a corollary of Theorem 1. Cor o llary 2 Let P be a guarded progr am. Then, a concept or role e xpression p is satisﬁable w .r .t. P iff p is satisﬁable w .r .t. ( ∅ , P ) . Theor em 3 Satisﬁability checking w .r .t. g-h ybrid knowledge ba ses where th e DL par t is a D LRO −{≤} knowledge base is 2- E X P T I M E -complete. G-Hybrid Knowledge Bases 13 Pr oof Membership in 2- E X P T I M E follows fro m Corollary 1. Hardness follows from 2- E X P T I M E - hardne ss of satisﬁ ability check ing w .r .t. GPs and the reduction to satisﬁability checking in Corollary 2. 5 Relation with DL + lo g and other Relat ed W ork In (Rosati 20 06), so-called D L + lo g kn owledge bases c ombine a Description Lo gic knowl- edge base with a weakly -safe d isjunctive lo gic p rogram . Formally , fo r a particular Des crip- tion Lo gic D L , a D L + lo g knowledge base is a pair (Σ , P ) where Σ is a D L k nowledge base consisting o f a TBo x (a set of terminolog ical a xioms) and an ABox (a set o f assertional axioms ), and P c ontains rules α ← β suc h that for ev ery rule r : α ← β ∈ P : • α − = ∅ , • β − does not contain DL atoms ( DL-po sitiveness ), • each variable i n r occurs in β + ( Datalog safeness ), and • each variable in r which occur s in a non-D L atom, occur s in a no n-DL atom in β + ( weak safeness ). The semantics for DL + lo g is th e same as that of g-hy brid k nowledge bases 4 , with the following e xceptions: • W e do n ot re quire the stand ar d name assumption , which basically says th at the d o- main of ev ery interpr etation is essentially the same inﬁnitely coun table set of con- stants. Neither do we h ave the implied unique name a ssumption , making the s eman- tics for g-h ybrid knowledge b ases more in line with cur rent Sem antic W eb s tandards such as OWL ( Dean and Schreiber 2004) where neith er the standard names assump- tion nor the unique names assumption ap plies. No te tha t Rosati also presented a version of hybrid knowledge bases which does not adhere to th e un ique na me as- sumption in an earlier w ork (Rosati 2005b). Ho wev er , the ground ing of the p rogra m part is with the constant sym bols explicitly appearing in the program or DL pa rt only , wh ich yields a less tigh t integration of th e program and the DL part than in (Rosati 2006) or in g-hybr id knowledge ba ses. • W e deﬁne an interpretation as a triple ( U, I , M ) instead of a pair ( U, I ′ ) where I ′ = I ∪ M ; this is, howe v er , e quiv alent to DL + lo g . The ke y differences o f the two approach es ar e: • The p rogram s considered in D L + lo g may ha ve multiple positi ve literals in the h ead, whereas we allow at most one. Ho wev er , we allow negative literals in the head, whereas this is not allowed in D L + lo g . Additionally , since DL-ato ms are interp reted classically , we may simulate po siti ve DL-atom s in the head through negative DL- atoms in the body . 4 Strictl y speaking , we did not deﬁne answer sets of disjunct i ve programs, howe v er , the deﬁnitions of Subsection 2.1 can serve for disjunct i ve pr ograms without modiﬁcation. Al so, we did not consider ABo xes in our deﬁniti on of DL s in Subsection 2.2. Ho we ve r , the extension of the semantics to DL knowl edge bases with ABoxes is straight forward . 14 S. Heymans e t al. • Instead of Datalog safeness we require gu ar dedness . Whereas with Datalog safeness ev ery variable in the ru le should ap pear in som e po siti ve atom of the body o f the rule, guarde dness requires that there is a po siti ve atom that contain s e very v ariable in the rule, with the exception of free rules. E.g., a ( X ) ← b ( X ) , c ( Y ) is Datalog safe since X app ears in b ( X ) and Y app ears in c ( Y ) , but this ru le is not g uarded since there is no atom that contains both X and Y . Note that we could easily extend the approa ch taken in this paper to loosely gu ar ded p r ograms wh ich re quire that every two variables in the rule should appear tog ether in a p ositiv e atom , Howev er , this would still be less expressi ve th an Datalog safeness. • W e do no t have the requ irement fo r weak safen ess, i. e., head variables do n ot need to app ear positiv ely in a non-D L atom . The gu ardedn ess may be provid ed by a DL atom. Example 3 Example 1 contains the rule pr oblematic ( X ) ← s o cialDrinker ( X ) , knowsF r o mAA ( X , Y ) This allows to deduc e that X might be a pr oblem case even if X knows someo ne from the AA but is not drinking with th at person. Indeed, as illustrated by the model in Example 1, john is dr inking wine with som e anonymous x a nd k nows michael from the AA. More correct would be the rule pr oblematic ( X , Z ) ← drinks ( X , Y , Z ) , knowsF r omAA ( X , Y ) where we explicitly say that X and Y in th e drinks and knowsF r omAA relations should be the same, and we e xtend the p r o blematic predicate with the kind of drink that X has a problem with. Then, the h ead variable Z is guarded by the DL atom drinks and the rule is thus n ot weakly- safe, but is g uarded no netheless. Thu s, the resulting knowledge base is not a D L + lo g knowledge base, b ut is a g-hybrid knowl- edge base. • W e do no t have the requ irement for DL-p ositiv eness, i.e., DL atoms may app ear negated in the body of rules (and also in the head s of rules). Ho we ver , o ne c ould allow this in DL + lo g knowledge bases as well, since not A ( ~ X ) in the b ody of the rule has the same ef fect as A ( ~ X ) in the head, where the latter is allowed in (Rosati 2006). V ice versa, we can also loosen our restriction on the occurren ce of positive atoms in the head (which allows at most one positive atom in the head ), to allow for an arbitrary number of positi ve DL atoms in th e head (but still keep the number of positive non- DL atoms limited to one) . E. g., a rule p ( X ) ∨ A ( X ) ← β , where A ( X ) is a DL atom, is n ot a valid rule in the progr ams we considered since the head co ntains mor e than o ne positive atom. Howe ver , w e can always rewrite such a rule to p ( X ) ← β , not A ( X ) , wh ich contain s at most one positiv e atom in the head. Arguably , DL atoms sh ould not be allo wed to occ ur negativ ely , because DL pr ed- icates are interpreted classically and thus the negation in front of the DL atom is not nonmonoto nic. Ho we ver , D atalog predicates which depend o n DL predicates are also (partially) interp reted classically , an d DL atom s occurrin g negati vely in the G-Hybrid Knowledge Bases 15 body are equi v alent to DL atoms o ccurring positi vely in the head which allo ws u s to partly overcome our limitation of rule heads to one positi ve a tom. • W e do no t take into account ABoxes in the DL knowledge base. Howe ver , the DL we con sider includes n ominals such that one can simulate th e ABox using termino- logical axioms. Mo reover , e ven if the DL d oes n ot in clude no minals, the ABox can be written as groun d facts in a program and ground facts are al ways guarded. • Decidability for satisﬁability ch ecking 5 of D L + lo g knowledge bases is gu aranteed if decidab ility o f the conjun ctiv e query con tainment pr oblem is guaran teed f or the DL at hand. In contrast, we r elied on a translation of DLs to g uarded progr ams for establishing decidability , and, as explained in the p revious section, not all DLs (e.g. those with numb er restrictions) can be translated to such a GP . W e brieﬂy mention AL -log ( Donini et al. 1998), which is a predecessor of D L + lo g . AL -log co nsiders AL C kn owledge bases f or th e DL p art an d a set of positiv e Horn clauses for the prog ram par t. Every variable must appear in a po siti ve atom in th e body , and concep t names ar e the only DL predica tes which may b e used in the rules, and th ey may on ly b e