Botoeva, Elena

Lutz, Carsten

Ryzhikov, Vladislav

Wolter, Frank

Zakharyaschev, Michael

Kent Academic Repository

Kent Academic RepositoryFull text document (pdf)Copyright & reuseContent in the Kent Academic Repository is made available for research purposes. Unless otherwise stated allcontent is protected by copyright and in the absence of an open licence (eg Creative Commons), permissions for further reuse of content should be sought from the publisher, author or other copyright holder. Versions of researchThe version in the Kent Academic Repository may differ from the final published version. Users are advised to check http://kar.kent.ac.uk for the status of the paper. Users should always cite the published version of record.EnquiriesFor any further enquiries regarding the licence status of this document, please contact: researchsupport@kent.ac.ukIf you believe this document infringes copyright then please contact the KAR admin team with the take-down information provided at http://kar.kent.ac.uk/contact.htmlCitation for published versionBotoeva, Elena, Lutz, Carsten, Ryzhikov, Vladislav, Wolter, Frank and Zakharyaschev, Michael  (2016) Is Query Inseparability for ALC Ontologies Decidable?    In:  CEUR Workshop Proceedings.Proceedings of the 29th International Workshop on Description Logics, Cape Town, South Africa,April 22-25, 2016. CEUR Workshop Proceedings , 1577.  CEUR-WS.orgDOILink to record in KARhttps://kar.kent.ac.uk/91267/Document VersionPublisher pdfIs Query Inseparability for ALC Ontologies Decidable?E. Botoeva,1 C. Lutz,2 V. Ryzhikov,1 F. Wolter3 and M. Zakharyaschev41Univ. of Bozen-Bolzano, 2Univ. of Bremen, 3Univ. of Liverpool, 4Birkbeck, Univ. of London1 IntroductionWhile query answering with Description Logics (DLs) is now well-developed, this ismuch less the case for reasoning services that support ontology engineering and targetquery answering as an application. In ontology versioning, for example, one wouldlike to know whether two versions of an ontology give the same answers to all queriesformulated over a given vocabulary of interest, which means that they can be safelyreplaced in an application [5]. Similarly, if one wants to know whether a given ontologycan be safely replaced by a smaller subset (a module), it is the answers to all queries thatshould be preserved [7]. In this context, the fundamental relationship between ontologiesis thus not whether they are logically equivalent (have the same models), but whetherthey give the same answers to all relevant queries. The resulting entailment problem canbe formalized in two ways, with different applications. First, given a class Q of queries,knowledge bases (KBs) K1 = (T1,A1) and K2 = (T2,A2), and a signature Σ ofrelevant concept and role names, we say thatK1 Σ-Q-entailsK2 if the answers to anyΣ-query inQ over K2 are contained in the answers to the query over K1. K1 and K2 are Σ-Q-inseparable if they Σ-Q-entail each other. This notion of entailment is appropriate ifthe data is known and does not change frequently. Applications include data-oriented KBversioning and KB module extraction, KB forgetting [9], and knowledge exchange [1].If the data is not known or changes frequently, it is not KBs that should be compared,but TBoxes. Given a pair Θ = (Σ1, Σ2) specifying a relevant signature Σ1 for ABoxesand Σ2 for queries, we say that a TBox T1 Θ-Q-entails a TBox T2 if, for every Σ1-ABox A, the KB (T1,A) Σ2-Q-entails (T2,A). T1 and T2 are Θ-Q-inseparable ifthey Θ-Q-entail each other. Applications include data-oriented TBox versioning, TBoxmodularization and TBox forgetting [7].Most important choices for Q are conjunctive queries (CQs) and unions thereof(UCQs); we also consider the practically relevant classes of rooted CQs (rCQs) andUCQs (rUCQs), in which every variable is connected to an answer variable. So far, CQ-entailment has been studied for Horn DL KBs [3], EL TBoxes [8, 5], DL-Lite TBoxes [6],and also for OBDA specifications, that is, DL-Lite TBoxes with mappings [2]. No resultsare available for non-Horn DLs (neither in the KB nor in the TBox case) and forexpressive Horn DLs in the TBox case. In particular, query entailment in non-Horn DLshad the reputation of being a technically challenging problem.This paper makes a first breakthrough into understanding query entailment andinseparability in these cases, with the main results summarized in Figures 1 and 2 (thosemarked with (?) are from [3]). Most unexpected is the undecidability of CQ- and rCQ-entailments between ALC KBs, even when the first KB is formulated in Horn-ALC(in fact, EL) and without any signature restriction. This should be contrasted with theALC Horn-ALCto ALCALC toHorn-ALC Horn-ALCundecidable? =EXPTIME(?)?undecidable ≤2EXPTIME =EXPTIME(?)≤2EXPTIMEFig. 