How To Write A Theory Of The Concept Of The Mind In A Quey

Transcription

1 Jounal of Atificial Intelligence Reseach 31 (2008) Submitted 06/07; published 01/08 Conjunctive Quey Answeing fo the Desciption Logic SHIQ Bite Glimm Ian Hoocks Oxfod Univesity Computing Laboatoy, UK Casten Lutz Desden Univesity of Technology, Gemany Ulike Sattle The Univesity of Mancheste, UK Abstact Conjunctive queies play an impotant ole as an expessive quey language fo Desciption Logics (DLs). Although moden DLs usually povide fo tansitive oles, conjunctive quey answeing ove DL knowledge bases is only pooly undestood if tansitive oles ae admitted in the quey. In this pape, we conside unions of conjunctive queies ove knowledge bases fomulated in the pominent DL SHIQ and allow tansitive oles in both the quey and the knowledge base. We show decidability of quey answeing in this setting and establish two tight complexity bounds: egading combined complexity, we pove that thee is a deteministic algoithm fo quey answeing that needs time single exponential in the size of the KB and double exponential in the size of the quey, which is optimal. Regading data complexity, we pove containment in co-np. 1. Intoduction Desciption Logics (DLs) ae a family of logic based knowledge epesentation fomalisms (Baade, Calvanese, McGuinness, Nadi, & Patel-Schneide, 2003). Most DLs ae fagments of Fist-Ode Logic esticted to unay and binay pedicates, which ae called concepts and oles in DLs. The constuctos fo building complex expessions ae usually chosen such that the key infeence poblems, such as concept satisfiability, ae decidable and pefeably of low computational complexity. A DL knowledge base (KB) consists of a TBox, which contains intensional knowledge such as concept definitions and geneal backgound knowledge, and an ABox, which contains extensional knowledge and is used to descibe individuals. Using a database metapho, the TBox coesponds to the schema, and the ABox coesponds to the data. In contast to databases, howeve, DL knowledge bases adopt an open wold semantics, i.e., they epesent infomation about the domain in an incomplete way. Standad DL easoning sevices include testing concepts fo satisfiability and etieving cetain instances of a given concept. The latte etieves, fo a knowledge base consisting of an ABox A and a TBox T, all (ABox) individuals that ae instances of the given (possibly complex) concept expession C, i.e., all those individuals a such that T and A entail that a is an instance of C. The undelying easoning poblems ae well-undestood, and it is known that the combined complexity of these easoning poblems, i.e., the complexity measued in the size of the TBox, the ABox, and the quey, is ExpTime-complete fo SHIQ (Tobies, c 2008 AI Access Foundation. All ights eseved.

2 Glimm, Hoocks, Lutz, & Sattle 2001). The data complexity of a easoning poblem is measued in the size of the ABox only. Wheneve the TBox and the quey ae small compaed to the ABox, as is often the case in pactice, the data complexity gives a moe useful pefomance estimate. Fo SHIQ, instance etieval is known to be data complete fo co-np (Hustadt, Motik, & Sattle, 2005). Despite the high wost case complexity of the standad easoning poblems fo vey expessive DLs such as SHIQ, thee ae highly optimized implementations available, e.g., FaCT++ (Tsakov & Hoocks, 2006), KAON2 1, Pellet (Siin, Pasia, Cuenca Gau, Kalyanpu, & Katz, 2006), and RacePo 2. These systems ae used in a wide ange of applications, e.g., configuation (McGuinness & Wight, 1998), bio infomatics (Wolstencoft, Bass, Hoocks, Lod, Sattle, Tui, & Stevens, 2005), and infomation integation (Calvanese, De Giacomo, Lenzeini, Nadi, & Rosati, 1998b). Most pominently, DLs ae known fo thei use as a logical undepinning of ontology languages, e.g., OIL, DAML+OIL, and OWL (Hoocks, Patel-Schneide, & van Hamelen, 2003), which is a W3C ecommendation (Bechhofe, van Hamelen, Hendle, Hoocks, McGuinness, Patel-Schneide, & Stein, 2004). In data-intensive applications, queying KBs plays a cental ole. Instance etieval is, in some aspects, a athe weak fom of queying: although possibly complex concept expessions ae used as queies, we can only quey fo tee-like elational stuctues, i.e., a DL concept cannot expess abitay cyclic stuctues. This popety is known as the tee model popety and is consideed an impotant eason fo the decidability of most Modal and Desciption Logics (Gädel, 2001; Vadi, 1997). Conjunctive queies (CQs) ae well known in the database community and constitute an expessive quey language with capabilities that go well beyond standad instance etieval. Fo an example, conside a knowledge base that contains an ABox assetion ( hasson.( hasdaughte. ))(May), which infomally states that the individual (o constant in FOL tems) May has a son who has a daughte; hence, that May is a gandmothe. Additionally, we assume that both oles hasson and hasdaughte have a tansitive supe-ole hasdescendant. This implies that May is elated via the ole hasdescendant to he (anonymous) gandchild. Fo this knowledge base, May is clealy an answe to the conjunctive quey hasson(x, y) hasdaughte(y, z) hasdescendant(x, z), when we assume that x is a distinguished vaiable (also called answe o fee vaiable) and y, z ae non-distinguished (existentially quantified) vaiables. If all vaiables in the quey ae non-distinguished, the quey answe is just tue o false and the quey is called a Boolean quey. Given a knowledge base K and a Boolean CQ q, the quey entailment poblem is deciding whethe q is tue o false w..t. K. If a CQ contains distinguished vaiables, the answes to the quey ae those tuples of individual names fo which the knowledge base entails the quey that is obtained by eplacing the fee vaiables with the individual names in the answe tuple. The poblem of finding all answe tuples is known as quey answeing. Since quey entailment is a decision poblem and thus bette suited fo complexity analysis than quey answeing, we concentate on quey entailment. This is no estiction since quey answeing can easily be educed to quey entailment as we illustate in moe detail in Section

