DL-Lite with Attributes and Sub-Roles (Full Version)

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1 DL-Lite with Attributes and Sub-Roles (Full Version) A. Artale and V. Ryzhikov KRDB Research Centre Free University of Bozen-Bolzano, Italy R. Kontchakov Dept. of Comp. Science and Inf. Sys. Birkbeck College, London, UK Abstract We extend the tractable DL-Lite languages by (i) relaxing the restriction on the allowed interaction between cardinality constraints and role inclusions; (ii) extending the languages with attributes. On the one hand, we push the limit on the use of number restrictions over role hierarchies and also show the effect of the presence of the ABox on such constraints. On the other hand, attributes a notion borrowed from data models associate concrete values from datatypes to abstract objects and in this way complement DL-Lite roles that describe relationships between abstract objects. We present complexity results for two most important reasoning problems in DL-Lite: combined complexity of KB satisfiability and data complexity of query answering. 1 Introduction The DL-Lite family of description logics has recently been proposed and investigated in (Calvanese et al., 2005, 2006, 2007) and later extended in (Artale et al., 2007a; Poggi et al., 2008; Artale et al., 2009). The relevance of the DL-Lite family is witnessed by the fact that it forms the basis of OWL 2 QL, one of the three profiles of OWL 2 ( According to the official W3C profiles document, the purpose of OWL 2 QL is to be the language of choice for applications that use very large amounts of data. This paper extends the DL-Lite languages of (Artale et al., 2009) by: (i) relaxing the restriction on the interaction between cardinality constraints (or number restrictions, N ) and role inclusions (or hierarchies, H); (ii) having the so called attributes (A), i.e., the possibility to associate concrete values from datatypes to abstract objects. These extensions will be formalized in a new family of languages, DL-Lite HN α A, with α {core, krom, horn, bool}. Original and tight complexity results for both KB satisfiability and query answering will be presented in this paper. Role inclusion axioms were introduced in DL-Lite by (Calvanese et al., 2006). The possibility to combine them with cardinality constraints on roles has been studied by (Artale et al., 2009), which shows the dramatic impact of role inclusions, when combined with cardinality (or even functionality) constraints, on the computational complexity of reasoning. In particular, query answering becomes CONP-complete for the data complexity even for the simplest, core, logics and PTIME-complete for the core and Horn logics only with functionality constraints only; moreover, KB satisfiability, which is NLOGSPACE-complete for the combined complexity in the simplest, core, case when role inclusions and cardinalities are used separately, becomes EXPTIME-complete when they both are present and interact. The DL-Lite logic DL-Lite A, introduced in (Poggi et al., 2008), retains both role inclusions and functionality constraints and, to regain nice computational results, limits the interaction between them. A similar restriction is also used by (Artale et al., 2009) for limiting this kind of interaction thus preserving the computational properties of the DL-Lite fragments with only role inclusions or only cardinality constraints. The restriction called (inter) essentially forbids the use of cardinality constraints whenever a role is specialized. In this paper we push the limit by relaxing these restriction and allowing specialization of roles even when cardinalities are 1

2 Employee salary (Real) Professor salary ({55K 100K}) Researcher salary ({35K 70K}) Figure 1: Salary example specified on them. We present two new restrictions, (inter T ) and (inter KB ), whose difference lies in the fact that the latter takes account of the number of R-successors in the ABox while the former does not. We show that (inter KB ) does not lead to an increase of the complexity of KB satisfiability, but adopting (inter T ) brings the computational complexity up to EXPTIME. The notion of attributes, borrowed from conceptual modelling formalisms, introduces a distinction between (abstract) objects and concrete values (integers, reals, strings, etc.) and, consequently, between concepts (sets of objects) and datatypes (sets of values), and between roles (relating objects to objects) and attributes (relating objects to values). The language DL-Lite A (Poggi et al., 2008) was introduced with the aim of capturing the notion of attributes in DL-Lite in the setting of ontology-based data access (OBDA). The datatypes of DL-Lite A are modelled as pairwise disjoint sets of values that are also disjoint from concepts. A similar choice is made by various DLs encoding conceptual models (Calvanese et al., 1999; Berardi et al., 2005; Artale et al., 2007b). Furthermore, datatypes of DL-Lite A are used for typing attributes globally, i.e., even if associated to different concepts, attributes sharing the same name are forced to have the same range restriction e.g., to constrain the range of the attribute salary to the type Real the following DL-Lite A axiom is used: salary Real. In this work we consider a more expressive language for attributes and datatypes in DL-Lite. We present two main extensions of the original DL-Lite A : (i) datatypes are not necessarily mutually disjoint but Horn-like constraints can formalize relations between them; (ii) range restrictions for attributes are local (rather than global), i.e., concept inclusion axioms of the form C U.T specify that all values of the attribute U of instances of concept C belong to the datatype T. In this way, we capture a wider range of datatypes (e.g., intervals over the reals) and allow re-use of the very same attribute associated to different concepts, but with different range restrictions. For example, the Entity-Relationship diagram shown in Fig. 1 says that employees salary is of type Real: Employee salary.real; researchers salary is in the range {35K 70K} (interval type, subset of Real): Researcher salary.{35k 70K}; and professors salary in the range {55K 100K}: Professor salary.{55k 100K}; with researchers and professors being employees: Researcher Employee, Professor Employee. Local attributes are strictly more expressive than global attributes the axiom salary.real would force every salary value to be a real number. Using local attributes we can infer concept disjointness just from datatype disjointness for the same (existentially qualified) attribute. For example, assume that in the scenario of Fig. 1 we add the concept of ForeignEmployee as having at-least one salary that must be a String (to include also the currency). Then Employee and ForeignEmployee will be disjoint concepts i.e., Employee ForeignEmployee because of the disjointness of the respective datatypes and restrictions on the salary attribute. More generally, we allow Horn datatype inclusions, which, for instance, can express that an intersection of a number of datatypes is empty. 2