used in rule bodies. (Hustadt et al. 2004) and (Swift 2004) simulate reasoning in DLs with an LP formalism by using an intermediate translation to ﬁrst-order clauses. In (Hustadt et al. 2004), S HI Q knowledge bases are reduced to ﬁrst-or der formu las, to which the ba sic superpo sition cal- culus is applied . (Swift 2004) translates ALC QI concept expressions to ﬁrst-order for- mulas, grounds them with a ﬁ nite number of constants, and transforms the r esult to a logic progr am. On e can u se a ﬁnite number of constants by the ﬁn ite mod el prop erty o f ALC QI . In the presence of term inologica l axio ms this is no lon ger possible sinc e the ﬁnite mod el proper ty is not guaranteed to hold. In (Levy and Rousset 1996), the DL ALC N R ( R stands for role intersection) is e x- tended with Hor n clau ses q ( ~ Y ) ← p 1 ( ~ X 1 ) , . . . , p n ( ~ X n ) where the variables in ~ Y must appear in ~ X 1 ∪ . . . ∪ ~ X n ; p 1 , . . . , p n are either co ncept or r ole names, o r ordinary pred- icates not a ppearing in the DL part, and q is an ord inary predicate. There is no safeness in the sen se th at every variable must appear in a n on-DL ato m. The sema ntics is d eﬁned throug h e xtended interpr etations that satisfy both the DL and clauses p art (as FOL fo rmu- las). Query answer ing is un decidable if re cursive Horn clauses are allowed, but decidabil- ity can be regaine d by restricting the DL part o r by enf orcing that the clauses are role safe (each variable in a role atom R ( X , Y ) for a r ole R must appear in a non- DL atom). Note that the latter restriction is less strict than the DL-safeness 6 of (Motik et al. 20 04), where also v ariables in concept atoms A ( X ) need to appear in n on-DL atoms. On the other h and, (Motik et al. 2004) a llows fo r the m ore expressiv e DL S H OI N ( D ) , and the head pred i- cates m ay b e DL atoms as well. Finally , SWR L (Horrocks and P atel-Schneider 2004b) c an be seen as an extension of (Mo tik et al. 2004) witho ut any safen ess re striction, which re- sults in the loss of decidability of the fo rmalism. Comp ared to our w ork, we consider a slightly les s expressiv e Description Logic, b ut we consider lo gic programs with n onmo no- 5 (Rosati 2006) c onsiders checking sati sﬁabilit y of k no wledge ba ses ra ther than sat isﬁabil ity of pre dicat es. How- e ver , the former can easily be reduced to the latte r . 6 DL-safeness is a restricti on of the earlier m entione d weak safen ess. 16 S. Heymans e t al. tonic negation , and req uire gu ardedn ess, r ather than role- o r DL-safen ess, to guarantee decidability . In (Eiter et al. 2004) Description Logic pr o grams are introdu ced; atoms in the pr ogram compon ent m ay be dl-atoms with whic h one can query the kn owledge in th e DL com- ponen t. Such dl-atoms m ay specify inf ormation from the logic program which needs to be taken in to acc ount whe n ev aluating the q uery , yieldin g a b i-direction al ﬂo w of infor- mation. This leads to a minimal interface between the DL knowledge ba se and the logic progr am, enabling a very loose integration, based o n an entailment relation . I n co ntrast, we propose a much tigh ter integration betwee n the rules and the ontology , with interaction based on single models rather th an entailment. For a d etailed discussion of these tw o kinds of interaction, we refer to (de Bruijn et al. 2006). T wo recent ap proach es (Motik and Rosati 2007; de Bruijn et al. 2007) use an embed- ding in a nonmo notonic modal lo gic for integrating n onmon otonic logic programs an d ontolog ies based on classical logic (e.g. D L). (M otik and Rosati 2007) use th e no nmono - tonic log ic of Minim al Knowledge and Negation as Failure (MKNF) f or the comb ination, and show d ecidability of re asoning in case rea soning in the co nsidered descrip tion lo gic is decidable, and the DL safeness cond ition (M otik et al. 2004) holds for the rules in the logic pro gram. In our appr oach, we do n ot requ ire such a safen ess condition, but require the rules to be guar ded , and make a semantic distinc tion b etween DL predic ates and rule pre d- icates. (de Bruijn et al. 2007) introduce se veral em beddin gs of non- groun d logic progr ams in ﬁrst-orde r autoepistemic logic (FO-AEL), and comp are them und er combinatio n with classical t heories (ontolog ies). Howev er , (de Bruijn et al. 2007) do not address t he issue o f decidability or reasoning of such combinatio ns. Finally , (d e Bruijn et al. 2006) use Qu antiﬁed Equilibr ium Logic as a single unify ing languag e to capture dif ferent ap proach es to hybrid knowledge bases, including the ap- proach presented in this paper . Altho ugh we have presented a tran slation o f g-hy brid knowledge bases to guarded log ic p rogram s, our direct semantics is still b ased o n two modules, relying on separate interpr etations for the DL knowledge b ase and the logic pro- gram, wh ereas (de Bruijn et al. 2006) deﬁn e eq uilibrium mo dels, wh ich ser ve to give a unifyin g semantics to th e hy brid knowledge base. Th e app roach of (de Bruijn et al. 200 6) may b e used to d eﬁne a n otion of equ iv alence b etween, and ma y lead to new algo rithms for reasoning with, g-hybr id k nowledge bases. 6 Conclusions and Directions f or Further Research W e de ﬁned g-hyb rid kno wledge bases which combine Description Logic (DL) knowledge bases with guarded lo gic progr ams. In particu lar , we combin ed k nowledge bases of the DL D LRO −{≤} , which is close to OWL DL, with gu arded progr ams, and showed decid- ability of this fram ew ork by a r eduction to guar ded pro grams under the open answer set semantics (Heymans et al. 2005a; Heyman s et al. 2006b). W e discussed the relation with DL + lo g knowledge bases: g-hybrid knowledge b ases ov ercome som e of the limitations o f DL + lo g , such as the unique n ames assumption, Datalog safeness, and weak DL-safeness, but introd uce the r equireme nt of guardedness. At p resent, a signiﬁcan t disadv antage of our approa ch is the lack of suppo rt fo r DLs with num ber restrictio ns which is inher ent to the use of guar ded p rogram s as our decid ability veh icle. A solution for this would be to con- G-Hybrid Knowledge Bases 17 sider other typ es of prog rams, such as con ceptual logic pr ogr ams (Heymans et al. 2006a). This would allow for the deﬁnition o f a hybrid kn owledge b ase (Σ , P ) wher e Σ is a S HI Q knowledge base and P is a co nceptual log ic progr am sin ce S HI Q knowledge bases can be translated to conceptu al logic programs. Although the re are known complexity boun ds for several fragmen ts of open an swer set progr amming (O ASP), inclu ding th e guarde d frag ment con sidered in this p aper, there are no known effecti ve algorithm s f or O ASP . Addition ally , at p resence, there are n o imple- mented systems for open answer set program ming. Th ese are part of future work. 