3 Conjunctive Quey Answeing fo the DL SHIQ Devising a decision pocedue fo conjunctive quey entailment in expessive DLs such as SHIQ is a challenging poblem, in paticula when tansitive oles ae admitted in the quey (Glimm, Hoocks, & Sattle, 2006). In the confeence vesion of this pape, we pesented the fist decision pocedue fo conjunctive quey entailment in SHIQ. In this pape, we genealize this esult to unions of conjunctive queies (UCQs) ove SHIQ knowledge bases. We achieve this by ewiting a conjunctive quey into a set of conjunctive queies such that each esulting quey is eithe tee-shaped (i.e., it can be expessed as a concept) o gounded (i.e., it contains only constants/individual names and no vaiables). The entailment of both types of queies can be educed to standad easoning poblems (Hoocks & Tessais, 2000; Calvanese, De Giacomo, & Lenzeini, 1998a). The pape is oganized as follows: in Section 2, we give the necessay definitions, followed by a discussion of elated wok in Section 3. In Section 4, we motivate the quey ewiting steps by means of an example. In Section 5, we give fomal definitions fo the ewiting pocedue and show that a Boolean quey is indeed entailed by a knowledge base K iff the disjunction of the ewitten queies is entailed by K. In Section 6, we pesent a deteministic algoithm fo UCQ entailment in SHIQ that uns in time single exponential in the size of the knowledge base and double exponential in the size of the quey. Since the combined complexity of conjunctive quey entailment is aleady 2ExpTime-had fo the DL ALCI (Lutz, 2007), it follows that this poblem is 2ExpTime-complete fo SHIQ. This shows that conjunctive quey entailment fo SHIQ is stictly hade than instance checking, which is also the case fo simple DLs such as EL (Rosati, 2007b). We futhe show that (the decision poblem coesponding to) conjunctive quey answeing in SHIQ is co-npcomplete egading data complexity, and thus not hade than instance etieval. The pesented decision pocedue gives not only insight into quey answeing; it also has an immediate consequence on the field of extending DL knowledge bases with ules. Fom the wok by Rosati (2006a, Thm. 11), the consistency of a SHIQ knowledge base extended with (weakly-safe) Datalog ules is decidable iff the entailment of unions of conjunctive queies in SHIQ is decidable. Hence, we close this open poblem as well. This pape is an extended vesion of the confeence pape: Conjunctive Quey Answeing fo the Desciption Logic SHIQ. Poceedings of the Twentieth Intenational Joint Confeence on Atificial Intelligence (IJCAI 07), Jan 06-12, Peliminaies We intoduce the basic tems and notations used thoughout the pape. In paticula, we intoduce the DL SHIQ (Hoocks, Sattle, & Tobies, 2000) and (unions of) conjunctive queies. 2.1 Syntax and Semantics of SHIQ Let N C, N R, and N I be countably infinite sets of concept names, ole names, and individual names. We assume that the set of ole names contains a subset N tr N R of tansitive ole names. A ole is an element of N R { N R }, whee oles of the fom ae called invese oles. A ole inclusion is of the fom s with, s oles. A ole hieachy R is a finite set of ole inclusions. 159

4 Glimm, Hoocks, Lutz, & Sattle An intepetation I = ( I, I) consists of a non-empty set I, the domain of I, and a function I, which maps evey concept name A to a subset A I I, evey ole name N R to a binay elation I I I, evey ole name N tr to a tansitive binay elation I I I, and evey individual name a to an element a I I. An intepetation I satisfies a ole inclusion s if I s I and a ole hieachy R if it satisfies all ole inclusions in R. We use the following standad notation: 1. We define the function Inv ove oles as Inv() := if N R and Inv() := s if = s fo a ole name s. 2. Fo a ole hieachy R, we define * R as the eflexive tansitive closue of ove R {Inv() Inv(s) s R}. We use R s as an abbeviation fo * Rs and s * R. 3. Fo a ole hieachy R and a ole s, we define the set Tans R of tansitive oles as {s thee is a ole with R s and N tr o Inv() N tr }. 4. A ole is called simple w..t. a ole hieachy R if, fo each ole s such that s * R, s / Tans R. The subscipt R of * R and Tans R is dopped if clea fom the context. The set of SHIQconcepts (o concepts fo shot) is the smallest set built inductively fom N C using the following gamma, whee A N C, n IN, is a ole and s is a simple ole: C ::= A C C 1 C 2 C 1 C 2.C.C n s.c n s.c. Given an intepetation I, the semantics of SHIQ-concepts is defined as follows: I = I (C D) I = C I D I ( C) I = I \ C I I = (C D) I = C I D I (.C) I = {d I if (d, d ) I, then d C I } (.C) I = {d I thee is a (d, d ) I with d C I } ( n s.c) I = {d I (s I (d, C)) n} ( n s.c) I = {d I (s I (d, C)) n} whee (M) denotes the cadinality of the set M and s I (d, C) is defined as {d I (d, d ) s I and d C I }. A geneal concept inclusion (GCI) is an expession C D, whee both C and D ae concepts. A finite set of GCIs is called a TBox. An intepetation I satisfies a GCI C D if C I D I, and a TBox T if it satisfies each GCI in T.. An (ABox) assetion is an expession of the fom C(a), (a, b), (a, b), o a = b, whee C is a concept, is a ole, a, b N I. An ABox is a finite set of assetions. We use Inds(A) to denote the set of individual names occuing in A. An intepetation I satisfies an assetion C(a) if a I C I, (a, b) if (a I, b I ) I, (a, b) if (a I, b I ) / I., and a =b if a I b I. An 160

5 Conjunctive Quey Answeing fo the DL SHIQ intepetation I satisfies an ABox if it satisfies each assetion in A, which we denote with I = A. A knowledge base (KB) is a tiple (T, R, A) with T a TBox, R a ole hieachy, and A an ABox. Let K = (T, R, A) be a KB and I = ( I, I) an intepetation. We say that I satisfies K if I satisfies T, R, and A. In this case, we say that I is a model of K and wite I = K. We say that K is consistent if K has a model Extending SHIQ to SHIQ In the following section, we show how we can educe a conjunctive quey to a set of gound o tee-shaped conjunctive queies. Duing the eduction, we may intoduce concepts that contain an intesection of oles unde existential quantification. We define, theefoe, the extension of SHIQ with ole conjunction/intesection, denoted as SHIQ and, in the appendix, we show how to decide the consistency of SHIQ knowledge bases. In addition to the constuctos intoduced fo SHIQ, SHIQ allows fo concepts of the fom C ::= R.C R.C n S.C n S.C, whee R := 1... n, S := s 1... s n, 1,..., n ae oles, and s 1,...,s n ae simple oles. The intepetation function is extended such that ( 1... n ) I = 1 I... n I. 2.2 Conjunctive Queies and Unions of Conjunctive Queies We now intoduce Boolean conjunctive queies since they ae the basic fom of queies we ae concened with. We late also define non-boolean queies and show how they can be educed to Boolean queies. Finally, unions of conjunctive queies ae just a disjunction of conjunctive queies. Fo simplicity, we wite a conjunctive quey as a set instead of as a conjunction of atoms. Fo example, we wite the intoductoy example fom Section 1 as {hasson(x, y), hasdaughte(y, z), hasdescendant(x, z)}. Fo non-boolean queies, i.e., when we conside the poblem of quey answeing, the answe vaiables ae often given in the head of the quey, e.g., (x 1, x 2, x 3 ) {hasson(x 1, x 2 ), hasdaughte(x 2, x 3 ), hasdescendant(x 1, x 3 )} indicates that the quey answes ae those tuples (a 1, a 2, a 3 ) of individual names that, substituted fo x 1, x 2, and x 3 espectively, esult in a Boolean quey that is entailed by the knowledge base. Fo simplicity and since we mainly focus on quey entailment, we do not use a quey head even in the case of a non-boolean quey. Instead, we explicitly say which vaiables ae answe vaiables and which ones ae existentially quantified. We now give a definition of Boolean conjunctive queies. Definition 1. Let N V be a countably infinite set of vaiables disjoint fom N C, N R, and N I. A tem t is an element fom N V N I. Let C be a concept, a ole, and t, t tems. An atom is an expession C(t), (t, t ), o t t and we efe to these thee diffeent types of atoms as concept atoms, ole atoms, and equality atoms espectively. A Boolean conjunctive quey 161