3 Our work lies in between the DL-Lite A proposal and the extensions of DLs with concrete domains; see (Lutz, 2003) for an overview. According to the concrete domain terminology, we consider a pathfree extension with unary predicates predicates coincide with datatypes with a fixed interpretation, as in DL-Lite A. Differently from the concrete domain approach, we do not require attributes to be functional but we can specify generic number restrictions over them similarly to extensions of EL with datatypes (Baader et al., 2005; Magka et al., 2011) and the notion of datatype properties in OWL 2 (Pan and Horrocks, 2011; Cuenca Grau et al., 2008). Our approach works as far as datatypes are unbounded query answering is CONP in presence of datatypes of specific cardinalities (Franconi et al., 2011; Savković, 2011) and no covering constraints can hold between them (unless the DL-Lite fragments with the full Booleans are considered). We provide tight complexity results showing that for the Bool, Horn and core cases the addition of local range restrictions for attributes does not change the complexity of KB satisfiability. On the other hand, surprisingly, in the Krom case the complexity increases from NLOGSPACE to NP. These results reflect the intuition that universal restrictions over attributes as studied in this paper cannot introduce cyclic dependencies between concepts, while the (unrestricted) use of universal range restrictions ( R.C) together with sub-roles, by which we can encode qualified existential restrictions ( R.C), and arbitrary TBox axioms would result in EXPTIME-completeness. We complete our complexity results by showing that positive existential query answering (and so, conjunctive query answering) for core and Horn KBs with local attributes and Horn datatype constraints ) under the relaxed version of the restriction on sub-roles (and sub-attributes) and cardinalities is still FO rewritable and so in AC 0 for data complexity. The paper is organized as follows. Section 2 presents DL-Lite and its fragments. Section 3 investigates the complexity of deciding KB satisfiability when relaxing the restriction on the interaction between cardinalities and role hierarchies. Sections 4 and 5 study combined complexity of KB satisfiability and data complexity for answering positive existential queries, respectively, when attributes and datatypes are present. Section 6 concludes this paper. Complete proofs of all the results can be found in (Artale et al., 2012). (i.e., the logics DL-Lite HN A horn and DL-Lite HN A core 2 The Description Logic DL-Lite HN A bool The language of DL-Lite HN bool A contains object names a 0, a 1,..., value names v 0, v 1,..., concept names A 0, A 1,..., role names P 0, P 1,..., attribute names U 0, U 1,..., and datatype names T 0, T 1,.... Complex roles R, concepts C and datatypes T are defined as follows: R ::= P i P i, B ::= A i q R q U i C ::= B C C 1 C 2, T ::= T i1 T ik, where q is a positive integer. The concepts of the form B are called basic concepts. A DL-Lite HN bool A T, is a finite set of concept, role, attribute and datatype inclusion axioms of the form: C 1 C 2 and C U. T, R 1 R 2, U 1 U 2, T 1 T 2, and an ABox, A, is a finite set of assertions of the form: A k (a i ), A k (a i ), P k (a i, a j ), P k (a i, a j ), U k (a i, v j ), T k (v j ), T k (v j ). TBox, We standardly abbreviate 1 R and 1 U by R and U, respectively. Taken together, a TBox T and an ABox A constitute the knowledge base (KB) K = (T, A). Semantics. As usual in description logic, an interpretation, I = ( I, I), consists of a nonempty domain I and an interpretation function I. The interpretation domain I is the union of two nonempty disjoint sets: the domain of objects I O and the domain of values I V. We assume that all interpretations agree on the semantics assigned to each datatype T i, as well as on each value v j. In particular, Ti I = val(t i ) I V 3

4 Languages (inter) (inter T ) (inter KB) No Restrict. QA DL-Lite HN core NLOGSPACE NP [Th.1] NLOGSPACE [Th.3] AC 0 DL-Lite HN horn PTIME EXPTIME [Th.1] PTIME [Th.3] AC 0 DL-Lite HN krom NLOGSPACE EXPTIME NP [Th.1] NLOGSPACE [Th.3] CONP DL-Lite HN bool NP EXPTIME [Th.1] NP [Th.3] CONP DL-Lite HN core A NLOGSPACE [Th.5] NP [Th.1] NLOGSPACE [Th.5] AC 0 [Th.8] DL-Lite HN horn A PTIME [Th.5] EXPTIME [Th.1] PTIME [Th.5] AC 0 [Th.8] DL-Lite HN krom A EXPTIME NP [Th.6] NP [Th.1] NP [Th.6] CONP DL-Lite HN bool A NP [Th.5] EXPTIME [Th.1] NP [Th.5] CONP Table 1: Complexity of DL-Lite logics; = (Artale et al., 2009). is the set of values of the datatype T i, and each v j is interpreted as one specific value, denoted val(v j ), i.e., vj I = val(v j ) I V. Furthermore, I assigns to each object name a i an element a I i I O, to each concept name A k a subset A I k I O of the domain of objects, to each role name P k a binary relation Pk I I O I O over the domain of objects, and to each attribute name U k a binary relation Uk I I O I V. We adopt here the unique name assumption (UNA): ai i a I j, for all i j. The role, concept and datatype constructs are interpreted in I in the standard way: (P k )I = {(w, w) I O I O (w, w ) P I k }, I = I O, I =, ( C) I = I O \ C I, ( q R) I = { w I O {w (w, w ) R I } q }, ( q U) I = { w I O {v (w, v) U I } q }, ( U. T ) I = { w I O v. (w, v) U I v T I}, (C 1 C 2 ) I = C I 1 C I 2, (T 1 T 2 ) I = T I 1 T I 2, where X is the cardinality of X. The satisfaction relation = is also standard: I = C 1 C 2 iff C I 1 C I 2, I = R 1 R 2 iff R I 1 R I 2, I = T 1 T 2 iff T I 1 T I 2, I = U 1 U 2 iff U I 1 U I 2, I = A k (a i ) iff a I i A I k, I = P k (a i, a j ) iff (a I i, a I j ) P I k, I = A k (a i ) iff a I i / A I k, I = P k (a i, a j ) iff (a I i, a I j ) / P I k, I = T k (v j ) iff v I j T I k, I = U k (a i, v j ) iff (a I i, v I j ) U I k, I = T k (v j ) iff v I j / T I k. A KB K = (T, A) is said to be satisfiable (or consistent) if there is an interpretation, I, satisfying all the members of T and A. In this case we write I = K (as well as I = T and I = A) and say that I is a model of K (of T and A). A positive existential query q(x 1,..., x n ) is a first-order formula ϕ(x 1,..., x n ) constructed by means of conjunction, disjunction and existential quantification starting from atoms of the from A k (t 1 ), T k (t 1 ), P k (t 1, t 2 ) and U k (t 1, t 2 ), where A k is a concept name, T k a datatype name, P k a role name, U k an attribute name, and t 1, t 2 are terms taken from the list of variables y 0, y 1,..., the list of object names a 0, a 1,... and the list of value names v 0, v 1,.... We write q( x) for a query with free variables x = x 1,..., x n and q( a) for the result of replacing every occurrence of x i in ϕ( x) with the ith component a i of a vector of constants a = a 1,..., a n. A conjunctive query is a positive existential query that contains no disjunction. For a KB K = (T, A), we say that a tuple a of constant names from A is a certain answer to q( x) with respect to K, and write K = q( a), if I = q( a) (with I regarded as a first-order interpretation) whenever I = K. The query answering problem is: given a KB K = (T, A), a query q( x), and a tuple a of constant names from A, decide whether K = q( a). Fragments of DL-Lite HN bool A. We consider various syntactical restrictions on the language of DL-Lite HN bool A along two axes: (i) the Boolean operators ( bool ) on concepts and (ii) the attributes (A). Similarly to classical 4