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Under conside ratio n for public ation in Theory and Practice of Logic Pro grammi ng 1 Guar ded Hybrid Knowledge Bases ∗ STIJN HEYMANS 1 , JOS DE BR UIJN 2 , LIVIA PREDOIU 3 , CRISTIN A FEIER 1 , D A VY V AN NIEUWENBORGH 4 † 1 Digital Enterprise Resear ch Insti tute , Univer sity of Innsbruck, T ech nik erstr asse 21a, Innsbruc k, Austria ( e-mail: { stijn.heymans ,cristina.feie r } @deri.org ) 2 F aculty of Computer Science , F r ee Uni versit y of Bozen-Bolzano, Ital y ( e-mail: debruijn@in f.unibz.it ) 3 Institut e of Compute r Science , Univer sity of Mannheim, A5, 6 68159 Mannheim, Germany ( e-mail: livia@infor matik.uni-mannh eim.de ) 4 Dept. of Computer Scienc e, Vrije Universitei t Brussel, VUB, Pleinlaa n 2, B1050 Brussels, Belgiu m ( e-mail: dvnieuwe@vu b.ac.be ) submitte d 20 J une 2006; r e vised 3 J anuary 2007; acce pted 18 October 2007 Abstract Recently , there has been a lot of interest in the integration of Description Logics and rules on the Semantic W eb . W e deﬁne guar ded hybrid knowledg e bases (or g-hybrid knowledg e bases ) as kno wl- edge bases that c onsist of a Desc ription Logic kno wled ge base and a guar ded logic program , similar to t he D L + l o g kno wledge bases from ( ? ). G-hybrid knowledge bases enable an integration of De- scription Logics and Logic Programming where, un like in o ther approaches, v ariables in the rules of a guarded program do not need to appear in positiv e non-DL atoms of the body , i.e. DL atoms can act as guar ds as well. Decidability of satisﬁability checking of g-hybrid kno wledg e bases is shown for the particular DL DLRO −{≤} , which is close to O WL DL, by a reduction to guarded progra ms under the o pen answer set semantics. Moreo v er , we sho w 2- E X P T I M E -completeness for satisﬁability checking of such g-hybrid kno wledge ba ses. Finally , we discuss adv antag es and disadv antages of ou r approach compared with DL + l o g kno wledge bases. KEYWORDS : g-hybrid kno wledge bases, open answer set programming, guarded logic programs, description logics 1 Introduction The integration of Descr iption Logics with rules has recently received a lot of attention in the context of the Semantic W eb ( ? ; ? ; ? ; ? ; ? ; ? ; ? ). R-hybrid kn owledge bases ( ? ), and its extension D L + lo g ( ? ), is an ele gant for malism b ased on co mbined m odels for Description ∗ A preliminary version of this paper appe ared in the proceedin gs of the ICLP’06 W orkshop on Applica tions of Logic P r ogr amming in the Semantic W eb and Semantic W eb Servi ces (ALPSWS2006) pages 39-54, Seattle , W ashington, USA, August 16 2006. † The work is funded by the European Commission under t he projects ASG, DIP , enIRaF , InfraW ebs, Knowle dge W eb, Musing, Saler o, SEKT , SEEMP , SemanticGO V , Super , SW ING and TripCom; by Scienc e Foundatio n Ireland under the DERI-Lion Grant No.SFI/02/CE1/I13 ; by the F FG ( ¨ Osterreic hische Forschun gsfrderungs- geselle schaft mbH) under the project s Grisino, R W 2 , SemNetMan, SEnSE , TSC and OnT ourism. Da vy V an Nieuwenbo rgh was supported by the Flemish Fund for Scientiﬁc Research (FWO-Vlaan deren). 2 S. Heymans e t al. Logic knowledge bases and nonmono tonic logic progr ams. W e propo se a variant o f r - hybrid kn owledge bases, called g-h ybrid kno wledge ba ses , which do n ot require standard names or a special saf eness restriction on rules, b ut instead require the p rogram to be guarded . W e sho w sev eral comp utational prop erties by a reduction to gu arded open answer set progr amming ( ? ; ? ). Open answer set programm ing (O ASP) ( ? ; ? ) combines the log ic progr amming and ﬁrst-order logic paradig ms. From the logic progr amming paradigm it inherits a rule-b ased presentation and a non mono tonic sema ntics by means of negation as failure. In contrast with usual logic p rogram ming semantics, such as th e answer set seman tics ( ? ), O ASP al- lows fo r domains consisting o f other objects than those present in th e logic progr am at hand. Such open domains a re inspired by ﬁrst-order lo gic based languages such as De- scription Logics (DLs) ( ? ) and make OASP a v iable can didate for conceptu al reason ing. Due to its rule-based pr esentation and its support fo r non monoto nic reasonin g and ope n domains, OASP can be used to reaso n with bo th rule-based and concep tual knowledge on the Semantic W eb, as illustrated in ( ? ) . A major challen ge for O ASP is to control undecidability of satisﬁability ch ecking, a challenge it shares with DL-based languag es. In ( ? ; ? ), we identify a decidab le class of progr ams, the so-called guarded p r ograms , for which decidability of satisﬁability check ing is obtain ed by a tra nslation to guar ded ﬁxed poin t logic ( ? ). In ( ? ), we show the expres- si veness of such gu arded p rogram s by simulating a DL with n -ary roles and nom inals. In particular, we extend the DL D LR ( ? ) with both co ncept nominals { o } and r ole nominals { ( o 1 , . . . , o n ) } , resulting in D LRO . W e den ote the DL DLRO witho ut nu mber restric- tions as DLRO −{≤} . Satisﬁability checking of concept expressions w .r .t. DLRO −{≤} knowledge bases can be reduced to checking satisﬁability of guarded programs ( ? ). A g-hyb rid knowledge base c onsists of a Description Log ic knowledge base an d a guarde d p rogram . The DL + lo g knowledge bases fro m ( ? ) are weakly safe , which means that the interaction between the pr ogram an d the DL knowledge ba se is restricted by re- quiring that variables which appear in no n-DL atoms, ap pear in positi ve non -DL ato ms in the bod y , w here DL atom s ar e atom s inv olving a concep t or role sym bol f rom the D L knowledge base. G-hybrid kn owledge bases do no t require such a restriction; in stead, v ari- ables must appear in a gua r d of the rule, but th is gua rd can be a D L atom as well. I n this paper , we sho w d ecidability of g-hybrid knowledge bases f or DLRO −{≤} knowledge bases by a reduc tion to gu arded p rogram s, a nd show that satis ﬁability ch ecking o f g- hybrid knowledge bases is 2- E X P T I M E -co mplete. The DL DLRO −{≤} is close to S HO I N , the Description Logic underlying O WL DL ( ? ). Compared with S HOI N , D LRO −{≤} does not include transitiv e roles and number restrictions, but does i nclude n -ary roles and com- plex role e xpressions. T o see why a combin ation of rules and ontolo gies, as prop osed in g-hyb rid kn owledge bases, is usef ul, and why the safeness cond itions considered so far in the literatur e are not approp riate in all scenarios, consider the Description Logic ontolog y F r aternityMe mb er ⊑ Drinker ⊓ ∃ hasDrinkingBuddy . F r aternityMemb er which says that fraternity members are drinkers who have drinkin g buddies, whic h are also G-Hybrid Knowledge Bases 3 fraternity members. Now c onsider the logic progra m pr oblemDrinker ( X ) ← Drinker ( X ) , not so cialDrinker ( X ) so cialDrinker ( X ) ← Drinker ( X ) , not pr oblemDrinker ( Y ) , hasDrinkingBuddy ( X , Y ) F r aternityMe mb er ( John ) ← which says th at drin kers are by default pro blem drinkers, unless it is known that they are social drinkers; dr inkers with drin king b uddies who are no t problem drinkers are social drinkers; and Joh n is a fra ternity mem ber . From the combin ation of the ontolog y and the logic pro gram, one would expect to deriv e that Joh n is a social d rinker, and not a prob- lem drinker . This logic p rogram cannot b e expressed using r-hybrid knowledge bases, or DL + lo g , b ecause the rules in the pro gram are not weakly safe . Howe ver , the logic p ro- gram is guar ded , and thus part of a v alid g -hybr id knowledge base, which has th e e xpected consequen ces. The remainde r of the paper starts with an introduc tion to open answer set programmin g and De scription Logics in Section 2. Section 3 deﬁnes g-h ybrid knowledge bases, tra nslates them to gu arded progra ms when th e DL D LRO −{≤} is co nsidered, an d p rovides a com- plexity characteriz ation for satisﬁability che cking of these particular g-hybr id knowledge bases. In Section 5, we discuss the relation of g-hybrid kn owledge bases with DL + lo g and other related work. W e conclud e and gi ve directions for further research in Section 6. 2 Preliminaries In this sectio n we introduc e Open Answer Set Program ming, guard ed pro grams, and the Description Logic DLRO −{≤} . 2.1 Decidab le Open An swer Set Programming W e introd uce the open answer set semantics from ( ? ; ? ), m odiﬁed as in ( ? ) such th at it does not assume un iqueness of names b y default. Con stants , variables , terms , and atoms are deﬁned as usual. A literal is an atom p ( ~ t ) o r a n af-literal n ot p ( ~ t ) , with ~ t a tup le of terms. 