The Combined Approach to Query Answering in DL-Lite

Transcription

1 The ombined Approach to Query Answering in DL-Lite R Kontchakov, 1 Lutz, 2 D Toman, 3 F Wolter 4 and M Zakharyaschev 1 1 Department of S and Information Systems 2 Fachbereich Mathematik und Informatik Birkbeck ollege London, UK Universität Bremen, Germany {roman,michael}@dcsbbkacuk clu@informatikuni-bremende 3 DR heriton School of S 4 Department of omputer Science University of Waterloo, anada University of Liverpool, UK david@csuwaterlooca frank@csclivacuk Introduction Description logics (DLs), as well as DL-based dialects of the Web ontology languages OWL and OWL 2, have been tailored as knowledge representation languages supporting the classical reasoning tasks that are used for ontology design such as satisfiability and subsumption Modern DL reasoners are indeed able to classify large and complex real-world ontologies, the OWL version of the medical ontology Galen being the latest fallen stronghold (Kazakov 2009) Along with the growing popularity and availability of ontologies, novel ways of their use, which go far beyond classical reasoning, are emerging In particular, it is generally believed in the KR community that ontology languages can play a key role in the next generation of information systems The core idea is ontology-based data access, where ontologies enrich the data with additional background knowledge, thus facilitating the use and integration of incomplete and semistructured data from heterogeneous sources (Dolby et al 2008; Heymans et al 2008; Poggi et al 2008) In this context, the main reasoning task is to answer queries posed to the data while taking account of the knowledge provided by the ontology It has turned out, however, that this task does not scale well in traditional DLs and is dramatically less efficient than querying in standard relational database management systems (RDBMSs) An investigation of DLs for which ontology-based data access can be reduced to query answering in RDBMSs thus taking advantage of the decades of research invested to make RDBMSs scalable was launched in series of papers (alvanese et al 2005; 2006; 2008) with the ultimate aim to identify DLs for which every conjunctive query q over a data instance D, given an ontology T, can be rewritten independently of D into a first-order query q T over D alone and then executed by an RDBMS This effort gave birth to a new family of DLs, called the DL-Lite family, and subsequently to the OWL 2 QL profile of OWL 2 This rewriting approach to ontology-based data access has been implemented in various systems such as QuOnto, Owlgres and REQUIEM Unfortunately, experiments have revealed that these systems do not provide sufficient scalability even for medium-size ontologies (with a few hundred of axioms) opyright c 2009, Association for the Advancement of Artificial Intelligence (wwwaaaiorg) All rights reserved In a nutshell, the reason is that the rewritten queries are of size ( T q ) q in all known rewriting techniques, which can be prohibitive for efficient execution by an RDBMS when T is large (even if q is relatively small) A different combined approach to ontology-based data access using RDBMSs, with the main goal of overcoming the inherent limitation of the rewriting approach being applicable only to DLs for which conjunctive query answering is in A 0 for data complexity, has been proposed (Lutz, Toman, and Wolter 2009) This combined approach separates query answering into two steps: first, the data D is extended independently of possible queries by taking account of the ontology T, and then any given query over T and D is rewritten independently of D to an RDBMS query over the extended data The new technique was applied to (extensions of) the DL EL (underlying the OWL 2 EL profile), which is PTIME-complete for data complexity In this paper, we investigate the combined approach to conjunctive query answering for DL-Lite ontologies In particular, we present polynomial rewriting techniques for both data and query in the case when ontologies are formulated in DL-Lite N horn (properly containing DL-Lite F, of (alvanese et al 2006)) For ontologies in DL-Lite (HN horn ), extending DL-Lite N horn with role inclusions, we could not avoid an exponential blowup (but only in the number of roles) of the rewritten queries, while keeping the expanded data polynomial To evaluate the new techniques, we have conducted experiments with real-world DL-Lite ontologies, which demonstrate that the combined approach outperforms pure query rewriting It is to be noted, however, that these amenities come at a price: in general, the combined approach is applicable only if the information system is allowed to manipulate the source data, which is not the case in some information integration scenarios To explain our approach in more detail, suppose that we want to answer conjunctive queries over a data instance D given an ontology T As a first step, we expand D by applying the axioms of T which gives a new data instance D whose size is O( D T + T 2 ) in the worst case When given a conjunctive query q to be executed over D and T, we rewrite q into a first-order query q over D (independently of D and almost independently of T ) whose size is O( q 2 ) for DL-Lite N horn ontologies The rewriting q is drastically different from the rewriting of (Kontchakov

2 et al 2009), which involves an exponential blowup in the worst case For a DL-Lite (HN horn ) ontology T, we can reduce conjunctive query answering to the case without role inclusions, but the resulting query may have to be a union of r q queries of the form q, where r is the maximum number of subroles for role atoms in q We also show that the expansion of data can be implemented using views in the RDBMS, which has the nice effect that the expanded database is automatically and transparently adjusted when the underlying data is updated As a by-product of the view construction, we obtain a novel way of pure query rewriting for DL-Lite N horn When applied to DL-Lite F of (alvanese et al 2006), which disallows conjunction and all number restrictions but existential quantifiers and functionality constraints, this new technique blows up the rewritten query only polynomially As no views are involved in this case, we obtain a first polynomial query rewriting technique for a DL-Lite logic A full version of the paper is available at roman/ Preliminaries We briefly introduce DL-Lite (HN horn ), the most