6 Glimm, Hoocks, Lutz, & Sattle q is a non-empty set of atoms. We use Vas(q) to denote the set of (existentially quantified) vaiables occuing in q, Inds(q) to denote the set of individual names occuing in q, and Tems(q) fo the set of tems in q, whee Tems(q) = Vas(q) Inds(q). If all tems in q ae individual names, we say that q is gound. A sub-quey of q is simply a subset of q (including q itself). As usual, we use (q) to denote the cadinality of q, which is simply the numbe of atoms in q, and we use q fo the size of q, i.e., the numbe of symbols necessay to wite q. A SHIQ conjunctive quey is a conjunctive quey in which all concepts C that occu in a concept atom C(t) ae SHIQ-concepts. Since equality is eflexive, symmetic and tansitive, we define * as the tansitive, eflexive, and symmetic closue of ove the tems in q. Hence, the elation * is an equivalence elation ove the tems in q and, fo t Tems(q), we use [t] to denote the equivalence class of t by *. Let I = ( I, I) be an intepetation. A total function π: Tems(q) I is an evaluation if (i) π(a) = a I fo each individual name a Inds(q) and (ii) π(t) = π(t ) fo all t * t. We wite I = π C(t) if π(t) C I ; I = π (t, t ) if (π(t), π(t )) I ; I = π t t if π(t) = π(t ). If, fo an evaluation π, I = π at fo all atoms at q, we wite I = π q. We say that I satisfies q and wite I = q if thee exists an evaluation π such that I = π q. We call such a π a match fo q in I. Let K be a SHIQ knowledge base and q a conjunctive quey. If I = K implies I = q, we say that K entails q and wite K = q. The quey entailment poblem is defined as follows: given a knowledge base K and a quey q, decide whethe K = q. Fo bevity and simplicity of notation, we define the elation ove atoms in q as follows: C(t) q if thee is a tem t Tems(q) such that t * t and C(t ) q, and (t 1, t 2 ) q if thee ae tems t 1, t 2 Tems(q) such that t 1 * t 1, t 2 * t 2, and (t 1, t 2 ) q o Inv()(t 2, t 1 ) q. This is clealy justified by definition of the semantics, in paticula, because I = (t, t ) implies that I = Inv()(t, t). When devising a decision pocedue fo CQ entailment, most complications aise fom cyclic queies (Calvanese et al., 1998a; Chekui & Rajaaman, 1997). In this context, when we say cyclic, we mean that the gaph stuctue induced by the quey is cyclic, i.e., the gaph obtained fom q such that each tem is consideed as a node and each ole atom induces an edge. Since, in the pesence of invese oles, a quey containing the ole atom (t, t ) is equivalent to the quey obtained by eplacing this atom with Inv()(t, t), the diection of the edges is not impotant and we say that a quey is cyclic if its undelying undiected gaph stuctue is cyclic. Please note also that multiple ole atoms fo two tems ae not consideed as a cycle, e.g., the quey {(t, t ), s(t, t )} is not a cyclic quey. The following is a moe fomal definition of this popety. Definition 2. A quey q is cyclic if thee exists a sequence of tems t 1,...,t n with n > 3 such that 162

7 Conjunctive Quey Answeing fo the DL SHIQ 1. fo each i with 1 i < n, thee exists a ole atom i (t i, t i+1 ) q, 2. t 1 = t n, and 3. t i t j fo 1 i < j < n. In the above definition, Item 3 makes sue that we do not conside queies as cyclic just because they contain two tems t, t fo which thee ae moe than two ole atoms using the two tems. Please note that we use the elation hee, which implicitly uses the elation * and abstacts fom the diectedness of ole atoms. In the following, if we wite that we eplace (t, t ) q with s(t 1, t 2 ),...,s(t n 1, t n ) fo t = t 1 and t = t n, we mean that we fist emove any occuences of (ˆt, ˆt ) and Inv()(ˆt, ˆt) such that ˆt * t and ˆt * t fom q, and then add the atoms s(t 1, t 2 ),...,s(t n 1, t n ) to q. W.l.o.g., we assume that queies ae connected. Moe pecisely, let q be a conjunctive quey. We say that q is connected if, fo all t, t Tems(q), thee exists a sequence t 1,...,t n such that t 1 = t, t n = t and, fo all 1 i < n, thee exists a ole such that (t i, t i+1 ) q. A collection q 1,...,q n of queies is a patitioning of q if q = q 1... q n, q i q j = fo 1 i < j n, and each q i is connected. Lemma 3. Let K be a knowledge base, q a conjunctive quey, and q 1,...,q n a patitioning of q. Then K = q iff K = q i fo each i with 1 i n. A poof is given by Tessais (2001, 7.3.2) and, with this lemma, it is clea that the estiction to connected queies is indeed w.l.o.g. since entailment of q can be decided by checking entailment of each q i at a time. In what follows, we theefoe assume queies to be connected without futhe notice. Definition 4. A union of Boolean conjunctive queies is a fomula q 1... q n, whee each disjunct q i is a Boolean conjunctive quey. A knowledge base K entails a union of Boolean conjunctive queies q 1... q n, witten as K = q 1... q n, if, fo each intepetation I such that I = K, thee is some i such that I = q i and 1 i n. W.l.o.g. we assume that the vaiable names in each disjunct ae diffeent fom the vaiable names in the othe disjuncts. This can always be achieved by naming vaiables apat. We futhe assume that each disjunct is a connected conjunctive quey. This is w.l.o.g. since a UCQ which contains unconnected disjuncts can always be tansfomed into conjunctive nomal fom; we can then decide entailment fo each esulting conjunct sepaately and each conjunct is a union of connected conjunctive queies. We descibe this tansfomation now in moe detail and, fo a moe convenient notation, we wite a conjunctive quey {at 1,...,at k } as at 1... at k in the following poof, instead of the usual set notation. Lemma 5. Let K be a knowledge base, q = q 1... q n a union of conjunctive queies such that, fo 1 i n, qi 1,...,qk i i is a patitioning of the conjunctive quey q i. Then K = q iff K = (q i qn in ). (i 1,...,i n) {1,...,k 1 }... {1,...,k n} 163