1 The po sitive part of a set of literals α is α + = { p ( ~ t ) | p ( ~ t ) ∈ α } and the n e gative part of α is α − = { p ( ~ t ) | not p ( ~ t ) ∈ α } . W e a ssume the existence of the (in)eq uality predicates = and 6 = , u sually written in inﬁx nota tion; t = s is an atom and t 6 = s is short f or not t = s . A r e g ular atom is an ato m with out e quality . For a set A o f ato ms, not A = { not l | l ∈ A } . A pr ogram is a coun table set of rules α ← β , where α and β are ﬁnite sets of literals, | α + | ≤ 1 (b ut α − may b e of arbitrar y size), and every a tom in α + is regu lar , i.e. α contain s at most one po siti ve atom, which may not con tain the equality pr edicate. 2 The set α is the head o f the rule and repr esents a disjunction of literals, while β is the body and re presents 1 W e do not allo w “classical” negatio n ¬ , ho we ve r , programs with ¬ ca n be reduced to programs without it, see e.g. ( ? ). 2 The condit ion | α + | ≤ 1 makes the GL-reduct non-disjunct i ve, ensuring that the immediate consequence operat or is well-de ﬁned, s ee ( ? ). 4 S. Heymans e t al. a conjunction of literals. If α = ∅ , the rule is called a constraint . F r ee rules are rules of th e form q ( ~ X ) ∨ n ot q ( ~ X ) ← ; they enable a choice for the inclusion of atoms in a model. W e call a pr edicate p fr ee if there is a free ru le p ( ~ X ) ∨ n ot p ( ~ X ) ← . Atoms, literals, rules, an d progr ams that do not contain v ariables are gr ou nd . For a literal, rule, or program o , let cts ( o ) , vars ( o ) , pr e ds ( o ) b e the con stants, v ariables, and predicate s, respectively , in o . A p r e-interpr etation U fo r a pro gram P is a pair ( D , σ ) where D is a n on-em pty do main and σ : cts ( P ) → D is a f unction which maps all constants in P to elemen ts fro m D . 3 P U is the gro und p rogram obtained fro m P b y sub - stituting e v ery variable in P with e very possible element from D and e very constant c with σ ( c ) . E.g., f or a ru le r : p ( X ) ← f ( X , c ) and U = ( { x, y } , σ ) where σ ( c ) = x , we h av e that the ground ing w .r .t. U is: p ( x ) ← f ( x , x ) p ( y ) ← f ( y , x ) Let B P be the set of re gular atom s o btained f rom the languag e of th e ground prog ram P . An interpr etation I of a gr ound progra m P is a subset of B P . For a groun d regular atom p ( ~ t ) , we write I | = p ( ~ t ) if p ( ~ t ) ∈ I ; for an e quality atom t = s , we wr ite I | = t = s if s a nd t are equal terms. W e write I | = not p ( ~ t ) if I 6| = p ( ~ t ) , for p ( ~ t ) an atom. For a set of groun d literals A , I | = A holds if I | = l for every l ∈ A . A ground rule r : α ← β is satisﬁed w .r .t. I , deno ted I | = r , if I | = l for some l ∈ α when ev er I | = β . A g roun d constraint ← β is satisﬁed w .r .t. I if I 6| = β . For a ground progr am P withou t no t , an interpretation I of P is a mod el of P if I satisﬁes e very ru le in P ; it is an answer set of P if it is a subset minim al model o f P . For groun d programs P containing n ot , the reduct ( ? ) w .r .t. I is P I , whe re P I consists of α + ← β + for ev ery α ← β in P such that I | = not β − and I | = α − . I is an answer set of P if I is an answer set o f P I . Note that allowing negation in the head of rules leads to the loss of the anti-chain pr operty ( ? ) which states that no answer set can be a strict subset of ano ther answer set. E.g , a r ule a ∨ not a ← h as the answer sets ∅ and { a } . Howe ver , negation in the head is required to ensure ﬁrst-order b ehavior for certain pr edicates, e. g., when simulating Description Logic reasoning. In the following, a progra m is assumed to be a ﬁnite set of rules; inﬁnite p rogram s only appear as b yprod ucts of ground ing a ﬁnite program u sing an in ﬁnite pre-interpretation . An open interpretation of a p rogram P is a pair ( U, M ) where U is a pre-interp retation for P and M is an interpretation of P U . An open answer set of P is an open interp retation ( U, M ) of P with M an answer set of P U . An n -ary predicate p in P is satisﬁable if there is an open an swer set (( D , σ ) , M ) o f P and a ~ x ∈ D n such that p ( ~ x ) ∈ M . A pro gram P is satisﬁable i ff it has an open an swer set. Note that satisﬁability checking o f programs can be easily reduce d to satisﬁability checking of predicates: P is satisﬁable iff p is satisﬁable w .r .t. P ∪ { p ( ~ X ) ∨ not p ( ~ X ) ←} , where p is a pred icate symbo l no t used in P and ~ X is a tuple of v ariables. In the follo wing, when we s peak of satisﬁability checking, we refer to satisﬁability checking of predicates, unless speciﬁed otherwise. 3 In ( ? ), we only use the domain D whic h is there deﬁned as a non-empty superset of the constants in P . This correspond s to a pre-i nterpre tation ( D, σ ) w here σ is the iden tity function on D . G-Hybrid Knowledge Bases 5 Satisﬁability checking w .r .t. the open answer set seman tics is und ecidable in gen eral. In ( ? ) , we id entify a syntactically restricted fragm ent of pro grams, so-called gu ar ded pr o- grams , for which satisﬁability ch ecking is decidab le, which is shown thr ough a reduction to guard ed ﬁxed poin t logic ( ? ). T he decidab ility o f guard ed p rogra ms relies on the pres- ence of a guar d in each rule, wher e a guard is an atom that contains all variables of the rule. Formally , a rule r : α ← β is guarded if there is an atom γ b ∈ β + such that vars ( r ) ⊆ vars ( γ b ) ; γ b is th e guard of r . A pro gram P is a gua r ded pr ogram (GP) if ev- ery non-free rule in P is guarded . E. g., a rule a ( X, Y ) ← not f ( X , Y ) is not guarded, b ut a ( X , Y ) ← g ( X , Y ) , not f ( X, Y ) is gu arded with guard g ( X , Y ) . Satisﬁability ch ecking of p redicates w .r .t. g uarded pro grams is 2 - E X P T I M E -co mplete ( ? ) – a result that stems fr om the correspon ding complexity in gua rded ﬁxed point logic. 2.2 T he Description Logic D LRO −{≤} DLR ( ? ; ? ) is a DL wh ich sup ports roles of ar bitrary ar ity , whereas most DLs only sup port binary r oles. W e intro duce an extensio n of D LR with nomina ls, called D LRO ( ? ). Th e basic building bloc ks of D LRO ar e co ncept names A and relation names P where P denotes an arbitrary n - ary relation f or 2 ≤ n ≤ n max and n max is a gi ven ﬁnite non - negativ e in teger . Role expressions R and concept expressions C ar e deﬁned as: R → ⊤ n | P | ($ i/n : C ) | ¬ R | R 1 ⊓ R 2 | { ( o 1 , . . . , o n ) } C → ⊤ 1 | A | ¬ C | C 1 ⊓ C 2 | ∃ [$ i ] R | ≤ k [$ i ] R | { o } where i is between 1 and n in ($ i/n : C ) ; similarly in ∃ [$ i ] R and ≤ k [$ i ] R f or R an n -ary relation. Moreover, we a ssume that the above constructs are well-typed , e.g. , R 1 ⊓ R 2 is deﬁned only for relations of the same arity . The semantics of D LRO is given by interpre- tations I = (∆ I , · I ) where ∆ I is a no n-empty set, the domain , and · I is an interpr etation function such that C I ⊆ ∆ I , R I ⊆ (∆ I ) n for an n -ary relatio n R , and the following condition s ar e satisﬁed ( P , R , R 1 , and R 2 have arity n ): ⊤ I n ⊆ (∆ I ) n P I ⊆ ⊤ I n ( ¬ R ) I = ⊤ I n \ R I ( R 1 ⊓ R 2 ) I = R I 1 ∩ R I 2 ($ i/n : C ) I = { ( d 1 , . . . , d n ) ∈ ⊤ I n | d i ∈ C I } ⊤ I 1 = ∆ I A I ⊆ ∆ I ( ¬ C ) I = ∆ I \ C I ( C 1 ⊓ C 2 ) I = C I 1 ∩ C I 2 ( ∃ [$ i ] R ) I = { d ∈ ∆ I | ∃ ( d 1 , . . . , d n ) ∈ R I . d i = d } ( ≤ k [$ i ] R ) I = { d ∈ ∆ I | |{ ( d 1 , . . . , d n ) ∈ R I | d i = d }| ≤ k } { o } I = { o I } ⊆ ∆ I { ( o 1 , . . . , o n ) } I = { ( o I 1 , . . . , o I n ) } 6 S. Heymans e t al. Note that in D LRO the negation of role expressions is deﬁned w .r .t. ⊤ I n and not w .r .t. (∆ I ) n . A DLRO knowledge base Σ is a set of termino logical axioms and role ax ioms, which de- note subset relations be tween concept and ro le expressions ( of the same a rity), respectively . A te rminolog ical axiom C 1 ⊑ C 2 is satisﬁed by I iff C I 1 ⊆ C I 2 . A role axiom R 1 ⊑ R 2 is satisﬁed by I iff R I 1 ⊆ R I 2 . An interpretatio n I is a model of a knowledge base Σ (i.e. Σ is satisﬁed by I ) if all ax ioms in Σ ar e satisﬁed by I ; if Σ has a model, then Σ is satisﬁable . A con cept expression C is satisﬁable w .r .t. a knowledge base Σ if ther e is a model I of Σ such that C I 6 = ∅ . Note that for ev ery interpretation I , ( { ( o 1 , . . . , o n ) } ) I = (($1 /n : { o 1 } ) ⊓ . . . ⊓ ($ n/n : { o n } )) I . Therefo re, in the remainder of th e paper , we will restrict ourselves to nom inals of the f orm { o } . W e denote the fragme nt of DLRO with out the numb er restriction ≤ k [$ i ] R with DLRO −{≤} . 