5 logic, we adopt the following definitions. A TBox T will be called a Krom TBox from the Krom fragment of FOL if only negation is allowed in the construction of its complex concepts, i.e., if C ::= B B, (Krom) (here and below the B are basic concepts). A TBox T will be called a Horn TBox if its complex concepts are constructed by using only intersection: C ::= B 1 B k. Finally, we call T a core TBox if its concept and datatype inclusions are of the form: (Horn) B 1 B 2, B 1 B 2, T i1 T i2, T i1 T ik. (core) Note that the positive occurrences of B on the right-hand side of the above axioms can also have the form U.T. As B 1 B 2 is equivalent to B 1 B 2, core TBoxes can be regarded as sitting in the intersection of Krom and Horn TBoxes. In this paper, in addition to the full language of DL-Lite HN bool A, we study the following logics: DL-Lite HN A krom, DL-LiteHN A horn, DL-Lite HN A core are the fragments of DL-Lite HN bool A with Krom, Horn, and core TBoxes, respectively; DL-Lite HN A α, for α {core, krom, horn, bool}, is the fragment of DL-LiteHN α without attributes and datatypes. Table 1 summarizes the obtained complexity results (with numbers q coded in binary) for KB satisfiability (combined complexity) and query answering (data complexity). 3 Complexity of Reasoning in DL-Lite HN α As shown in (Artale et al., 2009), reasoning in DL-Lite HN α is already rather costly (EXPTIME-complete) due to the interaction between role inclusions and number restrictions. However, both of these constructs turn out to be useful for the purposes of conceptual modelling. By limiting their interplay one can get languages with better computational properties. In this section we formulate and study two syntactic restrictions that are weaker than the ones known in the literature (Poggi et al., 2008; Artale et al., 2009). In the following, we denote by role(k) the set of role names in K; let role ± (K) = {P k, P k P k role(k)}. For a role R, let inv(r) = P k if R = P k and inv(r) = P k if R = P k. Given a TBox T we denote by T the reflexive and transitive closure of the relation {(R, R ), (inv(r), inv(r )) R R T }. We say that R is a proper sub-role of R in T if R T R and R T R. A proper sub-role R of R is a direct sub-role of R if there is no other proper sub-role R of R such that R is a proper sub-role of R ; dsub T (R) denotes the set of direct sub-roles of R in T. An occurrence of a concept on the right-hand (left-hand) side of a concept inclusion is called negative if it is in the scope of an odd (even) number of negations ; otherwise it is called positive. 3.1 Counting Successors in Hierarchies The languages DL-Lite (HN α ) of (Artale et al., 2009) are the result of imposing the following syntactic restriction on DL-Lite HN α TBoxes T : (inter) if R has a proper sub-role in T then T contains no negative occurrences of number restrictions q R or q inv(r) with q 2. To formulate our subtler restrictions, we need the following parameters, for a TBox T and a role R role ± (T ): ub(r, T ) = min ( { } {q 1 q 2 and q R occurs negatively in T } ), lb(r, T ) = max ( {0} {q q R occurs positively in T } ), rank(r, T ) = max ( lb(r, T ), rank(r, T ) ). R dsub T (R) 5

6 Consider first the languages obtained from DL-Lite HN α role ± (T ): (inter T ) if R has a proper sub-role in T then ub(r, T ) rank(r, T ). by imposing the following restriction on all R It turns out, however, that these languages are too expressive to keep the same complexity of the satisfiability problem as their basic counterparts: THEOREM 1. Under (inter T ), KB satisfiability is NP-hard for DL-Lite HN core and DL-Lite HN krom and EXPTIMEcomplete for DL-Lite HN horn and DL-Lite HN bool. Proof. To prove NP-hardness, we show that graph 3-colourability can be reduced to DL-Lite HN core KB satisfiability. Let G = (V, E) be a graph with vertices V and edges E and {r, g, b} be three colours. Consider the following KB K = (T, A) with a role name S and its sub-roles R i, for each vertex v i V, and object names o, r, g, b and v i, for each vertex v i V : T ={ ( V + 4) S } {R i S, B 1 R i, B 2 R i v i V } { R i R j (v i, v j ) E}, A ={B 1 (o), S(o, r), S(o, g), S(o, b)} {S(o, v i ), B 2 (v i ) v i V }. Clearly, K enjoys (inter T ). It can be shown that K is satisfiable iff G is 3-colourable. Indeed, for every vertex v i, the individual v i is a an S-successor of o, which has another three S-successors: r, g and b. On the other hand, for each vertex v i, o must have an R i -successor (which is also an S-successor) but the total number of S-successors of o is bounded by V + 3. Since the v j cannot be R i -successors (for any pair i, j), all the R i -successors of o must be among r, g and b, which by the range disjointness axiom for R i and R j (provided that (v i, v j ) E) happens iff the graph is 3-colourable. EXPTIME-hardness can be proved by reduction of the complement of the state reachability problem for alternating Turing machines (ATMs). We only give an idea of the proof here. Suppose we are given an ATM that, on every input, requires only a polynomial number of cells on the tape. Without loss of generality we may assume that each state has exactly two successor states on each input symbol. Let n be the length of the input and l the number of cells required. Then we need the following 3 sets of roles, for 0 k < 3, S kai, for each symbol a Σ and position 1 i l, so that S kai says the symbol a is written at the position i ; H kqi, for each state q Q and head position 1 i l, so that H kqi says the current state is q and the head is over the position i ; (the three sets are required for the disjointness constraints below). Since each state has two successors, we also need two sub-roles (left and right) of each S kai : LS kai S kai, RS kai S kai and sub-roles LH kqi and RH kqi for each H kqi. With the help of these pairs of roles we can encode transitions of the form δ(a, q) = {(a 1, q 1, d 1 ), (a 2, q 2, d 2 )} in a natural way: S kai H kqi LH k+1 q 1(i+d 1) LS k+1 a1i RH k+1 q2(i+d 2) RS k+1 a2i, where k denotes the value of k modulo 3. We also need to say that cells that are not under the current position of the head do not change their symbols: for all j i, S kaj H kqi LS k+1 aj RS k+1 aj. 6