8 Glimm, Hoocks, Lutz, & Sattle Again, a detailed poof is given by Tessais (2001, 7.3.3). Please note that, due to the tansfomation into conjunctive nomal fom, the esulting numbe of unions of connected conjunctive queies fo which we have to test entailment can be exponential in the size of the oiginal quey. When analysing the complexity of the decision pocedues pesented in Section 6, we show that the assumption that each CQ in a UCQ is connected does not incease the complexity. We now make the connection between quey entailment and quey answeing cleae. Fo quey answeing, let the vaiables of a conjunctive quey be typed: each vaiable can eithe be existentially quantified (also called non-distinguished) o fee (also called distinguished o answe vaiables). Let q be a quey in n vaiables (i.e., (Vas(q)) = n), of which v 1,...,v m (m n) ae answe vaiables. The answes of K = (T, R, A) to q ae those m-tuples (a 1,...,a m ) Inds(A) m such that, fo all models I of K, I = π q fo some π that satisfies π(v i ) = a i I fo all i with 1 i m. It is not had to see that the answes of K to q can be computed by testing, fo each (a 1,...,a m ) Inds(A) m, whethe the quey q [v1,...,v m/a 1,...,a m] obtained fom q by eplacing each occuence of v i with a i fo 1 i m is entailed by K. The answe to q is then the set of all m-tuples (a 1,...,a m ) fo which K = q [v1,...,v m/a 1,...,a m]. Let k = (Inds(A)) be the numbe of individual names used in the ABox A. Since A is finite, clealy k is finite. Hence, deciding which tuples belong to the set of answes can be checked with at most k m entailment tests. This is clealy not vey efficient, but optimizations can be used, e.g., to identify a (hopefully small) set of candidate tuples. The algoithm that we pesent in Section 6 decides quey entailment. The easons fo devising a decision pocedue fo quey entailment instead of quey answeing ae twofold: fist, quey answeing can be educed to quey entailment as shown above; second, in contast to quey answeing, quey entailment is a decision poblem and can be studied in tems of complexity theoy. In the emainde of this pape, if not stated othewise, we use q (possibly with subscipts) fo a connected Boolean conjunctive quey, K fo a SHIQ knowledge base (T, R, A), I fo an intepetation ( I, I), and π fo an evaluation. 3. Related Wok Vey ecently, an automata-based decision pocedue fo positive existential path queies ove ALCQIb eg knowledge bases has been pesented (Calvanese, Eite, & Otiz, 2007). Positive existential path queies genealize unions of conjunctive queies and since a SHIQ knowledge base can be polynomially educed to an ALCQIb eg knowledge base, the pesented algoithm is a decision pocedue fo (union of) conjunctive quey entailment in SHIQ as well. The automata-based technique can be consideed moe elegant than ou ewiting algoithm, but it does not give an NP uppe bound fo the data complexity as ou technique. Most existing algoithms fo conjunctive quey answeing in expessive DLs assume, howeve, that ole atoms in conjunctive queies use only oles that ae not tansitive. As a consequence, the example quey fom the intoductoy section cannot be answeed. Unde this estiction, decision pocedues fo vaious DLs aound SHIQ ae known (Hoocks & Tessais, 2000; Otiz, Calvanese, & Eite, 2006b), and it is known that answeing conjunctive queies in this setting is data complete fo co-np (Otiz et al., 2006b). Anothe common 164

9 Conjunctive Quey Answeing fo the DL SHIQ estiction is that only individuals named in the ABox ae consideed fo the assignments of vaiables. In this setting, the semantics of queies is no longe the standad Fist-Ode one. With this estiction, the answe to the example quey fom the intoduction would be false since May is the only named individual. It is not had to see that conjunctive quey answeing with this estiction can be educed to standad instance etieval by eplacing the vaiables with individual names fom the ABox and then testing the entailment of each conjunct sepaately. Most of the implemented DL easones, e.g., KAON2, Pellet, and RacePo, povide an inteface fo conjunctive quey answeing in this setting and employ seveal optimizations to impove the pefomance (Siin & Pasia, 2006; Motik, Sattle, & Stude, 2004; Wessel & Mölle, 2005). Pellet appeas to be the only easone that also suppots the standad Fist-Ode semantics fo SHIQ conjunctive queies unde the estiction that the queies ae acyclic. To the best of ou knowledge, it is still an open poblem whethe conjunctive quey entailment is decidable in SHOIQ. Regading undecidability esults, it is known that conjunctive quey entailment in the two vaiable fagment of Fist-Ode Logic L 2 is undecidable (Rosati, 2007a) and Rosati identifies a elatively small set of constuctos that causes the undecidability. Quey entailment and answeing have also been studied in the context of databases with incomplete infomation (Rosati, 2006b; van de Meyden, 1998; Gahne, 1991). In this setting, DLs can be used as schema languages, but the expessivity of the consideed DLs is much lowe than the expessivity of SHIQ. Fo example, the constuctos povided by logics of the DL-Lite family (Calvanese, De Giacomo, Lembo, Lenzeini, & Rosati, 2007) ae chosen such that the standad easoning tasks ae in PTime and quey entailment is in LogSpace with espect to data complexity. Futhemoe, TBox easoning can be done independently of the ABox and the ABox can be stoed and accessed using a standad database SQL engine. Since the consideed DLs ae consideable less expessive than SHIQ, the techniques used in databases with incomplete infomation cannot be applied in ou setting. Regading the quey language, it is well known that an extension of conjunctive queies with inequalities is undecidable (Calvanese et al., 1998a). Recently, it has futhe been shown that even fo DLs with low expessivity, an extension of conjunctive queies with inequalities o safe ole negation leads to undecidability (Rosati, 2007a). A elated easoning poblem is quey containment. Given a schema (o TBox) S and two queies q and q, we have that q is contained in q w..t. S iff evey intepetation I that satisfies S and q also satisfies q. It is well known that quey containment w..t. a TBox can be educed to deciding quey entailment fo (unions of) conjunctive queies w..t. a knowledge base (Calvanese et al., 1998a). Hence a decision pocedue fo (unions of) conjunctive queies in SHIQ can also be used fo deciding quey containment w..t. to a SHIQ TBox. Entailment of unions of conjunctive queies is also closely elated to the poblem of adding ules to a DL knowledge base, e.g., in the fom of Datalog ules. Augmenting a DL KB with an abitay Datalog pogam easily leads to undecidability (Levy & Rousset, 1998). In ode to ensue decidability, the inteaction between the Datalog ules and the DL knowledge base is usually esticted by imposing a safeness condition. The DL+log famewok (Rosati, 2006a) povides the least estictive integation poposed so fa. Rosati 165