expressive dialect of DL-Lite for which conjunctive query answering is in A 0 for data complexity under the unique name assumption, along with its fragments that are considered in this paper, such as DL-Lite N horn For more details and the relation to other DL-Lite logics, we refer the interested reader to (Artale et al 2009) Let N I, N and N R be countably infinite sets of individual names, concept names and role names Roles R and concepts are built according to the following syntax rules, where P N R, A N and m > 0: R ::= P P, ::= A m R As usual, we write R for 1 R and identify (P ) with P We use N R to denote the set of all roles In DL, ontologies are represented as TBoxes A DL-Lite (HN horn ) TBox is a finite set T of concept and role inclusions (Is and RIs), which take the form 1 n and R 1 R 2, respectively Denote by T the transitive-reflexive closure of RIs in T It is required that if S T R with S R, then T does not contain a I with i = m R, for m 2, on its left-hand side Without this restriction, conjunctive query answering becomes conp-hard for data complexity (Artale et al 2009), which means that the combined approach is no longer possible without an exponential blowup of the data (Lutz, Toman, and Wolter 2009) DL-Lite N horn is the fragment of DL-Lite (HN horn ) in which RIs are disallowed An ABox is used to store instance data Formally, it is a finite set of concept assertions A(a) and role assertions P (a, b), where A N, P N R and a, b N I We denote by Ind(A) the set of individual names occurring in A A knowledge base (KB) is a pair (T, A) where T is a TBox and A an ABox The semantics of DL-Lite (HN horn ) is defined in the standard way based on interpretations I = ( I, I); for details consult (Baader et al 2003) Throughout the paper, we adopt the unique name assumption (UNA), ie, require that a I b I for distinct a, b N I In the context of OWL, the UNA is not adopted The combined approach to the case without the UNA is discussed later on in the paper Given an interpretation I and an inclusion or assertion α, we write I = α to say that α is true in I The interpretation I is a model of a KB K = (T, A) if I = α for all α T A K is consistent if it has a model We write K = α whenever I = α for all models I of K Let N V be a countably infinite set of variables Taken together, the sets N V and N I form the set N T of terms A firstorder (FO) query is a first-order formula q = ϕ( v) in the signature N N R with terms from N T, where the concept and role names are treated as unary and binary predicates, respectively, and the sequence v = v 1,, v k of variables from N V contains all the free variables of ϕ The variables v are called the answer variables of q, and q is k-ary if v comprises k variables A positive existential query is a first-order query of the form q = u ψ( u, v), where ψ is constructed using conjunction and disjunction from concept atoms A(t) and role atoms P (t, t ) with t, t N T A conjunctive query (Q) is a positive existential query containing no disjunction The variables in u are called the quantified variables of q We denote by qvar(q) the set of quantified variables u, by avar(q) the set of answer variables v, and by term(q) the set of terms in q Let q = ϕ( v) be a k-ary FO query and a = a 1,, a k a k-tuple of individual names We say that a is an answer to q in an interpretation I and write I = q[ a] if I satisfies q under the assignment π which sets π(v i ) = a I i, i k Such an assignment is called a match of q in I We use ans(q, I) to denote the set of all answers to q in I We say that a is a certain answer to q over a KB K = (T, A) if a Ind(A) and I = q[ a] for all models I of K The set of all certain answers to q over K is denoted by cert(q, K) Throughout the paper, we use K to denote the size of a KB K, that is, the number of symbols required to write K T, A and q are defined analogously ABox Extension First, we describe the ABox extension part of the combined approach to Q answering in DL-Lite N horn Semantically, the ABox extension means expanding the ABox A to a canonical interpretation I K for the given KB K = (T, A) More specifically, I K is constructed by (i) expanding the set Ind(A) of individual names in A with additional individuals to witness existential and number restrictions, and (ii) expanding the extensions of concept and role names as required by the Is in T The individual names in (i) are taken from the set N T I = {c P, c P P a role name in T }, which is assumed to be disjoint from Ind(A) The domain of I K will contain those witnesses c R that are really needed in any model of K To identify such witnesses, we require the following definition A role R is called generating in K if there exist a Ind(A) and R 0,, R n = R such that the following conditions hold: (agen) K = R 0 (a) but R 0 (a, b) / A for all b Ind(A) (written a c R0 ), (rgen) for i < n, T = R i R i+1 and R i R i+1 (written c Ri c Ri+1 )

3 It can be seen that R is generating in K if, and only if, every model I of K contains some point x I with an incoming R-arrow, but there is a model I where no such x is identified by an individual name in the ABox The canonical interpretation I K for K is defined as follows: I K a I K A I K P I K = Ind(A) {c R R N R, R is generating in K}, = a, for all a Ind(A), = {a Ind(A) K = A(a)} {c R I K T = R A}, = {(a, b) Ind(A) Ind(A) P (a, b) A} {(d, c P ) I K N T I d c P } {(c P, d) N T I I K d c P } learly, I K is an extension of the ABox A if we represent the concept and role memberships in the form of ABox assertions The number of domain elements in I K does not exceed K The canonical interpretation I K is not in general a model of K Indeed, this cannot be the case because DL-Lite N horn does not enjoy the finite model property (alvanese et al 2005) whereas I K is always finite Example 1 Let K = (T, A), where T = {A P, 2 P }, A = {A(a), A(b)} Then I K = {a, b, c P }, P I K = {(a, c P ), (b, c P )}, and so c P ( 2 P ) I K and I K = K As far as query answering is concerned, this is not a problem More important is that I K does not always give the correct answers to Qs, ie, it is not the case that I K = q[ a] iff K = q[ a] for all k-ary Qs q and k-tuples a Ind(A) We illustrate this by two examples Example 2 (i) onsider the cyclic query q = v P (v, v) over K = (T, A), where T = {A P, P P }, A = {A(a)} Then I K = {a, c P }, P I K = {(a, c P ), (c P, c P )} and A I K = {a} The assignment π defined by π(v) = c P shows that I K = q On the other