7 But now the main difficulty is to enforce that all the LS k+1 aj - and LH k+1 q1i 1 -successors coincide (and similarly, their right counterparts). We could introduce a new functional super-role for all of them but then the restriction (inter T ) would be violated. Instead, we will employ a role T k and its two subroles L k and R k, for each 0 k < 3, and super-roles LS kai, RSkai, LH kqi and RH kqi. Each of these super-roles contains its title role, L k and T k 1 as its sub-roles and has not more than 2 successors, e.g.: LS kai LS kai, L k LS kai, T k 1 LS kai, 3 LS kai. With the help of disjointness constraints of the form T k 1 T k and T k 1 S kai and an ABox, modelling the initial configuration and containing H 0q01(z, a), S 0a11(z, a),..., S 0al l(z, a) and T 0 (z, a), we can ensure that in all models of this TBox each point (but z) has a single T k 1 -predecessor and a single L k -successor, which is a T k -successor, and, by the cardinality constraints above, is also the LS k+1 aj - and LH k+1 q1i 1 -successor for the respective combination of subscripts. It is easily seen that the TBox enjoys (inter T ) and encodes the tree of computations of the ATM. In a similar way one can encode the condition that a certain state is never reached. The NP-hardness proof used the fact that the restriction (inter T ) does not impose any bounds on the number of R-successors in the ABox. And the EXPTIME-hardness proof also reveals that if we were to maintain the low complexity of reasoning, we would have to take into account not only the number of R-successors in the ABox, but also the number of R -predecessors (i.e., R-successors) that come to the unnamed individuals outside the ABox. In the next section, this intuition will drive our next attempt to weaken the restrictions on the interaction of role inclusions and cardinality constraints. 3.2 Taking the ABox into Account In this section, we formulate our second restriction, (inter KB ), and show that the complexity of KB satisfiability remains low under it. We need the following additional parameters, for an ABox A, a TBox T and R role ± (T ): rank(r, A) = max ( {0} {n R i (a, a i ) A, R i T R, for distinct a 1,..., a n } ), { 1, if lb(r, T ) 1, for some R T pred(r, T ) = R, 0, otherwise. Then our second restriction on role inclusions and cardinality constraints is as follows: for every R role ± (T ), (inter KB ) if R has a proper sub-role in T then ub(r, T ) rank(r, T ) + max ( pred(r, T ), rank(r, A) ). Both (inter T ) and (inter KB ) are weaker than (inter) and, for example, allow for the specialization of functional roles: T = { 2 R, R 1 R 2, R 2 R} and A = {R(a, b), R 1 (a 1, b 1 ), R 2 (a 2, b 2 )} do not satisfy (inter), but do satisfy both (inter T ) and (inter KB ). The above restrictions will also be applied to sub-attributes in the languages DL-Lite HN α A. To show that (inter KB ) matches the complexity of KB satisfiability of the basic languages, we adapt the proof presented in (Artale et al., 2009), where a DL-Lite HN bool KB K = (T, A) is encoded into a firstorder sentence K e with one variable. Every a i ob(a) is associated to the individual constant a i, and every concept name A i to the unary predicate A i (x). For each concept q R in K we introduce a fresh unary predicate E q R(x). We also introduce the set dr(k) = {dp k, dp k P k role ± (K)} 7

8 of individual constants, as representatives of the objects in the domain (dp k ) and the range (dp k ) of each role P k, respectively. The encoding C of a concept C is defined inductively: =, =, (A i ) = A i (x), ( C) = C (x), ( q R) = E q R(x), (C 1 C 2 ) = C 1 (x) C 2 (x). The following sentence encodes the knowledge base K: K e = x [T (x) T R (x) ( ɛr (x) δ R (x) )] A e, R role ± (K) where T ( (x) = C 1 (x) C2 (x) ), δ R (x) = T R (x) = C 1 C 2 T q,q Q R T, q >q ( Eq R(x) E q R(x) ), ( Eq R(x) E q R (x) ), R T R q Q R T and Q R T contains 1, all q such that q R occurs in T and all QR T, for R T R. Sentence encodes A e the ABox A: A e = A k (a i ) A k (a i ) E q e R,ai R(a i ), A k (a i) A A k (a i) A a i ob(a) R T R, R (a i,a j) A P k (a i,a j) A R(a i,a j) A, R T P k where qr,a e is the maximum number in QR T such that there are qe R,a many distinct a i with R i (a, a i ) A and R i T R. For each R role± (K), we also need a formula expressing the fact that the range of R is not empty whenever its domain is nonempty: ɛ R (x) = E 1 R(x) inv(e 1 R(dr)), with inv(e 1 R(dr)) denoting E 1 P k (dp k ) if R = P k and E 1 P k (dp k ) if R = P k. LEMMA 2. A DL-Lite HN bool KB K under (inter KB ) is satisfiable iff the one-variable sentence K e is satisfiable. Proof. The only challenging direction is ( ). To prove it, we adapt the proofs of Theorem 5.2 and Lemma 5.14 of (Artale et al., 2009). The idea of the proof is to construct a DL-Lite HN bool model I of K from the minimal Herbrand model M of K e. with domain D = ob(a) dr(k). The interpretation I = ( I, I) is defined inductively: I = m=0 W m, such that W 0 is the set ob(a), and each set W m+1, m 0, is constructed by adding to W m fresh copies of elements of dr(k). We write cp(w) for the element d D of which w is a copy, with cp(a) = a for a ob(a) = W 0. We define a I i = a i, for all individuals a i ob(a), and, for all concept names A k, A I k = {w I M = A k[cp(w)]}, The interpretation of each role P k, is defined inductively as Pk I = m=0 P k m, where P k m W m W m, along with the construction of I. The initial interpretation of P k is P 0 k = {(a M i, a M j ) W 0 W 0 R(a i, a j ) A and R T P k }. The required R-rank r(r, d) of d D is defined as: r(r, d) = max ( {0} {q Q R+ T M = E q R[d]} ), 8