10 Glimm, Hoocks, Lutz, & Sattle pesents an algoithm that decides the consistency of a DL+log knowledge base by educing the poblem to entailment of unions of conjunctive queies, and he poves that decidability of UCQs in SHIQ implies the decidability of consistency fo SHIQ+log knowledge bases. 4. Quey Rewiting by Example In this section, we motivate the ideas behind ou quey ewiting technique by means of examples. In the following section, we give pecise definitions fo all ewiting steps. 4.1 Foest Bases and Canonical Intepetations The main idea is that we can focus on models of the knowledge base that have a kind of tee o foest shape. It is well known that one eason fo Desciption and Modal Logics being so obustly decidable is that they enjoy some fom of tee model popety, i.e., evey satisfiable concept has a model that is tee-shaped (Vadi, 1997; Gädel, 2001). When going fom concept satisfiability to knowledge base consistency, we need to eplace the tee model popety with a fom of foest model popety, i.e., evey consistent KB has a model that consists of a set of tees, whee each oot coesponds to a named individual in the ABox. The oots can be connected via abitay elational stuctues, induced by the ole assetions given in the ABox. A foest model is, theefoe, not a foest in the gaph theoetic sense. Futhemoe, tansitive oles can intoduce shot-cut edges between elements within a tee o even between elements of diffeent tees. Hence we talk of a fom of foest model popety. We now define foest models and show that, fo deciding quey entailment, we can estict ou attention to foest models. The ewiting steps ae then used to tansfom cyclic subpats of the quey into tee-shaped ones such that thee is a foest-shaped match fo the ewitten quey into the foest models. In ode to make the foest model popety even cleae, we also intoduce foest bases, which ae intepetations that intepet tansitive oles in an unesticted way, i.e., not necessaily in a tansitive way. Fo a foest base, we equie in paticula that all elationships between elements of the domain that can be infeed by tansitively closing a ole ae omitted. In the following, we assume that the ABox contains at least one individual name, i.e., Inds(A) is non-empty. This is w.l.o.g. since we can always add an assetion (a) to the ABox fo a fesh individual name a N I. Fo eades familia with tableau algoithms, it is woth noting that foest bases can also be thought of as those tableaux geneated fom a complete and clash-fee completion tee (Hoocks et al., 2000). Definition 6. Let IN denote the non-negative integes and IN the set of all (finite) wods ove the alphabet IN. A tee T is a non-empty, pefix-closed subset of IN. Fo w, w T, we call w a successo of w if w = w c fo some c IN, whee denotes concatenation. We call w a neighbo of w if w is a successo of w o vice vesa. The empty wod ε is called the oot. A foest base fo K is an intepetation J = ( J, J ) that intepets tansitive oles in an unesticted (i.e., not necessaily tansitive) way and, additionally, satisfies the following conditions: T1 J Inds(A) IN such that, fo all a Inds(A), the set {w (a, w) J } is a tee; 166

11 Conjunctive Quey Answeing fo the DL SHIQ T2 if ((a, w), (a, w )) J, then eithe w = w = ε o a = a and w is a neighbo of w; T3 fo each a Inds(A), a J = (a, ε); An intepetation I is canonical fo K if thee exists a foest base J fo K such that I is identical to J except that, fo all non-simple oles, we have I = J s * R, s Tans R (s J ) + In this case, we say that J is a foest base fo I and if I = K we say that I is a canonical model fo K. Fo convenience, we extend the notion of successos and neighbos to elements in canonical models. Let I be a canonical model with (a, w), (a, w ) I. We call (a, w ) a successo of (a, w) if eithe a = a and w = w c fo some c IN o w = w = ε. We call (a, w ) a neighbo of (a, w) if (a, w ) is a successo of (a, w) o vice vesa. Please note that the above definition implicitly elies on the unique name assumption (UNA) (cf. T3). This is w.l.o.g. as we can guess an appopiate patition among the individual names and eplace the individual names in each patition with one epesentative individual name fom that patition. In Section 6, we show how the patitioning of individual names can be used to simulate the UNA, hence, ou decision pocedue does not ely on the UNA. We also show that this does not affect the complexity. Lemma 7. Let K be a SHIQ knowledge base and q = q 1... q n a union of conjunctive queies. Then K = q iff thee exists a canonical model I of K such that I = q. A detailed poof is given in the appendix. Infomally, fo the only if diection, we can take an abitay counte-model fo the quey, which exists by assumption, and unavel all non-tee stuctues. Since, duing the unaveling pocess, we only eplace cycles in the model by infinite paths and leave the intepetation of concepts unchanged, the quey is still not satisfied in the unavelled canonical model. The if diection of the poof is tivial. 4.2 The Running Example We use the following Boolean quey and knowledge base as a unning example: Example 8. Let K = (T, R, A) be a SHIQ knowledge base with, t N tr, k IN T = { } R = { t t, s } A = { } C k k p., C 3 3 p., D 2 s. t. (a, b), ( p.c k p.c.c 3 )(a), ( p.d 1.D 2 )(b) and q = {(u, x), (x, y), t(y, y), s(z, y), (u, z)} with Inds(q) = and Vas(q) = {u, x, y, z}. 167