hand, the interpretation I with I = {a, 1, 2, }, P I = {(a, 1)} {(n, n + 1) n 1} and A I = {a} is a model of K, but I = q Thus K = q (ii) onsider the query q = v 2 (P (v 1, v 2 ) P (v 3, v 2 )) over K = (T, A), where T = {A P }, A = {A(a), A(b)} Then I K = {a, b, c P }, P I K = {(a, c P ), (b, c P )} and A I K = {a, b} learly, I K = q[a, b] On the other hand, the interpretation I with I = {a, b, c 1, c 2 }, A I = {a, b}, and P I = {(a, c 1 ), (b, c 2 )} is a model of K such that I = q[a, b] Thus K = q[a, b] We overcome these problems in two steps First, we show that the unravelling of I K into a forest-shaped interpretation U K does give the right answers to queries However, U K may be infinite, and so we cannot store it as a database instance But, as shown in the next section, any given Q q can be rewritten into an FO query q in such a way that the answers to q over U K are identical to the answers to q over I K This enables us to use the finite interpretation I K as a relational instance and still obtain correct answers to queries The unravelling U K is defined as follows A path in I K is a finite sequence ac R1 c Rn, n 0, such that a Ind(A) and R 1,, R n satisfy (agen) and (rgen) (that is, a c R1 and c Ri c Ri+1 ) We denote by paths(i K ) the set of paths in I K and by tail(σ) the last element in σ paths(i K ) The interpretation U K is then defined by taking: U K = paths(i K ), a U K = a, for all a Ind(A), A U K = {σ tail(σ) A I K }, P U K = {(a, b) Ind(A) Ind(A) P (a, b) A} {(σ, σ c P ) σ c P paths(i K )} {(σ c P, σ) σ c P paths(i K )}, where denotes concatenation Notice that the interpretations I constructed in Example 2 are isomorphic to the respective models U K The interpretation U K is forest-shaped in the sense that the graph G = (V, E) with V = U K and E = {(σ, σ c R ) σ c R paths(i K )} is a forest The map τ : U K I K defined by taking τ(σ) = tail(σ) is a homomorphism from U K onto I K, and so U K = q implies I K = q for all Qs q (but, as shown in Example 2, not necessarily vice versa) Just like I K, the interpretation U K is not in general a model of K A simple example is given by K = (T, A), T = {A 2 P } and A = {A(a)}, where we have U K = A 2P because a is P -related only to a c P in U K Nevertheless, U K gives the right answers to all Qs, as shown by the following: Theorem 3 For every consistent DL-Lite N horn KB K and every Q q, we have cert(q, K) = ans(q, U K ) Note that Theorem 3 requires consistency of K We shall see below that there is an FO query q T such that ans(q T, A) = iff K = (T, A) is consistent This will allow us to check consistency using an RDBMS before building the canonical interpretation Query Rewriting for DL-Lite N horn We now present a polytime algorithm that rewrites every Q q into an FO query q such that ans(q, U K ) = ans(q, I K ), the length of q is O( q 2 + q T ) (and can be made O( q 2 ) by adding an auxiliary database table) By Theorem 3, we can compute the answers to q over K using an RDBMS to execute q over I K stored as a relational instance To simplify notation, we often identify a Q with the set of its atoms and assume that P (v, u) q iff P (u, v) q The rewriting q of a given Q q = u ϕ is defined by taking q = u (ϕ ϕ 1 ϕ 2 ϕ 3 ),

4 where ϕ 1, ϕ 2 and ϕ 3 are Boolean combinations of built-in equality predicates over terms in q and constants c R N T I (which means that they can be evaluated by a RDBMS over the result of evaluating ϕ) The formula ϕ 1 is defined as ϕ 1 = (v c R ), v avar(q) c R N T I and its purpose is to select only those matches where all answer variables receive values from Ind(A), as required by the definition of certain answers The intuition behind ϕ 2 and ϕ 3 is that the rewriting q of q has to select exactly those matches of q in I K that can be reproduced as matches in U K Due to the forest structure of U K, this requirement imposes strong constraints on the way in which variables can be matched to the non-abox elements of I K Essentially, the part of q that is mapped to the non-abox elements of I K must be homomorphically embeddable into a forest This intuition is captured by the following definition Let (N R ) be the set of all finite words over N R (including the empty word ε) Definition 4 Let q = u ϕ be a Q and R(t, t ) q A partial map f : term(q) (N R ) is a tree witness for R(t, t ) in q if its domain is minimal (wrt set-theoretic inclusion) such that the following conditions hold, where w (N R ) : f(t ) = R; if f(s) = w S and S (s, s ) q with S S, then f(s ) = w S S ; if f(s) = w S and S (s, s ) q, then f(s ) = w If a tree witness for R(t, t ) in q exists, then it is unique and denoted by f R(t,t ) Intuitively, the tree witness f R(t,t ) deals with matches π in I K where π(t ) = c R The reproduction π of this match in U K has to satisfy π (t ) = σ c R for some σ Due to the condition R i R i+1 in (rgen), we have σ c R c R / U K By the definition of U K, (π (t), σ c R ) R U K thus implies π (t) = σ Now, f R(t,t ) serves two purposes First, it identifies, via its domain D term(q), those terms in q that must be mapped by π to the subtree of U K with root π (t) Second, it describes a homomorphic embedding of q D (ie, q restricted to D) into that subtree: f R(t,t )(s) = R 1 R k means that π (s) = π(t) c R1 c Rk Due to the structure of U K, it can actually be seen that the described homomorphic embedding is the only possible embedding of this kind Therefore, if the tree witness for R(t, t ) does not exist, then no homomorphic embedding is possible, which means that t cannot be mapped to σ c R in U K, and so not to c R in I K This is precisely what ϕ 2 is for: ϕ 2 = (t c R ) R(t,t ) q f R(t,t ) does not exist Example 5 We illustrate ϕ 2 using the query q = v P (v, v) and KB K from Example 2 (i) As we saw, K = q but I K = q since (c P, c P ) P I K Now observe that there exists no witness tree f P (v,v) because otherwise we would have f P (v,v) (v) = P and, by P (v, v) q and P P, also f P (v,v) (v) = P P, contrary to f P (v,v) being a function Thus, v c P is a conjunct of ϕ 2, and so I K = q In general, the variable v can only be matched to an ABox individual no matter what K is: both v c P and v c P are conjuncts of ϕ 2 and, by the definition of I K, we have (c R, c R ) P I K for all K and R / {P, P } If f R(t,t ) exists then, in principle, t can be mapped to a c R in I K But then we still have to ensure that all terms in the domain D of f R(t,t ) are matched in I K in a way that can be reproduced in