9 where Q R+ T contains all q such that q R occurs positively in T. Note that: r(r, d) lb(r, T ). (1) The actual R-rank r m (R, w) of a point w I at step m is defined as follows: r m (R, w) = {w (w, w ) P m k P m+1 j, P j dsub T (P k )}, if R = P k ; replace (w, w ) by (w, w) if R = P k. Assume that W m and Pk m, m 0, have been already defined. Let W m+1 \ W m =. If the actual rank of some points is smaller than the required rank, then, we cure these defects by adding R-successors for them. For each P k role(k), we consider two sets of defects in Pk m: Λm k = {w W m \ W m 1 r m (P k, w) < r(p k, cp(w))} and Λ m k = {w W m \ W m 1 r m (P k, w) < r(p k, cp(w))}. In each equivalence class [R] = {S S T R, R T S} we select a single role, a representative. Let G = (Rep T, E) be a directed graph such that Rep T is the set of representatives and (R, R ) E iff R is a proper sub-role of R. We use the ascending total order induced on G when choosing an element P k in Rep T, and extend in that way W m and Pk m to W m+1 and P m+1 k, respectively. (Λ m k ) Let w Λm k, q = r(p k, d) r m (P k, w), d = cp(w). There is q q > 0 with M = E q P k [d] and so, M = E 1 P k [d] and M = E 1 P k [dp k ]. We take q fresh copies w 1,..., w q of dp k, add them to W m+1 and for each 1 i q, set cp(w i ) = dp k, add the pairs (w, w m+1 i ) to each Pj with P k T P j and the pairs (w i m+1, w) to each Pj with P k T P j; (Λ m k ) This rule is the mirror image of (Λ m k ): P k and dp k are replaced with P k and dp k, respectively. We now show that I = ϕ for each ϕ T A. From the construction of R I, it immediately follows that the interpretation of roles respects role inclusions, i.e., R I 1 R I 2 whenever R 1 R 2 T. For ϕ = A k (a i ) and ϕ = A k (a i ), the claim follows from the definition of A I k. For ϕ = P k(a i, a j ) and ϕ = P k (a i, a j ), we have (a i, a j ) P I k iff (a i, a j ) P 0 k iff R(a i, a j ) A and R T P k. The challenging part is, however, to show that I = C 1 C 2 whenever M = x (C 1 (x) C 2 (x)), for each ϕ = C 1 C 2. We need to prove that, for all w I and all q R in T, (a 1 ) M = E q R[cp(w)] implies w ( q R) I, for all q R that occur positively in T ; (a 2 ) w ( q R) I implies M = E q R[cp(w)], for all q R that occur negatively in T. In order to do that, we demonstrate the following property of the unravelling construction, for all w W m : { rank(r, A), if m = 0, r m (R, w) rank(r i, T ) + (2) pred(r, T ), if m > 0. R i dsub T (R) First, note that we have, for all w W m : where r m (R, w) = s R w + {w W m (w, w ) R m } s R w + { rank(r, A), if m = 0, pred(r, T ), if m > 0, s R w = {w W m+1 \ W m (w, w ) R m+1 i, R i dsub T (R)}. Indeed, the case m = 0 is immediate from the definition of the Pk 0 ; if m > 0 then the second component of the sum does not exceed 1 because every such w is introduced to cure a defect of another w W m 1 and can be 1 only if an R 1 -defect of w was cured, for R 1 T R and lb(r 1, T ) 1. Now, by induction on the topological order in G = (Rep T, E), we show that s R w R i dsub T (R) rank(r i, T ). For the basis of induction, dsub T (R) = and so, by definition, s R w = 0 and the inequality trivially holds. For the inductive step, let R 1,..., R k be the direct sub-roles of R. If w has an R i -successor w that does not belong to any of its sub-roles, i.e., (w, w ) R m+1 i \ R ij dsub T (R i) Rm+1 ij, then R i had a defect 9

10 on w, which was cured, and therefore, s Ri w r(r i, cp(w)) lb(r i, T ) rank(r i, T ), whence s Ri w of w come from its direct sub-roles, in which case s Ri w hypothesis, s Ri w r(r i, cp(w)). Then, by (1) and the definition of rank, rank(r i, T ). Otherwise, all R i -successors = R ij dsub T (R i) srij w, whence, by the induction rank(r i, T ). In R i dsub T (R) rank(r i, T ) and so, (2) holds. R ij dsub T (R i) rank(r ij, T ) and, by the definition of rank, s Ri w either case, s R w = R i dsub T (R) sri w We then proceed by showing (a 1 ) and (a 2 ) as follows: (a 1 ) If q R occurs positively in T and M = E q R[cp(w)] then, by the definition of the required rank, q r(r, cp(w)) and so, the construction ensures that w ( q R) I. (a 2 ) We consider the following three subcases: Let dsub T (R) =. Suppose w ( q R) I. If w W 0 and there are w 1,..., w q W 0 with q q and (w, w 1 ),..., (w, w q ) R I then, by A e, M = E q R[cp(w)] whence, by δ R (x), M = E q R[cp(w)]. Otherwise, some w I \ W 0 with (w, w ) R I was introduced to cure an R-defect of w and so q r(r, cp(w)). Let q = r(r, cp(w)). Then M = E q R[cp(w)] and, by δ R (x), we obtain M = E q R[cp(w)]. Let dsub T (R) and ub(r, T ) =. Since q R occurs negatively in T then, by definition, q = 1. Suppose w ( R) I. If w W 0 and there is w with w W 0 and (w, w ) R I then, by A e and δ R (x), M = E 1 R[cp(w)]. Otherwise, some w I \ W 0 was introduced to cure an R 1 -defect of w for some R 1 T R. It follows then that r(r 1, cp(w)) 1 and so, M = E 1 R 1 [cp(w)] whence, by T R (x), M = E 1 R[cp(w)]. Let dsub T (R) and ub(r, T ). We show ( q R) I =. Assume, to the contrary, there is w ( q R) I. Since q R occurs negatively in T and ub(r, T ), q > ub(r, T ). By (inter KB ) and the definition of the required rank, ub(r, T ) lb(r, T ) r(r, cp(w)), whence q > r(r, cp(w)). On the other hand, w W m, for some m 0, and, by (inter KB ) and (2), ub(r, T ) r m (R, w), whence q > r m (R, w). Then, since w ( q R) I, an R-defect was cured on w, and so, as the procedure (if applied) does not create more than r(r, cp(w))-many R-successors, we have q r(r, cp(w)), contrary to q > r(r, cp(w)). Finally, we can prove that, for all C 1 C 2 T, M = x ( C 1 (x) C 2 (x) ) implies I = C 1 C 2. It should be clear that each C 1 C 2 is equivalent to a set of concept inclusions in the following normal form D 1 D k, where each D i is either, A, A, q R or ( q R). It is to be noted that q R occurs positively in such concept inclusion if it occurs positively in C 1 C 2 and negatively if negatively in C 1 C 2. So, suffice it to prove that, for each concept inclusion, M = x ( D 1(x) D k(x) ) implies I = D 1 D k. Let w I. Then, we have M = D i [cp(w)], for some 1 i k. Obviously, D i is not. If D i is A or A then we clearly have w D I i. If D i is q R then q R occurs positively in T and, by (a 1 ), w ( q R) I. If D i is ( q R) then q R occurs negatively in T and, by (a 2 ), w / ( q R) I. In any case w D I i and so, I = D 1 D k. THEOREM 3. Under (inter KB ), KB satisfiability is NP-complete in DL-Lite HN bool, PTIME-complete in DL-Lite HN horn and NLOGSPACE-complete indl-lite HN krom and DL-Lite HN core. 10