12 Glimm, Hoocks, Lutz, & Sattle Fo simplicity, we choose to use a CQ instead of a UCQ. In case of a UCQ, the ewiting steps ae applied to each disjunct sepaately. p (a,1) (a,11) (a,12)... (a,1k) p C k p (a,ε) (b,ε) p p p t,t (a,2) C (a,3) C 3 D 1 (b,1) D 2 (b,2) p p p,s t,t t,t (a, 31) (a, 32) (a, 33) Figue 1: A epesentation of a canonical intepetation I fo K. (b, 21) (b, 22) Figue 1 shows a epesentation of a canonical model I fo the knowledge base K fom Example 8. Each labeled node epesents an element in the domain, e.g., the individual name a is epesented by the node labeled (a, ε). The edges epesent elationships between individuals. Fo example, we can ead the -labeled edge fom (a, ε) to (b, ε) in both diections, i.e., (a I, b I ) = ((a, ε), (b, ε)) I and (b I, a I ) = ((b, ε), (a, ε)) I. The shot-cuts due to tansitive oles ae shown as dashed lines, while the elationship between the nodes that epesent ABox individuals is shown in gey. Please note that we did not indicate the intepetations of all concepts in the figue. Since I is a canonical model fo K, the elements of the domain ae pais (a, w), whee a indicates the individual name that coesponds to the oot of the tee, i.e., a I = (a, ε) and the elements in the second place fom a tee accoding to ou definition of tees. Fo each individual name a in ou ABox, we can, theefoe, easily define the tee ooted in a as {w (a, w) I }. (a,ε) p p (a,1) (a,2) (a,3) (b,1) (b,2) p (b,ε) p p p p p p,s t,t (a, 11) (a, 12)... (a, 1k) (a, 31) (a, 32) (a, 33) (b, 21) (b, 22) Figue 2: A foest base fo the intepetation epesented by Figue 1. Figue 2 shows a epesentation of a foest base fo the intepetation fom Figue 1 above. Fo simplicity, the intepetation of concepts is no longe shown. The two tees, ooted in (a, ε) and (b, ε) espectively, ae now clea. A gaphical epesentation of the quey q fom Example 8 is shown in Figue 3, whee the meaning of the nodes and edges is analogous to the ones given fo intepetations. We call this quey a cyclic quey since its undelying undiected gaph is cyclic (cf. Definition 2). Figue 4 shows a match π fo q and I and, although we conside only one canonical model hee, it is not had to see that the quey is tue in each model of the knowledge base, i.e., K = q. 168

13 Conjunctive Quey Answeing fo the DL SHIQ x u s t y Figue 3: A gaph epesentation of the quey fom Example 8. z (a,1) (a,ε) (a,2) (a,3) u (b,1) x (b,ε) t,t y (b,2),s t,t t,t (a, 11) (a,12)... (a,1k) (a,31) (a,32) (a,33) (b, 21) z (b, 22) Figue 4: A match π fo the quey q fom Example 8 onto the model I fom Figue 1. The foest model popety is also exploited in the quey ewiting pocess. We want to ewite q into a set of queies q 1,...,q n of gound o tee-shaped queies such that K = q iff K = q 1... q n. Since the esulting queies ae gound o tee-shaped queies, we can exploe the known techniques fo deciding entailment of these queies. As a fist step, we tansfom q into a set of foest-shaped queies. Intuitively, foest-shaped queies consist of a set of tee-shaped sub-queies, whee the oots of these tees might be abitaily inteconnected (by atoms of the fom (t, t )). A tee-shaped quey is a special case of a foest-shaped quey. We will call the abitaily inteconnected tems of a foest-shaped quey the oot choice (o, fo shot, just oots). At the end of the ewiting pocess, we eplace the oots with individual names fom Inds(A) and tansfom the tee pats into a concept by applying the so called olling-up o tuple gaph technique (Tessais, 2001; Calvanese et al., 1998a). In the poof of the coectness of ou pocedue, we use the stuctue of the foest bases in ode to explicate the tansitive shot-cuts used in the quey match. By explicating we mean that we eplace each ole atom that is mapped to such a shot-cut with a sequence of ole atoms such that an extended match fo the modified quey uses only paths that ae in the foest base. 4.3 The Rewiting Steps The ewiting pocess fo a quey q is a six stage pocess. At the end of this pocess, the ewitten quey may o may not be in a foest shape. As we show late, this don t know non-deteminism does not compomise the coectness of the algoithm. In the fist stage, we deive a collapsing q co of q by adding (possibly seveal) equality atoms to q. Conside, 169

14 Glimm, Hoocks, Lutz, & Sattle fo example, the cyclic quey q = {(x, y), (x, y ), s(y, z), s(y, z)} (see Figue 5), which can be tansfomed into a tee-shaped one by adding the equality atom y y. x x y s s y y,y s z z Figue 5: A epesentation of a cyclic quey and of the tee-shaped quey obtained by adding the atom y y to the quey depicted on the left hand side. A common popety of the next thee ewiting steps is that they allow fo substituting the implicit shot-cut edges with explicit paths that induce the shot-cut. The thee steps aim at diffeent cases in which these shot-cuts can occu and we descibe thei goals and application now in moe detail: The second stage is called split ewiting. In a split ewiting we take cae of all ole atoms that ae matched to tansitive shot-cuts connecting elements of two diffeent tees and by-passing one o both of thei oots. We substitute these shot-cuts with eithe one o two ole atoms such that the oots ae included. In ou unning example, π maps u to (a,3) and x to (b, ε). Hence I = π (u, x), but the used -edge is a tansitive shot-cut connecting the tee ooted in a with the tee ooted in b, and by-passing (a, ε). Simila aguments hold fo the atom (u, z), whee the path that implies this shot-cut elationship goes via the two oots (a, ε) and (b, ε). It is clea that must be a non-simple ole since, in the foest base J fo I, thee is no diect connection between diffeent tees othe than between the oots of the tees. Hence, (π(u), π(x)) I holds only because thee is a ole s Tans R such that s * R. In case of ou example, itself is tansitive. A split ewiting eliminates tansitive shot-cuts between diffeent tees of a canonical model and adds the missing vaiables and ole atoms matching the sequence of edges that induce the shot-cut. ux u x s y t Figue 6: A split ewiting q s fo the quey shown in Figue 3. Figue 6 depicts the split ewiting q s = { (u, ux), (ux, x), (x, y), t(y, y), s(z, y), (u, ux), (ux, x), (x, z)} z 170