the relevant subtree of U K by a homomorphic embedding This is the case precisely when the terms in D are matched in I K as prescribed by f R(t,t ): if f R(t,t )(s) = R 1 R n, for n > 0, then π(s) = c Rn ; if f R(t,t )(s) = ε, then π(s) = π(t) Both of these conditions are in fact guaranteed by ϕ 3 : ( ϕ 3 = (t = c R ) (s = t) ) R(t,t ) q f R(t,t ) exists f R(t,t ) (s)=ε Example 6 We illustrate ϕ 3 using the fork-shaped query q = v 2 (P (v 1, v 2 ) P (v 3, v 2 )) from Example 2 (ii) For every KB K, if U K = q[a 1, a 3 ] and v 2 is not mapped to an ABox individual, then a 1 = a 3 This is not the case in I K as, depending on K, we might be able to map v 2 to c P Thus, it is sufficient to add (v 2 = c P ) (v 1 = v 3 ) as a conjunct to P (v 1, v 2 ) P (v 3, v 2 ) This is achieved using ϕ 3 as follows: we have f P (v1,v 2)(v 2 ) = P and f P (v1,v 2)(v 3 ) = ε Thus, ϕ 3 contains the conjunct (v 2 = c P ) (v 1 = v 3 ), as required It is easy to see that q can be computed in polynomial time in the size of the query q and the set N T I In particular, the required functions f R(t,t ) can be computed in polytime by a straightforward algorithm akin to breadth-first search, which also decides their existence The length of q is of size O( q 2 + q T ), where the T factor is solely due to ϕ 1 If we add a unary database table aux that identifies exactly the elements of N T I, then we can replace ϕ 1 with ϕ 1 = aux(v) v avar(q) and obtain q of size O( q 2 ) The main result of this paper is the following theorem: Theorem 7 For every DL-Lite N horn KB K and every Q q, ans(q, I K ) = ans(q, U K ) anonical Interpretation by FO Queries The aim of this section is to show that the canonical interpretation I K for a DL-Lite N horn KB K = (T, A) can be constructed by means of FO queries This allows us to implement the construction of I K in an RDBMS using (potentially materialised) views The benefit is that updates of the ABox A, such as insertions and deletions, are automatically reflected in those views, which solves the problem of updating I K in a simple and elegant way Given a DL-Lite N horn TBox T and concept and role names A and P, we construct FO queries qa T (x) and qt P (x, y) such

5 that the answers to qa T and qt P over A coincide with AI K and, respectively, P I K for all ABoxes A and K = (T, A) It will be convenient for us to regard A as the interpretation I A defined by taking: I A = Ind(A) N T I ; A I A = {a A(a) A}, for all A N ; P I A = {(a, b) P (a, b) A}, for all P N R Equivalently, we could rely on domain independence of queries and use the active domain semantics (Abiteboul et al 1995) We now construct qa T (x) in three steps First, for each concept in T, we define inductively a query exp T (x) which determines I A To simplify presentation, we assume that T contains all Is of the form m R m R, for pairs of concepts m R and m R in T such that m < m, and no m R, for m < m < m, occurs in T In any case, the extended TBox is only linearly larger than the original one Set exp T,0,0 (x) =, expt (x) = A(x), A exp T,0 R (x) = (x = c R ) y R(x, y), mr (x) = y ( 1,, y m R(x, y i ) i j(y i y j ) ), exp T,0 1 i m where m 2, and for j 1, set exp T,j,j 1 (x) = expt (x) k 1 k i=1 exp T,j 1 i (x) Thus, exp T,j,j 1 (x) adds to expt (x) those elements of I A that can be obtained by one inference step of SLD resolution (Kowalski and Kuehner 1971) As we do not need more than T inference steps, the formulas exp T,j (x) are all equivalent for j T We set exp T, T (x) = expt (x) Lemma 8 For a DL-Lite N horn TBox T and a concept in T, ans(exp T, I A) I K = I K for all KBs K = (T, A) Second, we construct queries qp T (x, y) computing P I K Let rgen T R n be the set of pairs (R 0, R n 1 ) of roles in T such that there is a sequence R 0,, R n satisfying (rgen) learly, rgen T R n depends only on T and can be computed in time polynomial in T For a role name P, set q T P (x, y) = P (x, y) ( gen T P (x) (y = c P ) ) ( gen T P (y) (x = c P ) ), gen T R(x) = agen T R(x) (R 0,S) rgen T R ( z agen T R0 (z) (x = c S ) ), agen T R(x) = exp T R(x) y R(x, y) (x c S ) c S N T I Lemma 9 For a DL-Lite N horn TBox T and a role name P, ans(qp T, I A) = P I K for all KBs K = (T, A) To define queries qa T (x) computing AI K, it is enough, by Lemma 8, to restrict exp T A (x) to the domain of I K So as the third step we set qa T (x) = expt A (x) D(x), where D(x) = ( ) (x = c R ) z gen T c R R(z) N T I Lemma 10 For a DL-Lite N horn TBox T and a concept name A, ans(q T A, I A) = A I K for all KBs K = (T, A) The queries for constructing I K also provide us with the previously announced query q T = x (expt (x) D(x)) which can be used to check consistency of K (see the remark after Theorem 3): KB K = (T, A) is consistent if and only if ans(q T, I A) = A very interesting observation is that we can combine the rewritten query q with the queries constructing I K The resulting FO query q T, can be executed directly over I A rather than I K Thus we obtain a novel technique for the pure query rewriting approach It involves only a polynomial blowup for DL-Lite F (alvanese et al 2006), the fragment of DL-Lite N horn with Is of the form 1 2, 2 R or 1 2 and the i of the form A or R More precisely, the length of q T, for a DL-Lite F TBox T is O( q T max R a role in T rgent R ), which is linear in q and at most cubic in T All the previously known query rewriting techniques for DL-Lite F produced exponential results Our technique can also be applied to DL-Lite N core, which extends DL-Lite F with arbitrary number restrictions m R In this case, however, we have to take account of the number encoding because the subformulas exp T,0 mr (x) are of length O(m2 ) If the numbers are represented in unary then q T, = O( q T 5 ) If the numbers are coded in binary and we are allowed to use aggregation functions (eg, OUNT in SQL), then q T, is O( q T 4 ) Furthermore, if the subqueries exp T,j (x) are defined as views and thus contribute with size 1 to the length of exp T,j+1 (x) then similar considerations also apply to TBoxes in the full language of DL-Lite N horn Without views or aggregation, the definition of I K is exponential Query Answering in DL-Lite (HN ) horn We now show how our (combined) approach to Q answering over DL-Lite N horn KBs can be extended to answering positive existential queries over DL-Lite (HN horn ) KBs, which can contain role inclusions subject to the constraint formulated in the preliminaries In fact, we show that this more general case can be reduced (at a price of an exponential blowup) to Q answering over DL-Lite N horn KBs Given a DL-Lite (HN horn ) KB K = (T, A), we first transform (in polytime) the TBox T into a DL-Lite N horn TBox T h by removing all RIs and adding the Is R S whenever R T S Since K h = (T h, A) is a DL-Lite N horn KB, it has a canonical interpretation I Kh, which can be stored as a relational instance in the RDBMS We then show that every positive existential query q can be rewritten into a union of Qs (UQ) q h such that the answers to q over K coincide with the answers to q h over I Kh We rely on Theorem 7 to answer the UQ q h in a component-wise fashion We construct q h in two steps First, we transform q into a positive existential query q by replacing each atom R(t, t ) in q with the disjunction S T R S(t, t ) Since the RIs were removed from T, this is now compensated by the construction of q learly, q can be constructed in polynomial time