11 4 Extending with Attributes In this section we study the effect of extending DL-Lite with attributes. We first define a class of datatypes that can be safely handled. DEFINITION 4. A set of datatypes {T 1,..., T n } is safe if: (i) each T i is unbounded; (ii) arbitrary conjunction of datatypes is also unbounded or the empty set; (iii) constraints between datatypes are expressible by Horn clauses. From now on we deal only with safe datatypes. We start by showing that for the Bool, Horn and core cases the addition of attributes does not change the complexity of KB satisfiability. THEOREM 5. Under restriction (inter KB ), checking KB satisfiability is NP-complete in DL-Lite HN A PTIME-complete in DL-Lite HN horn A and NLOGSPACE-complete in DL-Lite HN core A. bool, Proof. We encode a DL-Lite HN α A KB K = (T, A) in a first-order sentence K a with one variable in a way similar to the translation of Lemma 2. Denote by att(k) the set of all attribute names in K, and by val(a) the set of all value names in A. Similarly to roles, we define the sets Q U T containing 1 and all q for occurrences of q U (including sub-attributes). The set of all datatype names in K is denoted dt(k). We need a unary predicate E q U(x), for each attribute name U and q Q U T, denoting the set of objects with at least q values for the attribute U. We also need, for each attribute name U and each datatype name T, a unary predicate UT (x), denoting all objects such that all their U attribute values belong to the datatype T (if they have attribute U values at all). Following this intuition, we extend by the following statements: The following sentence encodes the KB K: K a = K e x [T U (x) ( q U) = E q U(x), ( U. ) = E 1 U(x), ( U.(T 1 T k )) = UT 1 (x) UT k (x). U att(k) ] (δ U (x) θ U (x)) β(x) A a A 2 a, where K e is as in Section 3.2, T U (x), δ U (x) and A a are similar to T R (x), δ R (x) and A e, but rephrased for attributes and their inclusions. The new types of ABox assertions require the following formula: A 2 a = U (a i,v j) A U T U ( ( U.T ) ) (a i ) T v j T v j T v j T (v j) A T (v j) A ( T1 v T k v T v ), T dt(k) v val(a) T 1...T k T T where T v j is a propositional variable for each datatype name T and each value v j val(a), and T v = in case T = and v j val(t ), otherwise T v =. The additional formulas capturing datatype and attribute inclusions are: ( θ U (x) = ( U.(T1 T k )) ( U.T ) ), β(x) = T 1 T k T T U 1 U 2 T T dt(k) (( U 2.T ) ( U 1.T ) ). We would like to note here that the formula θ U (x), in particular for disjoint datatypes, demonstrates a subtle interaction between attribute range constraints, U.T, and minimal cardinality constraints, U. We show that K is satisfiable iff the QL 1 -sentence K a is satisfiable. ( ) Let M = K a, we construct a model I = ( I O I V, I) of K similarly to the way we proved Lemma 2 but this time datatypes will have to be taken into account. Let I O be defined inductively as before and Ti I = val(t i ), for each datatype name T i. For each attribute name U, to cure its defects we begin with U 0 = {(a, v) U (a, v) A, U T U}. 11

12 For every attribute name U, we can define the required U-rank r(u, d) of d D and the actual U-rank r m (R, w) of a point w W m I O, m 0, as before, treating U as a role name. We can also consider the equivalence relation induced by the sub-attribute relation in T, then we can choose representatives and a linear order on them respecting the sub-attribute relation of T. We can start from the smaller attributes and cure their defects. Let U k be the smallest attribute name not considered so far. For each w W m, let q = r(u k, cp(w)) r m (U k, w). If q > 0, take q fresh values v 1,..., v q I V such that each v j val(t ), for all datatype names T with M = U k T [cp(w)] since the datatypes are safe, by Definition 4, such v j always exist. Then, for each 1 j q, add the pair (w, v j ) to all attribute relations U 0 with U k T U. Denote the relations resulting in applying the above procedure to all attributes by U I. Now, it can be shown that if M = K a then I = ϕ for every ϕ K. Consider C U.(T 1 T k ) T. And suppose w C I. Let (w, v) U I, for some v I V. It remains to show that v Tj I, for all 1 j k. By the unravelling construction, as showed in Lemma 2, M = C[cp(w)] and so, M = UT j [cp(w)], for all 1 j k. By the construction of U I, there are two possible cases. If v val(a) then w = cp(w) = a ob(a) and U (a, v) A, for U T U. Thus, by the first conjunct of A 2 a, for all 1 j k, we have M = Tj v, whence, by the definition of T j v, v val(t j ) = Tj I. Otherwise, by the construction of U I, we have v Tj I, for all 1 j k. Consider T 1 T k T T. Let v Tj I = val(t j ), for all 1 j k. If v val(a), then, by definition of T j v, M = T j v, whence, by the last conjunct of A 2 a, M = T v and so, v val(t ) = T I. Otherwise, by the unravelling construction, there is a unique w I O and attribute name U such that (w, v) U I and v has been introduced to cure the defects of U-successors of w. By U I construction, v Tj I, for each 1 j k, if M = UT j[cp(w)]. But then, by the formula θ U (x), M = UT [cp(w)], whence, by the construction of U I, v T I. For the other kinds of formulas the proof is similar to that on of Lemma 2. ( ) Conversely, if I is a model of K with the domain I = I O I V we construct a model M = (D, M) of K a with D = I O. The only difference with the proof of Lemma 2 is how to define UT M : for every attribute U and every datatype name T we set UT M = { w I O v T I, for all (w, v) U I}. Now, given a KB with a Bool or Horn TBox, K a is a universal one-variable formula or a universal one-variable Horn formula, respectively, which immediately gives the NP and PTIME upper complexity bounds for the Bool and Horn fragments. The NLOGSPACE upper bound for KBs with core TBoxes is not so straightforward because θ U (x) is not a binary clause. In this case we note that K a is still a universal onevariable Horn formula and therefore, K a is satisfiable iff it is true in the minimal model. The minimal model can be constructed in the bottom-to-top fashion by using only positive clauses of K a (i.e., clauses of the form x (B 1 (x) B k (x) H(x))) and then checking whether the negative clauses of K a (i.e., clauses of he form x (B 1 (x) B k (x) )) hold in the constructed model. By inspection of the structure of K a, one can see that all its positive clauses are in fact binary, and therefore, whether an atom is true in its minimal model or not can be checked in NLOGSPACE. It is of interest to note that the complexity of KB satisfiability increases in the case of Krom TBoxes: THEOREM 6. Satisfiability of DL-Lite HN krom A (and so, under (inter KB )). KBs is NP-hard even without role and attribute inclusions Proof. The proof is by reduction of 3SAT to the KB satisfiability problem. It exploits the structure of the formula θ U (x) in K a : if T T T then the concept inclusion U.T U.T U, although not in the syntax of DL-Lite HN krom A, is a logical consequence of T. Using such ternary intersections with the full negation of the Krom fragment one can encode 3SAT. Let ϕ = m i=1 C i be a 3CNF, where the C i are ternary clauses over variables p 1,..., p n. Now, suppose p i1 p i2 p i3 is the ith clause of ϕ. It is equivalent to p i1 p i2 p i3 and so, can be encoded as follows: T 1 i T 2 i, A i1 U i.t 1 i, A i2 U i.t 2 i, A i3 U i, 12