15 Conjunctive Quey Answeing fo the DL SHIQ of q that is obtained fom q by eplacing (i) (u, x) with (u, ux) and (ux, x) and (ii) (u, z) with (u, ux), (ux, x), and (x, z). Please note that we both intoduced a new vaiable (ux) and e-used an existing vaiable (x). Figue 7 shows a match fo q s and the canonical model I of K in which the two tees ae only connected via the oots. Fo the ewitten quey, we also guess a set of oots, which contains the vaiables that ae mapped to the oots in the canonical model. Fo ou unning example, we guess that the set of oots is {ux, x}. ux (a,ε) x (b,ε) t,t (a,1) (a,2) (a,3) u (b,1) y (b,2) s, (a,11) (a,12)... (a,1k) (a, 31) (a, 32) (a, 33) (b, 21) z (b, 22) Figue 7: A split match π s fo the quey q s fom Figue 6 onto the canonical intepetation fom Figue 1. In the thid step, called loop ewiting, we eliminate loops fo vaiables v that do not coespond to oots by eplacing atoms (v, v) with two atom (v, v ) and (v, v), whee v can eithe be a new o an existing vaiable in q. In ou unning example, we eliminate the loop t(y, y) as follows: q l = { (u, ux), (ux, x), (x, y), t(y, y ), t(y, y), s(z, y), (u, ux), (ux, x), (x, z)} is the quey obtained fom q s (see Figue 6) by eplacing t(y, y) with t(y, y ) and t(y, y) fo a new vaiable y. Please note that, since t is defined as tansitive and symmetic, t(y, y) is still implied, i.e., the loop is also a tansitive shot-cut. Figue 8 shows the canonical intepetation I fom Figue 1 with a match π l fo q l. The intoduction of the new vaiable y is needed in this case since thee is no vaiable that could be e-used and the individual (b, 22) is not in the ange of the match π s. ux (a,ε) x (b,ε) (a, 11) (a,1) (a,12)... (a,1k) (a,2) (a,3) u (a, 31) (a, 32) (a, 33) (b,1) y (b,2) s, t,t (b, 21) z y (b,22) Figue 8: A loop ewiting q l and a match fo the canonical intepetation fom Figue 1. The foth ewiting step, called foest ewiting, allows again the eplacement of ole atoms with sets of ole atoms. This allows the elimination of cycles that ae within a single 171

16 Glimm, Hoocks, Lutz, & Sattle tee. A foest ewiting q f fo ou example can be obtained fom q l by eplacing the ole atom (x, z) with (x, y) and (y, z), esulting in the quey q f = { (u, ux), (ux, x), (x, y), t(y, y ), t(y, y), s(z, y), (u, ux), (ux, x), (x, y), (y, z)}. Clealy, this esults in tee-shaped sub-queies, one ooted in ux and one ooted in x. Hence q f is foest-shaped w..t. the oot tems ux and x. Figue 9 shows the canonical intepetation I fom Figue 1 with a match π f fo q f. ux (a,ε) x (b,ε) (a, 11) (a, 12) (a,1) (a,2) (a,3) u (b,1) y (b,2),s t,t... (a,1k) (a, 31) (a, 32) (a, 33) z (b,21) y (b,22) Figue 9: A foest ewiting q f and a foest match π f fo the canonical intepetation fom Figue 1. In the fifth step, we use the standad olling-up technique (Hoocks & Tessais, 2000; Calvanese et al., 1998a) and expess the tee-shaped sub-queies as concepts. In ode to do this, we tavese each tee in a bottom-up fashion and eplace each leaf (labeled with a concept C, say) and its incoming edge (labeled with a ole, say) with the concept.c added to its pedecesso. Fo example, the tee ooted in ux (i.e., the ole atom (u, ux)) can be eplaced with the atom (. )(ux). Similaly, the tee ooted in x (i.e., the ole atoms (x, y), (y, z), s(z, y), t(y, y ), and t(y, y)) can be eplaced with the atom (.(( ( Inv(s)). ) ( (t Inv(t)). ))(x). Please note that we have to use ole conjunctions in the esulting quey in ode to captue the semantics of multiple ole atoms elating the same pai of vaiables. Recall that, in the split ewiting, we have guessed that x and ux coespond to oots and, theefoe, coespond to individual names in Inds(A). In the sixth and last ewiting step, we guess which vaiable coesponds to which individual name and eplace the vaiables with the guessed names. A possible guess fo ou unning example would be that ux coesponds to a and x to b. This esults in the (gound) quey {(. )(a), (a, b), (.(( ( Inv(s)). ) ( (t Inv(t)). )))(b)}, which is entailed by K. Please note that we focused in the unning example on the most easonable ewiting. Thee ae seveal othe possible ewitings, e.g., we obtain anothe ewiting fom q f by eplacing ux with b and x with a in the last step. Fo a UCQ, we apply the ewiting steps to each of the disjuncts sepaately. 172

17 Conjunctive Quey Answeing fo the DL SHIQ At the end of the ewiting pocess, we have, fo each disjunct, a set of gound queies and/o queies that wee olled-up into a single concept atom. The latte queies esult fom foest ewitings that ae tee-shaped and have an empty set of oots. Such tee-shaped ewitings can match anywhee in a tee and can, thus, not be gounded. Finally, we check if ou knowledge base entails the disjunction of all the ewitten queies. We show that thee is a bound on the numbe of (foest-shaped) ewitings and hence on the numbe of queies poduced in the ewiting pocess. Summing up, the ewiting pocess fo a connected conjunctive quey q involves the following steps: 1. Build all collapsings of q. 2. Build all split ewitings of each collapsing w..t. a subset R of oots. 3. Build all loop ewitings of the split ewitings. 4. Build all (foest-shaped) foest ewitings of the loop ewitings. 5. Roll up each tee-shaped sub-quey in a foest-ewiting into a concept atom and 6. eplace the oots in R with individual names fom the ABox in all possible ways. Let q 1,...,q n be the queies esulting fom the ewiting pocess. In the next section, we define each ewiting step and pove that K = q iff K = q 1 q n. Checking entailment fo the ewitten queies can easily be educed to KB consistency and any decision pocedue fo SHIQ KB consistency could be used in ode to decide if K = q. We pesent one such decision pocedue in Section Quey Rewiting In the pevious section, we have used seveal tems, e.g., tee- o foest-shaped quey, athe infomally. In the following, we give definitions fo the tems used in the quey ewiting pocess. Once this is done, we fomalize the quey ewiting steps and pove the coectness of the pocedue, i.e., we show that the foest-shaped queies obtained in the ewiting pocess can indeed be used fo deciding whethe a knowledge base entails the oiginal quey. We do not give the detailed poofs hee, but athe some intuitions behind the poofs. Poofs in full detail ae given in the appendix. 5.1 Tee- and Foest-Shaped Queies In ode to define tee- o foest-shaped queies moe pecisely, we use mappings between queies and tees o foests. Instead of mapping equivalence classes of tems by * to nodes in a tee, we extend some well-known popeties of functions as follows: Definition 9. Fo a mapping f : A B, we use dom(f) and an(f) to denote f s domain A and ange B, espectively. Given an equivalence elation * on dom(f), we say that f is injective modulo * if, fo all a, a dom(f), f(a) = f(a ) implies a * a and we say that f is bijective modulo * if f is injective modulo * and sujective. Let q be a quey. A tee mapping fo q is a total function f fom tems in q to a tee such that 173