6 Second, we convert q into disjunctive normal form q h, ie, into a UQ Of course, this results in an exponential blowup More precisely, we obtain a union of at most r q conjunctive queries, where r is the maximum over {S S T R}, for role atoms R(t, t ) in q, which is an improvement over the known ( T q ) q bound for the pure query rewriting approach Theorem 11 For every consistent DL-Lite (HN horn ) KB K and positive existential query q, cert(q, K) = ans(q h, I K h ) It is to be noted that if RIs are only used to express symmetry of roles then the exponential blowup can be avoided by modifying the definition of tree witnesses Query Answering in DL-Lite (HF) horn without UNA The Web ontology language OWL does not adopt the unique name assumption (UNA), but allows instead equality and inequality constraints of the form a b and a b for individual names Without UNA, Q answering in DL-Lite N core (with or without and ) becomes ONPhard for data complexity (Artale et al 2009) If, however, DL-Lite (HN horn ) TBoxes contain concepts m R with m 2 only in the form of functionality constraints 2 R (this fragment is called DL-Lite (HF) horn ) then query answering is only PTIME-complete for data complexity If concepts m R with m 2 are disallowed then query answering is LOGSPAE-complete (due to equalities a b) In the combined approach, both equality and functionality constraints can be eliminated at the stage of constructing the canonical interpretation Indeed, given a DL-Lite F horn KB K = (T, A), denote by eq K the minimal equivalence relation such that (a, b) eq K, for each a b in K, 2 R T, R(a, b), R(a, b ) A, (a, a ) eq K implies (b, b ) eq K To construct I K, we first compute the relation eq K and then take it into account in the definition of A I K and P I K above by saying in the exp T,j (x) that they can also be obtained from exp T,j 1 (y) and eq K (x, y); the queries qp T (x, y) are modified accordingly The RIs in DL-Lite horn (HF) can be treated at the query rewriting stage similarly to DL-Lite (HN horn ) under UNA Note also that DL-Lite (HF) horn TBoxes (and OWL 2 QL ontologies) may contain role disjointness, (a)symmetry and (ir)reflexivity constraints The presented approach can be extended to handle these features Experiments We evaluate the performance of the combined FO rewriting technique by comparing it with the pure query rewriting approach introduced in (alvanese et al 2005; 2006; 2007; 2008) and implemented in the QuOnto system (Acciarri et al 2005; Poggi, Rodriguez, and Ruzzi 2008) The experiments use several DL-Lite ontologies formulated in DL-Lite core, the common fragment of DL-Lite N horn and the logic underlying QuOnto Among the ontologies considered are the DL-Lite core approximation Galen-Lite of the medical ontology Galen (consisting of the DL-Lite core Is implied by Galen), the ore ontology (a representation of a fragment of a supply-chain management system used by the bookstore chain Waterstone s), the Stockexchange ontology (an EU financial institution s ontology), and the University ontology (a DL-Lite core version of the LUBM ontology developed at Lehigh University to describe the university organisational structure) The sample ontologies cover a wide spectrum of DL-Lite ontologies, ranging from complex concept hierarchies (as in Galen-Lite) to ontologies with rich role interactions (such as ore) The data was stored and the test queries were executed using DB2-Express version 95 running on Intel ore 2 Duo 25GHz PU, 4GB memory and 500GB storage under Linux 2628 Figure 1 summarises the running times for several test queries and randomly generated ABoxes of various size For each ABox, we report the number of individuals (Ind, in thousands), the numbers of concept assertions (As, in millions) and role assertions (RAs, in millions) in the original ABox and in the canonical interpretation For each query, we then show the execution times in the columns UN (the unmodified query over the original ABox, which does not give correct answers and serves as an ultimate lower bound ), RW (the rewritten query executed over the canonical interpretation), and QO (the query produced by QuOnto executed over the original ABox) The queries reported in the table are sample Qs with 3-6 atoms of various topologies (the exact shape of the queries for the Galen-lite and ore ontologies is given in the full paper at roman/ and, for the Stockexchange and University, in (Pérez-Urbina, Motik, and Horrocks 2009)); the size of the queries is limited by the feasibility of creating a QuOnto rewriting; the technique proposed here scales to considerably larger conjunctive queries In the case of the University ontology, RIs were incorporated into the query rewriting as outlined above, without a significant impact on the overall results The results can be summarised as follows: (1) query answering in our approach is competitive in performance with executing the original queries over the data (indeed, the query rewriting simply introduces additional selection conditions on top of the original Q that are executed in a pipelined fashion by the RDBMS); (2) query answering using the QuOnto approach is often prohibitively expensive even for relatively small ontologies; and (3) the construction of canonical interpretations can be performed off-line within 2 hours even for the largest data sets (where loading the data into the RDBMS alone takes tens of minutes) The need for constructing the canonical interpretation, similarly to creating additional auxiliary data structures in standard relational databases (eg, indices and materialised views), speeds up query processing at the expense of updates Incremental updates of the ABox can be supported by relying on techniques developed for efficient materialised view maintenance (olby et al 1996) onclusion We presented a combined approach to Q answering in DL-Lite N horn and some of its variants and demonstrated that this approach often allows more efficient query execution