13 where the A 1,..., A n are concept names for the variables p 1,..., p n, and U i is an attribute and Ti 1 and Ti 2 are datatypes for the ith clause (note that Krom concept inclusions of the form B B are required, which is not allowed in the core TBoxes). Let T consist of all such inclusions for clauses in ϕ. It can be seen that ϕ is satisfiable iff T is satisfiable. 5 Query Answering: Data Complexity In this section we study the data complexity of answering positive existential queries over a KB expressed in languages with attributes and datatypes. In the following, we slightly abuse notation and use H for an attribute name or a role. REMARK 7. It follows from the proofs of Theorems 5 and 6 and Lemma 2 that, for a DL-Lite HN bool A KB K = (T, A) under restriction (inter KB ), every model M of K a induces a forest-shaped model I M of K with the following properties: (ABox) For all a i, a j ob(a) val(a) and all roles (attributes) H, we have (a i, a j ) H I M iff there is H T H with H (a i, a j ) A. (forest) The object names a ob(a) induce a partitioning of I M into disjoint labelled trees T a = (T a, E a, l a ) with nodes T a, edges E a, root a I M, and a labelling function l a : E a role ± (K) att(k). (copy) There is a function, cp: I M ob(a) dr(k) such that cp(a I M ) = a, for a ob(a), and cp(w) = dr, if (w, w) E a and l a (w, w) = inv(r), for w T a. (role) For every role (attribute) name H, H I M = { (a i, a j ) H (a i, a j ) A, H T H } { (w, w ) E a l a (w, w ) = H, H T H }. a ob(a) THEOREM 8. The positive existential query answering problem for DL-Lite HN horn A restriction (inter KB ), is in AC 0 for data complexity. and DL-Lite HN core A, under Proof. Suppose that we are given a consistent DL-Lite HN horn A KB K = (T, A) and a positive existential query in prenex form q( x) = y ϕ( x, y) in the signature of K. Since K a is a Horn sentence, it is enough to consider just one special model I 0 of K. Let M 0 be the minimal Herbrand model of (the universal Horn sentence) K a. We remind the reader (for details consult, e.g., (Apt, 1990; Rautenberg, 2006)) that M 0 can be constructed by taking the intersection of all Herbrand models for K a, that is, of all models based on the domain that consists of the constant symbols from K a i.e., ob(a) val(a) dr(k). Let I 0 be the canonical model of K, i.e., the model induced by M 0 along the construction presented in Theorem 5. Denote the domain of I 0 by I0. The following are true from the construction of the canonical model: a I0 i B I0 iff K = B(a i ), for basic concepts B and a i ob(a), (3) w B I0 iff T = R B, for basic concepts B and w I0 O with cp(w) = dr, (4) v I0 i T I0 iff K = T (v i ), for datatype names T and v i val(a), (5) v T I0 iff there are B 1,..., B k such that T = B 1 B k U.T and w B I0 1,..., BI0 k, (6) for datatype names T, attribute names U and (w, v) U I0 with v / val(a). Then the canonical model I 0 provides answers to all queries: LEMMA 9. K = q( a) iff I 0 = q( a). 13

14 Proof. Suppose I 0 = K. As q( a) is a positive existential sentence, it is enough to construct a homomorphism h: I 0 I. By property (forest) of Remark 7, the domain I0 of I 0 is partitioned into disjoint trees T a, for a ob(a). Define the depth of a point w I0 to be the length of the shortest path in the respective tree to its root. Denote by W m the set of points of depth m. In the following we extend the meaning of sets W m to include also values v I0 V that were taken for objects in W m 1; in particular, W 0 = ob(a) val(a). We construct h as the union of maps h m, m 0, where each h m is defined on W m and has the following properties: h m+1 (w) = h m (w), for all w W m, and (a m ) for all w W m, if w B I0 then h m (w) B I, for each basic concept B; (b m ) for all u, w W m, if (u, w) R I0 then (h m (u), h m (w)) R I, for each R role ± (K). (t m ) for all v W m, if v T I0 then h m (v) T I, for each datatype name T ; (v m ) for all u, v W m, if (u, v) U I0 k then (h m(u), h m (v)) U I k, for each U k att(k). For the basis of induction, we set h 0 (a i ) = a I i, for a i ob(a), and h 0 (v i ) = v I i, for v i val(a). Property (a 0 ) follows then from (3), (t 0 ) from (5) and (b 0 ) and (v 0 ) from (ABox). For the induction step, suppose that h m has already been defined for W m, m 0. Set h m+1 (w) = h m (w) for all w W m. Consider an arbitrary w W m+1 \ W m. By (forest), there is a unique u W m such that (u, w) E a, for some T a. Let l a (u, w) = S role ± (K). Then, by (copy), cp(w) = inv(ds). By (role), u ( S) I0 and, by (a m ), h m (u) ( S) I, which means that there is w 1 I with (h m (u), w 1 ) S I. Set h m+1 (w) = w 1. As cp(w) = inv(ds) and ( inv(s)) I0, it follows from (4) that if w B I0 then w B I whenever we have w ( inv(s)) I. As w 1 ( inv(s)) I, we obtain (a m+1 ) for w. To show (b m+1 ), we notice that, by (role), we have (u, w) R I0 just when S T R. Thus, since (u, w) S I0 and (h m+1 (u), h m+1 (w)) S I and, as S T R, then, (h m+1(u), h m+1 (w)) R I. Let l a (u, w) = U att(k), then w = v W m+1 I0 V. By (role), u and, by (a ( U)I0 m ), h m (u) ( U) I, which means that there is v 1 I V with (h m(u), v 1 ) U I. Set h m+1 (v) = v 1. To show (v m+1 ), we notice that, by (role), we have (u, v) U I0 k just when U T U k; but then we have (h m+1 (u), h m+1 (v)) U I Uk I. Next, we show (t m+1). By definition, v / val(a). Then, by (6), v T I0 just in case there are basic concepts B 1,..., B k such that K = B 1 B k U.T and u B I0 1,..., BI0 k. By (a m+1), we have h m+1 (u) B I 1,..., B I k, whence h m+1(u) ( U.T ) I and so, as (h m+1 (u), h m+1 (v)) U I, we obtain h m+1 (v) T I. This completes the proof of the lemma. Our next lemma shows that in this case to check whether I 0 = q( a) it suffices to consider only the points of depth m 0 in I0, for some m 0 that does not depend on A : LEMMA 10. If I 0 = y ϕ( a, y) then there is an assignment a 0 such that I 0 = a0 ϕ( a, y) and a 0 (y i ) W m0, for all y i y, where m 0 = y + role ± (T ) + 1. Proof. The proof is similar to that one of Lemma 7.4 in (Artale et al., 2009) observing that attributes cannot be nested and cannot have role successors either. To complete the proof of Theorem 8, we encode the problem K = q( a)? as a model checking problem for first-order formulas. We fix a signature that contains unary predicates A, A, for concept names A, and T and T, for datatype names T, and binary predicates P, P, for role names P and U for attribute names U. Then we represent the ABox A of K as a first-order model A A with domain ob(a) val(a) (the predicates encode negative assertions of the ABox). Now we define a first-order formula ϕ T,q ( x) in the above signature such that (i) ϕ T,q ( x) depends on T and q but not on A, and (ii) A A = ϕ T,q ( a) iff I 0 = q( a). Denote by con(t ) the set of basic concepts in T together with all concepts of the form U.T, for attribute names U and datatypes T from T. 14