3 G-hybrid Knowledge Bases G-hybr id knowledge bases are com binations o f Description Log ic (DL ) knowledge bases and guarded lo gic p rogram s (GP). They are a v ariant o f the r -hybrid kn owledge bases introdu ced in ( ? ). Deﬁnition 1 Giv en a Descriptio n Logic DL , a g -hybrid kn owledge base is a pair (Σ , P ) wh ere Σ is a DL knowledge base and P is a guar ded program. Note that in the above deﬁnition there are n o restrictions on th e use of predica te symbols. W e call the atom s and literals in P that h ave un derlying pred icate symbols which cor- respond to concept o r role names in the DL k nowledge base DL atoms an d DL literals , respectively . V a riables i n rules are not required to appear in positi ve non-DL atoms, which is the ca se in , e.g., the DL + lo g knowledge b ases in ( ? ), the r-hybrid kn owledge bases in ( ? ), and th e DL-saf e rules in ( ? ). DL -atoms can a ppear in the h ead of r ules, thereb y en- abling a bi-direc tional ﬂow o f in formation between the DL kn owledge base an d th e logic progr am. Example 1 Consider the D LRO −{≤} knowledge b ase Σ wher e so cialDrinker is a co ncept, dri nks is a ternary role such th at, intuitively , ( x, y , z ) is in the interpr etation of drinks if a person x drinks some drink z with a perso n y . Σ consists of the single axiom so cialDrinker ⊑ ∃ [$ 1 ]( drinks ⊓ ($ 3 / 3 : { wine } )) which ind icates that social drinkers dr ink wine with someon e. Consider a GP P that indicates that someone has an increased risk of alc oholism if the person is a social drinker and knows someo ne f rom the association of Alcoh olics Anonymous (AA). Fur thermor e, we state that john is a social drinker and knows michael from AA: pr oblematic ( X ) ← s o cialDrinker ( X ) , knowsF r o mAA ( X , Y ) knowsF r omA A ( john , michael ) ← so cialDrinker ( john ) ← G-Hybrid Knowledge Bases 7 T o gether, Σ and P for m a g-hy brid knowledge base. The literals so cialDrinker ( X ) and so cialDrinker ( john ) are DL ato ms wher e the latter appears in the hea d of a ru le in P . The literal knowsF r omAA(X,Y) appears only in the program P (and is thus not a DL atom). Giv en a DL interpretation I = (∆ I , · I ) and a grou nd program P , we deﬁne Π( P, I ) as the pr ojection of P with respect to I , which is obta ined as follo ws: for e very rule r in P , • if there exists a DL literal in the head of the form — A ( ~ t ) with ~ t ∈ A I , or — not A ( ~ t ) with ~ t 6∈ A I , then delete r , • if there exists a DL literal in the body of the form — A ( ~ t ) with ~ t 6∈ A I , or — not A ( ~ t ) with ~ t ∈ A I , then delete r , • otherwise, delete all DL literals from r . Intuitively , th e pro jection “ev aluates” the progr am with respect to I by removin g ( ev alu- ating) rules and DL literals consistently with I ; concep tually this is similar to the r educt, which removes r ules and negative literals consistently with an in terpretation of the pro- gram. Deﬁnition 2 Let (Σ , P ) b e a g-hybrid kno wledge base. An interpretation o f (Σ , P ) is a tuple ( U, I , M ) such that • U = ( D, σ ) is a pr e-interpr etation for P , • I = ( D, · I ) is an interpr etation of Σ , • M is an interpr etation of Π( P U , I ) , and • b I = σ ( b ) f or ev ery constant symbol b appearin g both in Σ and in P . Then, ( U = ( D , σ ) , I , M ) is a model o f a g-hyb rid k nowledge base (Σ , P ) if I is a model of Σ and M is an answer set of Π( P U , I ) . For p a con cept expression from Σ o r a pred icate from P , we say th at p is satisﬁable w .r .t. (Σ , P ) if there is a model ( U, I , M ) su ch that p I 6 = ∅ or p ( ~ x ) ∈ M fo r some ~ x from D , respectively . Example 2 Consider the g- hybrid knowledge base in Examp le 1. T ake U = ( D , σ ) with D = { j ohn, michael , w ine, x } and σ the identity fun ction on the constant symbo ls in (Σ , P ) . Further- more, deﬁne · I as follows: • so cialDrinker I = { john } , • drinks I = { ( john , x , wine ) } , • wine I = wine . If M = { knowsfr omAA ( john , michael ) , pr oblematic ( john ) } , t hen ( U, I , M ) is a model of th is g -hybr id knowledge base. No te that the projection Π( P , I ) doe s n ot co ntain th e ru le so cialDrinker ( john ) ← . 8 S. Heymans e t al. 4 T ranslation to Guarded Logic Programs In th is section we introd uce a translation of g -hyb rid knowledge bases to g uarded logic progr ams (GP) under th e o pen answer set semantics, show tha t th is tran slation p reserves satisﬁability , and use this translation to obtain co mplexity r esults for reasoning i n g-hybrid knowledge bases. Before intr oducin g the tran slation to gu arded pro grams f ormally , we introdu ce the translation t hroug h an example. Consider the knowledge b ase in Example 1. The axiom so cialDrinker ⊑ ∃ [$ 1 ]( drinks ⊓ ($ 3 / 3 : { wine } )) translates to the constraint ← so cialDrinker ( X ) , not ( ∃ [$ 1 ]( drinks ⊓ ($ 3 / 3 : { wine } )))( X ) Thus, th e concept expressions on either side of the ⊑ symbol ar e associated with a new unary p redicate name. F or con venience, we name the new p redicates according to the orig- inal co ncept e xpressions. Th e constraint simulates the beh avior of the DLRO −{≤} axiom. If the left- hand side of th e axiom hold s and the r ight-han d side d oes not h old, there is a contradictio n. It remains to ensure that those newly introduced predic ates behav e according to th e DL semantics. First, all the concept and role names occu rring in the axiom above need to b e deﬁned as fr ee predicates, in order to simulate the ﬁrst-order semantics of co ncept and r ole names in DLs. I n DLs, a tuple is eit her true or fals e in a g iv en interpretation (cf. the law of the excluded middle); this beha vior can be captured e xactly by the free predicates: so cialDrinker ( X ) ∨ not so cia lDrinker ( X ) ← drinks ( X , Y , Z ) ∨ not dri nks ( X , Y , Z ) ← Note that concep t names are translated to u nary free p redicates, while n - ary role names are translated to n -ary free predicates. The d eﬁnition of the truth symbols ⊤ 1 and ⊤ 3 which a re imp licit in ou r D LRO −{≤} axiom (since the axio m co ntains a conc ept n ame and a ter nary role) are tr anslated to free predicates as we ll. Note that we do not need a pr edicate for ⊤ 2 since the axiom do es not contain binary predicates. ⊤ 1 ( X ) ∨ not ⊤ 1 ( X ) ← ⊤ 3 ( X , Y , Z ) ∨ n ot ⊤ 3 ( X , Y , Z ) ← W e en sure that, for the ternary DLRO −{≤} role drinks , drin k s I ⊆ ⊤ I 3 holds by adding the constraint: ← drinks ( X , Y , Z ) , not ⊤ 3 ( X , Y , Z ) T o ensure that ⊤ I 1 = ∆ I , we add the constraint: ← not ⊤ 1 ( X ) For rules contain ing only one variable, we ca n alw ays assum e that X = X is in the body and a cts as the gu ard of the rule, so th at th e latter r ule is gua rded; cf. the equiv alent rule ← not ⊤ 1 ( X ) , X = X . G-Hybrid Knowledge Bases 9 W e tran slate the nominal { wine } to the rule { wine } ( wine ) ← Intuitively , since this r ule will be the only rule with the predicate { wine } in the head, every open answer set of the tr anslated pr ogram will contain { wine } ( x ) with σ ( wine ) = x if and only if the correspond ing interpretation { wine } I = { x } for wine I = x . The D LRO −{≤} role expression ($3 / 3 : { wine } ) indicates the tern ary tuples f or which the thir d argument belongs to the e xtension of { win e } , w hich is translated to the follo wing rule: ($ 3 / 3 : { wine } )( X , Y , Z ) ← ⊤ 3 ( X , Y , Z ) , { wine } ( Z ) Note that the above rule is guarded by the ⊤ 3 literal. Finally , the concept expression ( drinks ⊓ ($ 3 / 3 : { wine } )) can be rep resented by the following rule: ( drinks ⊓ ($ 3 / 3 : { wine } ))( X , Y , Z ) ← drinks ( X , Y , Z ) , ($3 / 3 : { wine } )( X , Y , Z ) As we can see, the DL construct ⊓ is translated to con junction in the body of a rule. The D LRO −{≤} role ∃ [$1]( dr ink s ⊓ ($ 3 / 3 : { wine } )) can be rep resented using the following rule: ( ∃ [$1]( dri nk s ⊓ ($ 3 / 3 : { wine } )))( X ) ← ( drink s ⊓ ($3 / 3 : { wine } ))( X , Y , Z ) Indeed , the elemen ts which be long to the extension of ∃ [$1 ]( drink s ⊓ ($3 / 3 : { wine } )) are exactly those that are connected to the role ($3 / 3 : { wine } ) , as speciﬁed in the rule. This conclu des the translation of the DL kn owledge base in the g -hybr id k nowledge base of Exam ple 1 . The pr ogram can be co nsidered as is, since, by deﬁnitio n o f g-hy brid knowledge bases, it is already a guarded program. W e no w proceed with the f ormal tran slation. The closure clos (Σ) of a D LRO −{≤} knowl- edge base Σ is deﬁned as the smallest set satisfying the following conditions: • ⊤ 1 ∈ clos (Σ) , • for each C ⊑ D an axiom in Σ (role or terminolo gical), { C, D } ⊆ clos ( Σ) , • for every D in clos (Σ) , clos (Σ) contains every subform ula which is a concept ex- pression or a role expression, • if clos (Σ) contains an n -ary relation name, it contains ⊤ n . W e de ﬁne Φ(Σ) as the smallest logic pro gram satisfying t he following conditions: • For each terminological axiom C ⊑ D ∈ Σ , Φ(Σ) contains the constraint: ← C ( X ) , not D ( X ) (1) • For each role axiom R ⊑ S ∈ Σ where R and S are n -ary , Φ(Σ) con tains: ← R ( X 1 , . . . , X n ) , not S ( X 1 , . . . , X n ) (2) • For each ⊤ n ∈ clos (Σ) , Φ(Σ) c ontains the free rule: ⊤ n ( X 1 , . . . , X n ) ∨ not ⊤ n ( X 1 , . . . , X n ) ← (3) 10 S. Heymans e t al. Furthermo re, for each n -ary relation name P ∈ clos (Σ) , Φ(Σ) con tains: ← P ( X 1 , . . . , X n ) , not ⊤ n ( X 1 , . . . , X n ) (4) Intuitively , the latter rule ensures that P I ⊆ ⊤ I n . Add itionally , Φ(Σ) has to contain the constraint: ← not ⊤ 1 ( X ) (5) which ensures that, for every element x in the pre- interpretatio n, ⊤ 1 ( x ) is true in the open an swer set. The latter rule ensures that ⊤ I 1 = D fo r the corresp onding interpretatio n. The rule is implicitly guarded with X = X . • Next, we disting uish b etween the ty pes of concept an d r ole expr essions that appea r in clos (Σ) . For each D ∈ clos (Σ) : — if D is a concep t nominal { o } , Φ(Σ) contain s t he fact: D ( o ) ← (6) This fact ensures that { o } ( x ) ho lds in any open answer set iff x = σ ( o ) = o I for an interpretation of (Σ , P ) . — if D is a concep t name, Φ( Σ) con tains: D ( X ) ∨ n ot D ( X ) ← (7) — if D is an n -ary relation name, Φ(Σ) contain s: D ( X 1 , . . . , X n ) ∨ not D ( X 1 , . . . , X n ) ← (8) — if D = ¬ E for a concept expression E , Φ(Σ) contains the rule: D ( X ) ← not E ( X ) (9) Note that we can again assume that such a rule is guarded by X = X . — if D = ¬ R for an n -ary role expression R , Φ(Σ) contains: D ( X 1 , . . . , X n ) ← ⊤ n ( X 1 , . . . , X n ) , not R ( X 1 , . . . , X n ) (10) Note that if n egation wou ld have been deﬁn ed w .r .t. D n instead of ⊤ I n , we would not be able to write the above as a guar ded rule. — if D = E ⊓ F for concep t e xpressions E and F , Φ(Σ) co ntains: D ( X ) ← E ( X ) , F ( X ) (11) — if D = E ⊓ F for n -ary role e xpressions E and F , Φ(Σ) co ntains: D ( X 1 , . . . , X n ) ← E ( X 1 , . . . , X n ) , F ( X 1 , . . . , X n ) (12) — if D = ($ i/n : C ) , Φ(Σ) conta ins: D ( X 1 , . . . , X i , . . . , X n ) ← ⊤ n ( X 1 , . . . , X i , . . . , X n ) , C ( X i ) (13) — if D = ∃ [$ i ] R , Φ(Σ) co ntains: D ( X ) ← R ( X 1 , . . . , X i − 1 , X , X i +1 , . . . , X n ) (14) The following theorem sho ws that this translation preserves satisﬁability . G-Hybrid Knowledge Bases 11 Theor em 1 Let (Σ , P ) be a g-hyb rid kno wledge base with Σ a D LRO −{≤} knowledge base. Then, a predicate or conce pt expre ssion p is satisﬁ able w .r . t. (Σ , P ) iff p is satisﬁable w .r .t. Φ(Σ) ∪ P . Pr oof ( ⇒ ) Assume p is satisﬁ able w .r .t. (Σ , P ) , i.e., there exists a mod el ( U, I , M ) o f (Σ , P ) , with U = ( D , σ ) , in which p has a n on-em pty e xtension. Now , we c onstruct the open interpretatio n ( V , N ) of Φ(Σ , P ) as follows. V = ( D , σ ′ ) with σ ′ : cts (Φ(Σ) ∪ P ) → D , and σ ′ ( x ) = σ ( x ) for every c onstant symbol x from P and σ ′ ( x ) = x I for every c onstant symbol x from Σ . Note th at σ ′ is well-deﬁned, since, for a constant symbo l x which o ccurs in both Σ and P , we hav e that σ ( x ) = x I . W e deﬁne the set N as f ollows: N = M ∪ { C ( x ) | x ∈ C I , C ∈ clos (Σ) } ∪ { R ( x 1 , . . . , x n ) | ( x 1 , . . . , x n ) ∈ R I , R ∈ clos (Σ) } with C and R con cept expressions and role e xpressions respectively . It is easy to verify that ( V , N ) is an open a nswer set o f Φ(Σ) ∪ P and ( V , N ) satisﬁes p . ( ⇐ ) Assume ( V , N ) is an open answer set of Φ(Σ) ∪ P with V = ( D , σ ′ ) such that p is satisﬁed. W e d eﬁne the interpretation ( U, I , N ) of (Σ , P ) as fo llows. • U = ( D , σ ) wher e σ : cts ( P ) → D with σ ( x ) = σ ′ ( x ) ( note that this is po ssible since cts ( P ) ⊆ cts (Φ(Σ) ∪ P ) ). U is then a pre-in terpretation for P . • I = ( D, · I ) is deﬁned such that A I = { x | A ( x ) ∈ N } fo r conce pt names A , P I = { ( x 1 , . . . , x n ) | P ( x 1 , . . . , x n ) ∈ N } for n -ary role names P and o I = σ ′ ( o ) , for o a constant symbol in Σ ( note that σ ′ is indeed deﬁned on o ). I is then an interp retation of Σ . • M = N \ { p ( ~ x ) | p ∈ clos (Σ) } , such that M is an interp retation of Π( P U , I ) . Moreover , for every constant symb ol b which ap pears in both Σ and P , b I = σ ( b ) . As a consequen ce, ( U , I , M ) is an interp retation of (Σ , P ) . It is easy to verify that ( U, I , M ) is a model of (Σ , P ) which satisﬁes p . Theor em 2 Let (Σ , P ) be a g -hybr id knowledge base where Σ is a D LRO −{≤} knowledge base. Then, Φ(Σ) ∪ P is a guard ed program with a size polynomial in the size of (Σ , P ) . Pr oof The r ules in Φ(Σ) are obviou sly g uarded . Since P is a guarded p rogra m, Φ(Σ) ∪ P is a guarde d program as well. The size of clos (Σ) is of the o rder n log n where n is the size o f Σ . Intu itiv ely , g iv en that th e size of an expression is n , we have that the size of the set of its subexpressions is at most the size of a tree with depth log n where the size of the subexpr essions at a certain lev el of the tree is at most n . The size of Φ(Σ) is clearly polyno mial in the size of clos (Σ) , assumin g th at the arity n of an added role e xpression is p olyno mial in the size o f the maximal arity of role e xpressions in Σ . If we were to add a relation n ame R with arity 2 n , where n is the maximal arity of 12 S. Heymans e t al. relation names in C and Σ , the size of Σ would increase li nearly , but the s ize of Φ(Σ) ∪ P would increase exponentially: o ne needs to add, e.g., rules ⊤ 2 n ( X 