7 Galen-lite 2733 concepts, 207 roles, and 4888 axioms ore 81 concepts, 58 roles, and 381 axioms University 31 concepts, 25 roles, and 103 axioms Stockexchange 17 concepts, 12 roles, and 62 axioms ABox size (in M) query Ind original canonical Q1 Q2 Q3 Q4 (in K) A RA A RA UN RW QO UN RW QO UN RW QO UN RW QO m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m >2 h m m m >2 h m m m >2 h m m m >2 h Figure 1: Query processing times (in seconds) than pure query rewriting There are several open issues for future work In particular, we do not know whether the combined approach can be implemented for DL-Lite (HN horn ) without an exponential blowup in the rewritten queries A closely related open problem is whether the combined approach for DL-Lite N horn can be extended to positive existential queries without such a blowup Finally, our polynomial rewriting for DL-Lite N core in the pure query rewriting approach raises the question whether the exponential blowup can also be avoided in other variants of DL-Lite References Abiteboul, S; Hull, R; Vianu, V 1995 Foundations of Databases Addison-Wesley Acciarri, A; alvanese, D; De Giacomo, G; Lembo, D; Lenzerini, M; Palmieri, M; Rosati, R 2005 QUONTO: Querying ontologies In Proc of AAAI, Artale, A; alvanese, D; Kontchakov, R; and Zakharyaschev, M 2009 The DL-Lite family and relations J of Artificial Intelligence Research 36:1 69 Baader, F; alvanese, D; McGuinness, D; Nardi, D; and Patel-Schneider, P F, eds 2003 The Description Logic Handbook: Theory, Implementation and Applications ambridge University Press alvanese, D; De Giacomo, G; Lembo, D; Lenzerini, M; and Rosati, R 2005 DL-Lite: Tractable description logics for ontologies In Proc of AAAI, alvanese, D; De Giacomo, G; Lembo, D; Lenzerini, M; and Rosati, R 2006 Data complexity of query answering in description logics In Proc KR, alvanese, D; De Giacomo, G; Lembo, D; Lenzerini, M; and Rosati, R 2007 Tractable reasoning and efficient query answering in description logics: The DL-Lite family J of Automated Reasoning 39(3): alvanese, D; De Giacomo, G; Lembo, D; Lenzerini, M; and Rosati, R 2008 Inconsistency tolerance in P2P data integration: an epistemic logic approach Information Systems 33(4): olby, L S; Griffin, T; Libkin, L; Mumick, I S; and Trickey, H 1996 Algorithms for deferred view maintenance In AM SIGMOD: Management of Data, Dolby, J; Fokoue, A; Kalyanpur, A; Ma, L; Schonberg, E; Srinivas, K; and Sun, X 2008 Scalable grounded conjunctive query evaluation over large and expressive knowledge bases In Proc of ISW, Heymans, S et al 2008 Ontology reasoning with large data repositories In Ontology Management, Semantic Web, Semantic Web Services, and Business Applications, Springer Kazakov, Y 2009 onsequence-driven reasoning for Horn-SHIQ ontologies In Proc of IJAI, Kontchakov, R; Lutz, ; Toman, D; Wolter, F; and Zakharyaschev, M 2009 ombined FO rewritability for conjunctive query answering in DL-Lite In Proc of DL, vol 477 of EUR-WS Kowalski, R, and Kuehner, D 1971 Linear resolution with selection function Artificial Intelligence 2: Lutz, ; Toman, D; and Wolter, F 2009 onjunctive query answering in the description logic EL using a relational database system In Proc of IJAI, Pérez-Urbina, H; Motik, B; and Horrocks, I 2009 A comparison of query rewriting techniques for DL-Lite In Proc of DL, vol 477 of EUR-WS Poggi, A; Lembo, D; alvanese, D; De Giacomo, G; Lenzerini, M; and Rosati, R 2008 Linking data to ontologies J on Data Semantics 10: Poggi, A; Rodriguez, M; and Ruzzi, M 2008 Ontologybased database access with DIG-Mastro and the OBDA Plugin for Protégé In Proc of OWLED 2008 D, vol 496 of EUR-WS

8 Appendix Proof of Theorem 3 Theorem 3 For every satisfiable DL-Lite N horn KB K and every Q q, cert(q, K) = ans(q, U K ) Proof Suppose K = (T, A) As shown in (Artale et al 2009), K = q[ a] if, and only if, J K = q[ a], where J K is the (canonical or minimal) model of K constructed inductively as follows Step 0 Set W 0 = Ind(A) and, for all concept and role names A and P, set A 0 = {a W 0 K = A(a)} and P 0 = {(a, b) P (a, b) A}; P0 is the inverse of P 0 In parallel with the construction of J K we also define a map h: J K U K At step 0, we set h 0 (a) = a for a W 0 The domain J K of J K will consist of Ind(A) and multiple copies of certain virtual points y R for some roles R, which are supposed to serve as witnesses for incoming R- arrows If w is a copy of y R then we write cp(w) = y R Step n+1 For a role R and a point w W n, let r n (R, w) be the number of distinct R-successors of w in W n, that is, r n (R, w) = {u Wn (w, u) R n } Let r(r, a), for a Ind(A), be the maximum m for which K = m R(a) and, for cp(w) = y S, let r(r, w) be the maximum number m for which K = S m R If such an m does not exist then we set r(r, a) = 0 or, respectively, r(r, w) = 0 For each w W n with r(r, w) r n (R, w) = l > 0, we add l new points u 1,, u l to W n, set cp(u i ) = y R, add the u i to A n if K = R A, and add the pairs (w, u i ) to R n This defines W n+1, A n+1 and P n+1, for all concept and role names A and P Let us now define h n+1 Suppose that h n (w) = a Ind(A) If a c R then a c R paths(i K ), (a, a c R ) R U K, and we set h n+1 (u i ) = a c R, for i l If a c R then, by (agen), there is b Ind(A) such that R(a, b) A, ie, (a, b) R U K Set h n+1 (u i ) = b Assume now that h n (w) = σ c S for some S By IH, cp(w) = y S If c S c R then σ c S c R paths(i K ) and (σ c S, σ c S c R ) R U K We set h n+1 (u i ) = σ c S c R, for i l Otherwise, by (rgen), we must have S = R and then (σ c S, σ) R U K We then set h n+1 (u i ) = σ, i l Step ω Finally, set J K = i<ω W i, A J K = i<ω A i and P J K = i<ω P i, for all role and concept names A and P in K, and a J K = a for all individual names a (Note that J K = K) And let h = i<ω h i It follows immediately from the definition that U K is a substructure of J K On the other hand, the map h is clearly a homomorphism from J K onto U K Therefore, J K = q[ a] if, and only if, U K = q[ a] Proof of Theorem 7 To begin