15 We begin by defining formulas ψ B (x), for B con(t ), that describe the types of elements of the model I 0 (see also (3)): for all a i ob(a), A A = ψ B (a i ) iff a I0 i B I0, if B is a basic concept, (7) A A = ψ U.T (a i ) iff a I0 i B I0 1,..., BI0 k and T = B 1 B k U.T. (8) These formulas are defined as the fixed-points of sequences ψb 0 (x), ψ1 B (x),... : A(x), if B = A, ( ψb(x) 0 = y 1... y q (y i y j ) H T (x, y i ) ), if B = q H, 1 i<j q 1 i q, if B = U.T, ψb(x) i = ψb(x) 0 ( ψ i 1 (x) ψ i 1 B k (x) ), B 1 B 1 B k B ext(t ) where H T (x, y) = H T H H (x, y), and ext(t ) denotes the extension of T with the following concept inclusions, for H role ± (T ) att(t ): q H q H, for all q, q Q H T with q > q, q H q H, for all q Q H T and H T H. and the following concept inclusions, for all attributes names U and datatypes T in T : U.T U.T, for all U U T, U.T 1 U.T k U.T, for all T 1 T k T T. It should be clear that there is N with ψb N (x) ψn+1 B (x), for all B at the same time, and that N does not exceed the cardinality of con(t ). We set ψ B (x) = ψb N (x). Next we introduce sentences θ B,dr, for B con(t ) and dr dr(t ), that describes the types of the dr(t ) (cf. (4)): for all w with cp(w) = dr, A A = θ B,dr iff w B I0, if B is a basic concept, (9) A A = θ U.T,dr iff T = R U.T. (10) Note that, formula (10) uses that fact that the type of every w s.t. cp(w) = dr is generated by a single basic concept, R, and therefore, we do not need to consider conjunctions as above. We inductively define a sequence θb,dr 0, θ1 B,dr,... by taking θ0 B,dr =, if B = R, and θ0 B,dr =, otherwise, and θb,dr i = θb,dr 0 ( θ i 1 B 1,dr θi 1 B k,dr). B 1 B k B ext(t ) As with the ψ B, we set θ B,dr = θ N B,dr. Sentences ν T (x), for datatype names T, describe the types of an element of val(a) (cf. (5)): A A = ν T (v j ) iff v j T I0. (11) They are defined as the fixed-points of the following sequences: νt 0 (x) = T (x) y ( U(y, x) ψ U.T (y) ), ν i T (x) = ν 0 T (x) U att(t ) ( ν i 1 (x) ν i 1 T k (x) ). T 1 T 1 T k T T 15

16 Note that, for (11) to hold the ABox, A, must contain an assertion T (v) for every v val(a) s.t. v val(t ). Furthermore, the datatypes of a named value, say v j, are only partly described by the predicates T : concept inclusions of the form B 1 B k U.T can imply additional datatypes for v j, which is reflected by the second disjunct of νt 0 (x). We set ν T (x) = νt M (x), for M that does not exceed the number of datatypes. Now we consider the directed graph G T = (V T, E T ), where V T is the set of equivalence classes [H] = {H H T H and H T H}, and E T is the set of all pairs ([R], [H]) such that (p) T = inv(r) q H and either inv(r) T H or q 2, and H has no proper sub-role/attribute satisfying (p). Recall now that we are given a query q( x) = y ϕ( x, y), where y = y 1,..., y k. Let Σ T,m0 be the set of all paths in the graph G T of length m 0. More precisely, Σ T,m0 = { ε } V T { ([H 1 ],..., [H n ]) 2 n m 0 and ([H j ], [H j+1 ]) E T, for 1 j < n }. When evaluating the query y ϕ( x, y) over I 0, each bound variable y i is mapped to a point w in W m0. However, the first-order model A A does not contain the points from W m0 \ W 0, and to represent them, we use the following trick. By (forest), every w in W m0 \ W 0 is uniquely determined by the pair (a, σ), where a is the root of the tree T a containing w, and σ is the sequence of labels l a (u, v) on the path from a to w. It follows from the unravelling procedure and (p) that σ Σ T,m0. Let Σ k T,m 0 be the set of k-tuples σ = (σ 1,..., σ k ), with σ i Σ T,m0. In the formula ϕ T,q we are about to define we assume that the y i range over W 0 and represent the first component of the pairs (a, σ), whereas the second component is encoded in the ith member σ i of σ (these y i should not be confused with the y i in the original query q, which range over W m0 ). In order to treat arbitrary terms t occurring in ϕ( x, y) in a uniform way, we set t σ = ε, if t ob(a) val(a) or t = x i, and t σ = σ i, if t = y i. (the distinguished variables x i, the object names a and the value names v are mapped to W 0 and do not require the second component of the pairs). Given an assignment a 0 in W m0, we denote by split(a 0 ) the pair (a, σ) made of an assignment a in A A and σ Σ k T,m 0 such that (i) for each free variable x i, a 0 (x i ) ob(a) val(a), (ii) for each bound variable y i, a(y i ) = a and σ i = ([H 1 ],..., [H n ]), n m 0, with a being the root of the tree containing a 0 (y i ) and H 1,..., H n being the sequence of labels l a (u i, u i+1 ) on the path from a to a 0 (y i ). Not every pair (a, σ), however, corresponds to an assignment in W m0 because some paths in σ may not exist in the I 0 for a given ABox A. As follows from the unravelling procedure and (p), a point in W m0 \ W 0 corresponds to a ob(a) and σ = ([H],... ) Σ T,m0 iff a has not enough H-witnesses in A. Thus, for every (a, σ), there is an assignment a 0 in W m0 with split(a 0 ) = (a, σ) iff A A = a η σ ( y), where η σ ( ( y) = ψ 0 q Hi (y i ) ψ q Hi (y i ) ). 1 i k q Q H i T σ i=([h i],... ) ε We define now, for every σ Σ k T,m 0, concept name A, role or attribute name H and datatype name T : { A σ ψ A (t), if t σ = ε, (t) = θ A,inv(ds), if t σ = σ.[s], H T (t 1, t 2 ), if t σ 1 = t σ 2 = ε, H σ (t 1, t 2 ) = (t 1 = t 2 ), if either t σ 1.[S] = t σ 2 or t σ 2 = t σ 1.[inv(S)], for S T H,, otherwise, ν T (t), if t σ = ε, T σ (t) = ψ U.T (t), if t σ = [U], θ U.T,inv(ds), if t σ = σ.[s].[u]. 16