1 , . . . , X 2 n ) ∨ not ⊤ 2 n ( X 1 , . . . , X 2 n ) ← which introduce an expo nential number o f arguments while the size of the r ole R does not depend on its arity . Note th at in g-hybr id knowledge bases, we consider D LRO −{≤} , wh ich is D LRO without expressions of the form ≤ k [$ i ] R , since such exp ressions cannot be simulated with guarded pro grams. E .g., consider the concep t expression ≤ 1[$1] R where R is a binary role. One can simulate the ≤ by negation as f ailure: ≤ 1[$1] R ( X ) ← not q ( X ) for some new q , with q deﬁned such that there are at least 2 different R -successors: q ( X ) ← R ( X , Y 1 ) , R ( X , Y 2 ) , Y 1 6 = Y 2 Howe ver , the latter rule is not gu arded – the re is n o atom th at contain s X , Y 1 , and Y 2 . So, in gener al, expressing number restrictions such as ≤ k [$ i ] R is out of r each fo r GPs. From Theorem s 1 and 2 we obtain the following corollary . Cor o llary 1 Satisﬁability check ing w .r .t. g-h ybrid knowledge bases (Σ , P ) , with Σ a DLRO −{≤} knowledge base, can be polynomially reduced to satis ﬁability check ing w .r .t. GPs. Since satisﬁability checkin g w .r .t. GPs is 2- E X P T I M E -co mplete ( ? ), we obtain the same 2- E X P T I M E characterization for g-hybrid knowledge bases. W e ﬁrst make explicit a corol- lary of Theorem 1. Cor o llary 2 Let P be a guarded progr am. Then, a concept or role e xpression p is satisﬁable w .r .t. P iff p is satisﬁable w .r .t. ( ∅ , P ) . Theor em 3 Satisﬁability checking w .r .t. g-h ybrid knowledge ba ses where th e DL par t is a D LRO −{≤} knowledge base is 2- E X P T I M E -complete. Pr oof Membership in 2- E X P T I M E follows fro m Corollary 1. Hardness follows from 2- E X P T I M E - hardne ss of satisﬁ ability check ing w .r .t. GPs and the reduction to satisﬁability checking in Corollary 2. 5 Relation with DL + lo g and other Relat ed W ork In ( ? ), so-c alled D L + lo g knowledge bases co mbine a Descriptio n Lo gic knowledge base with a weakly-safe disjunctive lo gic progr am. F ormally , fo r a particular Description Log ic DL , a DL + lo g knowledge base is a pair (Σ , P ) where Σ is a DL knowledge base consist- ing of a TBox (a set of termin ological axio ms) and an ABox (a set of assertional axioms ), and P contains rules α ← β such that for every rule r : α ← β ∈ P : G-Hybrid Knowledge Bases 13 • α − = ∅ , • β − does not contain DL atoms ( DL-po sitiveness ), • each variable i n r occurs in β + ( Datalog safeness ), and • each variable in r which occur s in a non-D L atom, occur s in a no n-DL atom in β + ( weak safeness ). The semantics for DL + lo g is th e same as that of g-hy brid k nowledge bases 4 , with the following e xceptions: • W e do n ot re quire the stand ar d name assumption , which basically says th at the d o- main of ev ery interpr etation is essentially the same inﬁnitely coun table set of con- stants. Neither do we h ave the implied unique name a ssumption , making the s eman- tics for g-h ybrid knowledge b ases more in line with cur rent Sem antic W eb s tandards such as OWL ( ? ) where n either the stan dard names assump tion nor the u nique names assumption ap plies. Note th at Rosati also pre sented a version of hybrid kn owledge bases which doe s not adher e to th e unique name assumption in an earlier work ( ? ). Howe ver , the grou nding of the p rogram pa rt is with the con stant symb ols explicitly appearin g in the program o r DL p art only , which yield s a less tight integration of the progr am and the DL part than in ( ? ) or in g-hybrid kno wledge bases. • W e deﬁne an interpretation as a triple ( U, I , M ) instead of a pair ( U, I ′ ) where I ′ = I ∪ M ; this is, howe v er , e quiv alent to DL + lo g . The ke y differences o f the two approach es ar e: • The p rogram s considered in D L + lo g may ha ve multiple positi ve literals in the h ead, whereas we allow at most one. Ho wev er , we allow negative literals in the head, whereas this is not allowed in D L + lo g . Additionally , since DL-ato ms are interp reted classically , we may simulate po siti ve DL-atom s in the head through negative DL- atoms in the body . • Instead of Datalog safeness we require gu ar dedness . Whereas with Datalog safeness ev ery variable in the ru le should ap pear in som e po siti ve atom of the body o f the rule, guarde dness requires that there is a po siti ve atom that contain s e very v ariable in the rule, with the exception of free rules. E.g., a ( X ) ← b ( X ) , c ( Y ) is Datalog safe since X app ears in b ( X ) and Y app ears in c ( Y ) , but this ru le is not g uarded since there is no atom that contains both X and Y . Note that we could easily extend the approa ch taken in this paper to loosely gu ar ded p r ograms wh ich re quire that every two variables in the rule should appear tog ether in a p ositiv e atom , Howev er , this would still be less expressi ve th an Datalog safeness. • W e do no t have the requ irement fo r weak safen ess, i. e., head variables do n ot need to app ear positiv ely in a non-D L atom . The gu ardedn ess may be provid ed by a DL atom. 4 Strictl y speaking , we did not deﬁne answer sets of disjunct i ve programs, howe v er , the deﬁnitions of Subsection 2.1 can serve for disjunct i ve pr ograms without modiﬁcation. Al so, we did not consider ABo xes in our deﬁniti on of DL s in Subsection 2.2. Ho we ve r , the extension of the semantics to DL knowl edge bases with ABoxes is straight forward . 14 S. Heymans e t al. Example 3 Example 1 contains the rule pr oblematic ( X ) ← s o cialDrinker ( X ) , knowsF r o mAA ( X , Y ) This allows to deduc e that X might be a pr oblem case even if X knows someo ne from the AA but is not drinking with th at person. Indeed, as illustrated by the model in Example 1, john is dr inking wine with som e anonymous x a nd k nows michael from the AA. More correct would be the rule pr oblematic ( X , Z ) ← drinks ( X , Y , Z ) , knowsF r omAA ( X , Y ) where we explicitly say that X and Y in th e drinks and knowsF r omAA relations should be the same, and we e xtend the p r o blematic predicate with the kind of drink that X has a problem with. Then, the h ead variable Z is guarded by the DL atom drinks and the rule is thus n ot weakly- safe, but is g uarded no netheless. Thu s, the resulting knowledge base is not a D L + lo g knowledge base, b ut is a g-hybrid knowl- edge base. • W e do no t have the requ irement for DL-p ositiv eness, i.e., DL atoms may app ear negated in the body of rules (and also in the head s of rules). Ho we ver , o ne c ould allow this in D L + lo g knowledge bases as well, since not A ( ~ X ) in the b ody o f the rule has the same effect as A ( ~ X ) in the head, where th e latter is allowed in ( ? ). V ice versa, we c an also lo osen our re striction on the occurrence of positive atoms in the head (which allows at mo st one positi ve ato m in the head), to allow for an ar bitrary nu mber of positive DL atoms in the h ead (but still keep th e n umber of positive n on-DL atoms limited to o ne). E.g., a ru le p ( X ) ∨ A ( X ) ← β , wher e A ( X ) is a DL atom, is not a v alid rule in the progra ms we considered si nce the head contains mor e than on e positive atom . Howe ver , we can always rewrite such a rule to p ( X ) ← β , not A ( X ) , which contains at most one positive atom in the head . Arguably , DL atoms sh ould not be allo wed to occ ur negativ ely , because DL pr ed- icates are interpreted classically and thus the negation in front of the DL atom is not nonmonoto nic. Ho we ver , D atalog predicates which depend o n DL predicates are also (partially) interp reted classically , an d DL atom s occurrin g negati vely in the body are equi v alent to DL atoms o ccurring positi vely in the head which allo ws u s to partly overcome our limitation of rule heads to one positi ve a tom. • W e do no t take into account ABoxes in the DL knowledge base. Howe ver , the DL we con sider includes n ominals such that one can simulate th e ABox using termino- logical axioms. Mo reover , e ven if the DL d oes n ot in clude no minals, the ABox can be written as groun d facts in a program and ground facts are al ways guarded. • Decidability for satisﬁability ch ecking 5 of D L + lo g knowledge bases is gu aranteed if decidab ility o f the conjun ctiv e query con tainment pr oblem is guaran teed f or the DL at hand. In contrast, we r elied on a translation of DLs to g uarded progr ams for establishing decidability , and, as explained in the p revious section, not all DLs (e.g. those with numb er restrictions) can be translated to such a GP . 