with, we give some more intuition regarding how witness trees are computed Suppose that we want to compute f R1(t 0,t 1)(t n ) Assume first that the terms t 0 and t n are connected in q in the sense that there is a sequence of the form c = t 0 R 1 t 1 R 2 t 2 t n 1 R n t n, R i (t i 1, t i ) q (1) We will call c a computation of f R1(t 0,t 1)(t n ) We can compute f R1(t 0,t 1)(t n ) by taking first the word R 1 R 2 R n and then successively removing from it all leftmost pairs of the form R R (cf reductions in free groups) If in this removal process we obtain a word of the form R i R j w with R i = R j then f R1(t 0,t 1)(t j ) = ε and, if j < n, f R1(t 0,t 1)(t n ) is not defined Otherwise the resulting word is the value of f R1(t 0,t 1)(t n ), provided that all computations of f R1(t 0,t 1)(t n ) give the same result; if this is not the case then the tree witness for R 1 (t 0, t 1 ) does not exist If t 0 and t n are not connected by a computation then f R1(t 0,t 1)(t n ) is not defined Note also that if we use the computation c to determine f Rn(t n,t n 1)(t 0 ) then we can take the same word R 1 R 2 R n, successively remove from it all rightmost pairs of the form R R, and then take the inverse The example below shows that f Rn(t n,t n 1)(t 0 ) may be undefined even if f R1(t 0,t 1)(t n ) is defined Example 12 Let q = {R(t 1, t 2 ), S(t 2, t 3 ), S(t 4, t 3 )} Then f R(t1,t 2)(t 2 ) = R, f R(t1,t 2)(t 1 ) = ε, f R(t1,t 2)(t 3 ) = R S, f R(t1,t 2)(t 4 ) = R, f S(t4,t 3)(t 3 ) = S, f S(t4,t 3)(t 4 ) = ε, f S(t4,t 3)(t 2 ) = ε, and f S(t4,t 3)(t 1 ) is not defined In this case ϕ 1 = ϕ 2 = and ϕ 3 = ((t 3 = c S ) (t 2 = t 4 )) The following easily proved lemma will be used throughout this section Lemma 13 Suppose that f R1(t 0,t 1)(t n ) is defined Let c be a computation of f R1(t 0,t 1)(t n ) of the form (1) Then there are 0 = m 0 < m 1 < < m k = n, k 1, such that f Rmi (t mi,t mi 1)(t mi 1 ) = ε, for 1 < i k, and f Rm1 (t m1,t m1 1)(t 0 ) is the inverse of f R1(t 0,t 1)(t n ) Moreover, if f R1(t 0,t 1)(t n ) = ε then k = 1 In what follows, we exploit the following structural properties of I K and U K that are immediate from the definitions: (p1) If (d, d ) R I K, then one of the following holds: (i) d, d Ind(A); (ii) d = c R and d c R ; or (iii) d = c R and d c R (p2) If (σ, σ ) R U K, then one of the following holds: (i) σ, σ Ind(A); (ii) σ = σ c R and tail(σ) c R ; or (iii) σ = σ c R and tail(σ ) c R We now prove the main result of this paper: Theorem 7 For every DL-Lite N horn KB K and Q q, we have ans(q, I K ) = ans(q, U K ) Proof ( ) Let τ be an a-match for U K and q Define a map π : term(q) I K by taking π(t) = tail(τ(t)) for all t term(q) By the definitions of π and U K, we have I K = π ϕ, and so π is an a-match for I K and q Thus, it remains to show that I K = π ϕ 1 ϕ 2 ϕ 3 To do this, we require the following notion Let R(t, t ) q Define a relation X R(t,t ) term(q) (N R ) by taking X R(t,t ) = i 0 X (i) R(t,t ),

9 where X (0) R(t,t ) ={(t, R)}, X (i+1) R(t,t ) =X(i) R(t,t ) {(s, w S S ) (s, w S) X (i) R(t,t ), S (s, s ) q and S S } {(s, w) (s, w S) X (i) R(t,t ), S (s, s ) q} X R(t,t ) is clearly well-defined Moreover, it should also be clear that if it is a partial function then f R(t,t ) = X R(t,t ) Lemma 14 Suppose that R(t, t ) q and π(t ) = c R Then (s, S 1 S k ) X R(t,t ) implies τ(s) = τ(t) c S1 c Sk It follows that f R(t,t ) = X R(t,t ) Proof By the definition of X R(t,t ), it suffices to show that, for every i 0, if (s, S 1 S k ) X (i) R(t,t ) then τ(s) = τ(t) c S1 c Sk We proceed by induction on i For the basis of induction we consider (t, R) X (0) R(t,t ) Since π(t ) = c R, we have tail(τ(t )) = c R So, using (τ(t), τ(t )) R U K and (p2), we obtain τ(t ) = τ(t) c R as required For the induction step, we consider two cases (i) Assume that (s, R 1 R m S ) X (i+1) R(t,t ), (s, R 1 R m ) X (i) R(t,t ), S (s, s ) q and S R m Then, by IH, τ(s ) = τ(t) c R1 c Rm Since S R m and (τ(s), τ(s )) (S ) U K, (p2) gives us τ(s ) = τ(s) c S as required (ii) Suppose that (s, R 1 R m ) X (i+1) R(t,t ), (s, R 1 R m S) X (i) R(t,t ) and S (s, s ) q Then, by IH, τ(s) = τ(t) c R1 c Rm c S In view of (p2) and (τ(s ), τ(s)) S U K, we obtain τ(s ) = τ(t) c R1 c Rm as required We can now show that I K = π ϕ 1 ϕ 2 ϕ 3 ϕ 1 : By the definition of matches, we have τ(v) c R for any v avar(q) and c R N T I And by the definition of π, π(v) c R for any such v Thus, I K = π ϕ 1 ϕ 2 : Let R(t, t ) q and π(t ) = c R To prove I K = π ϕ 2, it is enough to show that f R(t,t ) exists, which follows from Lemma 14 Thus, I K = π ϕ 2 ϕ 3 : Let R(t, t ) q, f R(t,t )(s) = ε and π(t ) = c R By Lemma 14, τ(s) = τ(t), from which π(s) = π(t) Thus, I K = π ϕ 3 This completes the proof of ( ) ( ) Assume now that π is an a-match for I K and q Our aim is to show that there is an a-match τ for U K and q Obviously, we can set τ(t) = π(t) whenever π(t) Ind(A) Defining τ(t) for other t term(q) that is, for the terms t that are mapped by π to points of the form c R in I K is a bit more problematic all a term t term(q) a root (of q under π) if either π(t) Ind(A) or π(t) = c R and there is no atom R(t, t) q Lemma 15 For every non-root t term(q), there is some R(s, s ) q such that s is a root, π(s ) = c R and f R(s,s )(t) is defined Proof Suppose t term(q) is not a root By the definition of roots, this means that there is some R 0 (t 1, t 0 ) q with t 0 = t and π(t 0 ) = c R0 Further, either t 1 is a root or there is some R 1 (t 2, t 1 ) q with π(t 1 ) = c R1 We iterate this argument until we reach a root To show that this indeed eventually happens, suppose otherwise Then there is an infinite sequence R 0 (t 1, t 0 ), R 1 (t 2, t 1 ), of atoms in q such that t = t 0 and π(t i ) = c Ri for all i 0 By (p1), we have R i R i+1 for all i 0 Since term(q) is finite, there are j, k with j < k and t j = t k Due to I K = π ϕ 2, the tree witness f Rk (t k+1,t k ) exists Since R i R i+1 for all i 0, it is easy to see that f Rk (t k+1,t k )(t i ) = R k R k 1 R i for all i k We thus obtain f Rk (t k+1,t k )(t j ) = R k R j, contrary to f Rk (t k+1,t k )(t k ) = R k and t k = t j So there is a sequence R 0 (t 1, t 0 ),, R l 1 (t l, t l 1) q such that t 0 = t, t l is a root, R i R i+1, for i < l 1, and π(t i ) = c Ri, for i l 1 By the definition of tree witnesses, we then have f Rl 1 (t l,t l 1)(t) = R l 1 R 0 As π(t l 1 ) = c Rl 1, it follows that the atom R l 1 (t l, t l 1 ) is as required Example 16 onsider the KB K = (T, A) with