18 Glimm, Hoocks, Lutz, & Sattle 1. f is bijective modulo *, 2. if (t, t ) q, then f(t) is a neighbo of f(t ), and, 3. if a Inds(q), then f(a) = ε. The quey q is tee-shaped if (Inds(q)) 1 and thee is a tee mapping fo q. A oot choice R fo q is a subset of Tems(q) such that Inds(q) R and, if t R and t * t, then t R. Fo t R, we use Reach(t) to denote the set of tems t Tems(q) fo which thee exists a sequence of tems t 1,...,t n Tems(q) such that 1. t 1 = t and t n = t, 2. fo all 1 i < n, thee is a ole such that (t i, t i+1 ) q, and, 3. fo 1 < i n, if t i R, then t i * t. We call R a oot splitting w..t. q if eithe R = o if, fo t i, t j R, t i * t j implies that Reach(t i ) Reach(t j ) =. Each tem t R induces a sub-quey subq(q, t) := {at q the tems in at occu in Reach(t)}\ {(t, t) (t, t) q}. A quey q is foest-shaped w..t. a oot splitting R if eithe R = and q is tee-shaped o each sub-quey subq(q, t) fo t R is tee-shaped. Fo each tem t R, we collect the tems that ae eachable fom t in the set Reach(t). By Condition 3, we make sue that R and * ae such that each t Reach(t) is eithe not in R o t * t. Since queies ae connected by assumption, we would othewise collect all tems in Reach(t) and not just those t / R. Fo a oot splitting, we equie that the esulting sets ae mutually disjoint fo all tems t, t R that ae not equivalent. This guaantees that all paths between the sub-queies go via the oot nodes of thei espective tees. Intuitively, a foest-shaped quey is one that can potentially be mapped onto a canonical intepetation I = ( I, I) such that the tems in the oot splitting R coespond to oots (a, ε) I. In the definition of subq(q, t), we exclude loops of the fom (t, t) q, as these pats of the quey ae gounded late in the quey ewiting pocess and between gound tems, we allow abitay elationships. Conside, fo example, the quey q s of ou unning example fom the pevious section (cf. Figue 6). Let us again make the oot choice R := {ux, x} fo q. The sets Reach(ux) and Reach(x) w..t. q s and R ae {ux, u} and {x, y, z} espectively. Since both sets ae disjoint, R is a oot splitting w..t. q s. If we choose, howeve, R := {x, y}, the set R is not a oot splitting w..t. q s since Reach(x) = {ux, u, z} and Reach(y) = {z} ae not disjoint. 5.2 Fom Gaphs to Foests We ae now eady to define the quey ewiting steps. Given an abitay quey, we exhaustively apply the ewiting steps and show that we can use the esulting queies that ae foest-shaped fo deciding entailment of the oiginal quey. Please note that the following definitions ae fo conjunctive queies and not fo unions of conjunctive queies since we apply the ewiting steps fo each disjunct sepaately. 174

19 Conjunctive Quey Answeing fo the DL SHIQ Definition 10. Let q be a Boolean conjunctive quey. A collapsing q co of q is obtained by adding zeo o moe equality atoms of the fom t t fo t, t Tems(q) to q. We use co(q) to denote the set of all queies that ae a collapsing of q. Let K be a SHIQ knowledge base. A quey q s is called a split ewiting of q w..t. K if it is obtained fom q by choosing, fo each atom (t, t ) q, to eithe: 1. do nothing, 2. choose a ole s Tans R such that s * R and eplace (t, t ) with s(t, u), s(u, t ), o 3. choose a ole s Tans R such that s * R and eplace (t, t ) with s(t, u), s(u, u ), s(u, t ), whee u, u N V ae possibly fesh vaiables. We use s K (q) to denote the set of all pais (q s, R) fo which thee is a quey q co co(q) such that q s is a split ewiting of q co and R is a oot splitting w..t. q s. A quey q l is called a loop ewiting of q w..t. a oot splitting R and K if it is obtained fom q by choosing, fo all atoms of the fom (t, t) q with t / R, a ole s Tans R such that s * R and by eplacing (t, t) with two atoms s(t, t ) and s(t, t) fo t N V a possibly fesh vaiable. We use l K (q) to denote the set of all pais (q l, R) fo which thee is a tuple (q s, R) s K (q) such that q l is a loop ewiting of q s w..t. R and K. Fo a foest ewiting, fix a set V N V of vaiables not occuing in q such that (V ) (Vas(q)). A foest ewiting q f w..t. a oot splitting R of q and K is obtained fom q by choosing, fo each ole atom (t, t ) such that eithe R = and (t, t ) q o thee is some t R and (t, t ) subq(q, t ) to eithe 1. do nothing, o 2. choose a ole s Tans R such that s * R and eplace (t, t ) with l (Vas(q)) ole atoms s(t 1, t 2 ),..., s(t l, t l+1 ), whee t 1 = t, t l+1 = t, and t 2,...,t l Vas(q) V. We use f K (q) to denote the set of all pais (q f, R) fo which thee is a tuple (q l, R) l K (q) such that q f is a foest-shaped foest ewiting of q l w..t. R and K. If K is clea fom the context, we say that q is a split, loop, o foest ewiting of q instead of saying that q is a split, loop, o foest ewiting of q w..t. K. We assume that s K (q), l K (q), and f K (q) contain no isomophic queies, i.e., diffeences in (newly intoduced) vaiable names only ae neglected. In the next section, we show how we can build a disjunction of conjunctive queies q 1 q l fom the queies in f K (q) such that each q i fo 1 i l is eithe of the fom C(v) fo a single vaiable v Vas(q i ) o q i is gound, i.e., q i contains only constants and no vaiables. It then emains to show that K = q iff K = q 1 q l. 5.3 Fom Tees to Concepts In ode to tansfom a tee-shaped quey into a single concept atom and a foest-shaped quey into a gound quey, we define a mapping f fom the tems in each tee-shaped subquey to a tee. We then incementally build a concept that coesponds to the tee-shaped quey by tavesing the tee in a bottom-up fashion, i.e., fom the leaves upwads to the oot. 175