5 ( ? ) considers checking satisﬁabil ity of kno wledge bases rather than sati sﬁabilit y of predi cates. Howe ve r , the former can easily be reduced to the latte r . G-Hybrid Knowledge Bases 15 W e brieﬂy men tion AL -log ( ? ), which is a p redecessor of DL + lo g . AL -log co nsiders ALC k nowledge bases for the DL p art an d a set o f positive Horn clauses fo r th e pr ogram part. Every variable must ap pear in a positive atom in the body , and conc ept names are the only DL pred icates which may be u sed in the rules, an d they may on ly be used in rule bodies. ( ? ) a nd ( ? ) simulate r easoning in DLs with a n LP formalism by using an inter mediate translation to ﬁrst-order clauses. In ( ? ), S H I Q knowledge bases are reduced to ﬁrst-order formu las, to wh ich the basic superposition calculus is ap plied. ( ? ) translates A LC QI con- cept expressions to ﬁrst-ord er fo rmulas, ground s them with a ﬁnite number of constants, and transform s the result to a logic pro gram. One can use a ﬁnite n umber of co nstants b y the ﬁnite mode l proper ty of ALC QI . In the presence of termin ological axioms this is n o longer possible since the ﬁnite model proper ty is not guaranteed to hold. In ( ? ), the DL ALC N R ( R stands f or role in tersection) is extended with Ho rn clauses q ( ~ Y ) ← p 1 ( ~ X 1 ) , . . . , p n ( ~ X n ) wh ere the variables in ~ Y must appear in ~ X 1 ∪ . . . ∪ ~ X n ; p 1 , . . . , p n are either conce pt or role names, o r or dinary p redicates not appear ing in the DL part, a nd q is an or dinary p redicate. There is no safeness in the sense that e very variable must appea r in a n on-DL atom. The semantics is deﬁned thro ugh extended interpr etations that satisfy both the DL and clau ses par t (as FOL f ormulas). Qu ery answering is und ecid- able if re cursive Horn clauses are allowed, but d ecidability can be regained by restricting the DL part or by enforcin g th at the clauses are role safe (each v ariable in a role atom R ( X , Y ) fo r a role R mu st app ear in a n on-DL ato m). Note that th e latter restriction is less strict than the DL-safeness 6 of ( ? ) , where also variables in conce pt atoms A ( X ) need to appea r in no n-DL atoms. On the other h and, ( ? ) allo ws for the m ore expressive DL S HO I N ( D ) , and th e head predicates ma y b e DL atom s a s well. Finally , SWRL ( ? ) can be seen as an extension of ( ? ) withou t any safeness restriction, which results in the loss of decidability of the for malism. Compar ed to o ur work, we consid er a slightly less expr es- si ve Description Logic, but we consider log ic p rogram s wit h n onmon otonic ne gation, and require guardedn ess, r ather than role- or DL-safeness, to guaran tee decidability . In ( ? ) D escription Logic pr ogr ams are in troduc ed; atom s in the program co mpon ent may be d l-atoms with which one can query the knowledge in th e DL comp onent. Such dl-atoms may specify in formatio n from the log ic prog ram which need s to b e taken into account when ev aluating the query , yieldin g a b i-direction al ﬂo w of information . Th is leads to a minimal interface between the DL kno wledge base and the logic program, enabling a v ery loose integration, based on an entailm ent r elation. In contr ast, we propo se a mu ch tighter integration betwe en th e rules a nd the o ntolog y , with interactio n ba sed on single mod els rather than entailment. For a detailed discussion of these tw o kinds of interaction, we refer to ( ? ). T wo recent appro aches ( ? ; ? ) use an embeddin g in a no nmon otonic modal log ic fo r in - tegrating nonmo noton ic logic program s and ontologies b ased on c lassical logic (e.g. DL). ( ? ) use the n onmo notonic logic of Minimal Knowledge and Negation as Failure ( MKNF) for the co mbination , an d show decidability of reason ing in case reasoning in the c onsidered description log ic is decidable, and the DL safene ss cond ition ( ? ) ho lds for the ru les in the 6 DL-safeness is a restricti on of the earlier m entione d weak safen ess. 16 S. Heymans e t al. logic p rogram . I n our ap proach , we d o n ot req uire su ch a safen ess co ndition, but r equire the rules to be guarded , and make a sema ntic distinction between DL p redicates and rule predicates. ( ? ) introduce several embed dings of no n-gro und lo gic pr ograms in ﬁrst-or der autoepistemic logic (FO-AE L), and compa re them under combin ation with classical theo- ries (ontologies). Howe ver , ( ? ) do not ad dress the issue of decidability o r reaso ning of such combinatio ns. Finally , ( ? ) use Quantiﬁed Equilibrium Logic as a single unify ing langu age to captu re different app roache s to hy brid kn owledge bases, including the ap proach presented in th is paper . Althoug h we have presented a translation of g- hybr id knowledge ba ses to g uarded logic progr ams, o ur direct seman tics is still based on two mo dules, relying on separate interpretatio ns fo r the DL knowledge base and the logic program, whereas ( ? ) deﬁne equi- librium mode ls, wh ich serve to g iv e a unify ing sema ntics to the hy brid kn owledge base. The appro ach of ( ? ) may be used to deﬁn e a notion of equ iv alence b etween, and may lead to new algorithms for reasoning with, g-hybrid knowledge bases. 6 Conclusions and Directions f or Further Research W e de ﬁned g-hyb rid kno wledge bases which combine Description Logic (DL) knowledge bases with guar ded log ic p rogra ms. In particular , we combined knowledge bases of th e DL DLRO −{≤} , which is c lose to O WL DL, with gu arded program s, and showed d ecidability of this fr amework by a reduction to guarded pro grams under the open answer set seman- tics ( ? ; ? ). W e discu ssed the relation with D L + lo g kn owledge b ases: g-hyb rid kno wledge bases o vercome some of the limitations of DL + lo g , such as the un ique names assumption, Datalog safene ss, and weak DL-safe ness, but introdu ce the requirement of gu ardedn ess. At present, a sign iﬁcant d isadvantage of our appro ach is the lack o f suppo rt for DLs with number restrictions which is inherent to the use of guarde d prog rams as ou r de cidability vehicle. A solution for this would be to consider oth er types of pro grams, such as conce p- tual logic pr o grams ( ? ). This would allow for the d eﬁnition of a hybrid k nowledge base (Σ , P ) wh ere Σ is a S H I Q kno wledge base and P is a co nceptua l lo gic program since S HI Q knowledge ba ses can be translated to conceptual logic program s. Although the re are known complexity boun ds for several fragmen ts of open an swer set progr amming (O ASP), inclu ding th e guarde d frag ment con sidered in this p aper, there are no known effecti ve algorithm s f or O ASP . Addition ally , at p resence, there are n o imple- mented systems for open answer set program ming. Th ese are part of future work.

Guarded Hybrid Knowledge Bases

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