T = {A 1 A, A 2 A, P S R, R S, A P }, A = {A 1 (a), A 2 (b)} Then the canonical interpretation I K is as follows: I K = {a, b, c P, c S, c R }, A I K = {a, b}, A I K 1 = {a}, A I K 2 = {b}, P I K = {(a, c P ), (b, c P )}, R I K = {(c P, c R )}, S I K = {(c P, c S ), (c R, c S )} onsider again the query q from Example 12 It is readily checked that the map π defined by taking π(t 1 ) = c P, π(t 2 ) = π(t 4 ) = c R, π(t 3 ) = c S is an a-match for I K and q Both t 1 and t 4 are roots, while t 2 and t 3 are not roots Moreover, f R(t1,t 2)(t 4 ) = R, while f R(t4,t 3)(t 1 ) is not defined We shall also require the following two lemmas Lemma 17 If f S(s,s )(t) = w R and π(s ) = c S, then π(t) = c R Proof The proof is by induction on the number of steps required to compute f S(s,s )(t) The basis of induction, that is, the case t = s, is trivial For the induction step, we consider two cases (i) Assume that f S(s,s )(t ) = w Q, Q R, R(t, t) q, and so f S(s,s )(t) = w Q R By IH, π(t ) = c Q As I K = π ϕ, we have (c Q, π(t)) R I K So, by (p1), π(t) = c R (ii) Suppose that f S(s,s )(t ) = w Q, Q (t, t) q, and so f S(s,s )(t) = w = w R By IH, π(t ) = c Q Take the nearest predecessor t of t in the computation of f S(s,s )(t ) with f S(s,s )(t ) = w R, which must

10 exist by the definition of tree witnesses By IH, π(t ) = c R As I K = π ϕ 2 and π(t ) = c Q, f Q(t,t ) exists Moreover, by the choice of t, we clearly have f Q(t,t )(t ) = ε, from which, by I K = π ϕ 3, π(t) = π(t ) = c R Lemma 18 Suppose f S(s,s )(t) is defined and nonempty, π(s ) = c S, f R(t,t )(r) is defined and π(t ) = c R Then f S(s,s )(r) is defined and f S(s,s )(r) = f S(s,s )(t) f R(t,t )(r) Proof Let f S(s,s )(t) = w Q By Lemma 17, π(t) = c Q In view of π(t ) = c R and (p1), Q R We can now prove our claim by an easy induction on the number of steps required to compute f S(s,s )(r) The basis of induction ie, the case r = t follows from f S(s,s )(t) = w Q and Q R, as we have f S(s,s )(t ) = w Q R Suppose next that f R(t,t )(r ) = w T, T (r, r) q and T T, from which f R(t,t )(r) = w T T By IH, f S(s,s )(r ) = w Q w T, and so f S(s,s )(r) = w Q w T T Finally, if f R(t,t )(r ) = w T and T (r, r) q then f R(t,t )(r) = w By IH, f S(s,s )(r ) = w Q w T, and so f S(s,s )(r) = w Q w A root t of q under π is called initial if there is no S(s, s ) q such that s is a root, π(s ) = c S, f S(s,s )(t) is defined and f S(s,s )(t) ε Example 19 In the environment of Example 16, root t 1 is initial (although t 4 is a root with S(t 4, t 3 ) q, π(c 3 ) = c S and f S(t4,t 3)(t 1 ) not defined) On the contrary, root t 4 is not initial because f R(t1,t 2)(t 4 ) = R Lemma 20 If π(t) Ind(A) then t is an initial root Proof By definition, if π(t) Ind(A) then t is a root To show that t is initial, assume to the contrary that there is some S(s, s ) q such that s is a root, π(s ) = c S, f S(s,s )(t) is defined and f S(s,s )(t) ε By Lemma 17, we have π(t) = c R N T I contrary to π(t) Ind(A) We now strengthen Lemma 15 to the following: Lemma 21 For each t term(q), either t is an initial root or there is some R(s, s ) q such that s is an initial root, π(s ) = c R, f R(s,s )(t) is defined and nonempty Proof Let t term(q) If t is an initial root, we are done Suppose now that t is a root but not initial Then there is some R 0 (s 0, s 0) q such that s 0 is a root, π(s 0) = c R0, f R0(s 0,s 0 ) (t) is defined and f R0(s 0,s 0 ) (t) ε If s 0 is initial, we are done Otherwise, there is some R 1 (s 1, s 1) q such that s 1 is a root, π(s 1) = c R1, f R1(s 1,s 1 ) (s 0 ) is defined and f R1(s 1,s 1 ) (s 0 ) ε Moreover, by Lemma 18, f R1(s 1,s 1 ) (t) = f R1(s 1,s 1 ) (s 0 ) f R0(s 0,s 0 ) (t) We can repeat this argument, and each time the word f Ri(s i,s i ) (t) becomes strictly longer Due to finiteness of q, we thus eventually reach an initial root Finally, if t is not a root, then Lemma 15 gives some R(s, s ) q such that s is a root, π(s ) = c R and f R(s,s )(t) is defined We can then proceed as in the previous case We now show the following: Lemma 22 Suppose r term(q) and R(t, t ), S(s, s ) q are such that both t and s are initial roots, π(t ) = c R, π(s ) = c S and f R(t,t )(r), f S(s,s )(r) are defined Then f R(t,t )(r) = f S(s,s )(r) and π(t) = π(s) Proof onsider four possible cases ase 1: f S(s,s )(r) = f R(t,t )(r) = ε As I K = π ϕ 3, we have π(t) = π(r) = π(s) ase 2: f S(s,s )(r) = ε and f R(t,t )(r) ε onsider some shortest computation of f S(s,s )(r), say, sss S 2 s 2 s n 1 S n r As f S(s,s )(r) = ε, S n = S And in view of I K = π ϕ 3, we have π(s) = π(r) It follows that π(s n 1 ) = π(s ) = c S By Lemma 13, f S(r,sn 1)(s) = ε And by Lemma 18, f R(t,t )(s) = f R(t,t )(r) f S(r,sn 1)(s) ε, contrary to s being an initial root ase 3: f S(s,s )(r) ε and f R(t,t )(r) = ε is similar to ase 2 ase 4: f R(t,t )(r) ε and f S(s,s )(r) ε onsider again the shortest computation sss S 2 s 2 s n 1 S n r of f S(s,s )(r) We claim that f R(t,t )(s) = ε Indeed, as s is an initial root, f R(t,t )(s) cannot be defined and nonempty And if f R(t,t )(s) is not defined then there must exist some i, 1 i < n, such that f R(t,t )(s i ) = ε (here we set s = s 1 ) But, as we saw in ase 2, this is impossible because f S(s,s )(s i ) ε Thus, f R(t,t )(s) = ε and so, by ϕ 3, we have π(t) = π(s) It should also be clear from the discussion after the definition of tree witnesses that f R(t,t )(r) = f S(s,s )(r) We are now in a position to define an a-match τ for U K and q For each c R I K, we choose a γ(c R ) U K with tail(γ(c R )) = c R and define a map τ : term(q) U K as follows: (a) if π(t) Ind(A), then we set τ(t) = π(t); (b) if π(t) / Ind(A) and t is an initial root, then we set τ(t) = γ(π(t)); (c) if R(t, t ) q, t is an initial root, s term(q), π(t ) = c R, f R(t,t )(s) is defined and f R(t,t )(s) = S 1 S k, then set τ(s) = τ(t) c S1 c Sk By Lemma 21, τ is total To see that it is well-defined, it is suffices to observe that, by Lemma 20, cases (a) (c) are disjoint (ie, for each t term(q), τ(t) is defined in only one of them) and that (c) is well-defined by Lemma 22 It thus remains to show that τ is an a-match for U K and q Note first that, by the definition of τ and Lemma 17, we have tail(τ(t)) = π(t), for all t term(q) (2) By the definition of U K and (2), all concept atoms in q are satisfied by τ Now let R(t, t ) be a role atom in q The following six cases are possible: ase 1: π(t) and π(t ) are defined in (a) Then we have τ(t) = π(t) Ind(A) and τ(t ) = π(t ) Ind(A) Since