1. KB query entailment.QueriesCQUCQrCQrUCQALC Horn-ALCto ALCALC toHorn-ALC Horn-ALCundecidable? =2EXPTIME?undecidable=EXPTIME =EXPTIME?Fig. 2. TBox query entailment.decidability of subsumption-based entailment between ALC TBoxes [4] and of CQ-entailment between Horn-ALC KBs [3]. The second surprising result is that entailmentbetween ALC KBs becomes decidable when CQs are replaced with rUCQs. For ALCTBoxes, CQ- and rCQ-entailments are undecidable as well. We obtain decidability forHorn-ALC TBoxes (where CQ- und UCQ-entailments coincide) using the fact thatnon-entailment is always witnessed by tree-shaped ABoxes. As another surprise, CQ-entailment of Horn-ALC TBoxes is 2EXPTIME-complete while rCQ-entailment is onlyEXPTIME-complete. This should be contrasted with the EL case, where both problemsare EXPTIME-complete [8]. All upper bounds and most lower bounds hold also whenentailment is replaced with inseparability.2 Undecidability of CQ-Entailment for ALC KBsHere we sketch the construction showing that the problem whether an EL KB Σ-CQentails anALC KB is undecidable. The proof is by reduction of the undecidableN ×M -tiling problem: given a finite set T of tile types T with four colours up(T ), down(T ),left(T ) and right(T ), a tile type I ∈ T, and two colours W (for wall) and C (for ceiling),decide whether there exist N,M ∈ N such that the N ×M grid can be tiled using T insuch a way that (1, 1) is covered by a tile of type I; every (N, i), for i ≤M , is coveredby a tile of type T with right(T ) =W ; and every (i,M), for i ≤ N , is covered by a tileof type T with up(T ) = C. Given an instance of this problem, we first describe a KBK2 = (T2, {A(a)}) that uses (among others) three concept names Tk, k = 0, 1, 2, foreach tile type T ∈ T. If a point x in a model I of K2 is in Tk and right(T ) = left(T ′),then x has anR-successor in T ′k. Thus, branches of I define (possibly infinite) horizontalrows of tilings with T. If a branch contains a point y ∈ Tk with right(T ) = W , thenthis y can be the last point in the row, which is indicated by an R-successor z ∈ Row ofy. In turn, z has R-successors in all T(k+1) mod 3 that can be possible beginnings of thenext row of tiles. To coordinate the up and down colours between the rows—which willbe done by the CQs separating K1 and K2— we make every x ∈ Tk, starting from thesecond row, an instance of all T ′(k−1) mod 3 with down(T ) = up(T′). The row started byz ∈ Row can be the last one in the tiling, in which case we require that each of its tilesT has up(T ) = C. After the point in Row indicating the end of the final row, we add anR-successor in End for the end of tiling. The beginning of the first row is indicated by aP -successor in Start of the ABox element a, after which we add an R-successor in I0for the given initial tile type I; see the lowest branch in the figure below. To generate atree with all possible branches described above, we only require EL axioms of the formE v D and E v ∃S.D.T0 T0 T0 T1 T1 T1 T1T2T2RowEndT2∨ RowRowStartRow EndhlhlhrhrA Start I0 TN10 Start T121 TN21RowT 1M-11 TNM-11Row EndT 1M2 TNM2RowEndRowaI0. . . TN10 . . . T1M-20 . . . TNM-20 . . .T 1M-11 . . . TNM-11 . . .P ∨IlIrStart B1 BN BN+1 Bn−N Bn−1 Bn EndqnThe existence of a tiling of some N ×M grid for the given instance can be checkedby Boolean CQs qn that require an R-path from Start to End going through Tk- orRow-points: ∃x(Start(x0) ∧∧ni=0R(xi, xi+1) ∧∧ni=1Bi(xi) ∧ End(xn+1)), whereBi ∈ {Row} ∪ {Tk | T ∈ T, k = 0, 1, 2}. The key trick is—using an axiom of theform D v E t E′—to ensure that the Row-point before the final row of the tilinghas two alternative continuations: one as described above, and the other one havingjust a single R-successor in End where ∨ indicates an or-node. This or-node gives twomodels of K2 denoted Il and Ir in the picture. If K2 |= qn, then qn holds in both ofthem, and so there are homomorphisms hl : qn → Il and hr : qn → Ir. As hl(xn−1)and hr(xn−1) are instances of Bn−1, we have Bn−1 = TNM−11 in the picture, andso up(TNM−1) = down(TNM ). By repeating this argument until x0, we see that thecolours between horizontal rows match and the rows are of the same length. (For thistrick to work, we have to make the first Row-point in every branch an instance of Start.)In fact, an instance of the N ×M -tiling problem has a positive answer iff there exists qnsuch that K2 |= qn. It is to be noted that to construct T2 with the properties describedabove, one needs quite a few auxiliary concept names.Let K1 = (T1, {A(a)}) be an EL KB with the following canonical model:A Start, Σ0End, Σ0 End, Σ0Start, Σ0End, Σ0 End, Σ0Start, Σ0End, Σ0 End, Σ0aPwhere Σ0 = {Row} ∪ {Tk | T ∈ T, k = 0, 1, 2}. Note that the vertical R-successors ofthe Start-points are not instances of any concept name, and so K1 does not satisfy anyquery of the form qn. On the other hand, K2 |= q implies K1 |= q, for every Σ-CQ qwithout a subquery of the form qn and Σ = sig(K1). Therefore, K1 Σ-CQ entails K2iff there exists no tiling as specified above.3 Future WorkWe have made first steps towards understanding query entailment and inseparability forKBs and TBoxes in expressive DLs. From a theoretical viewpoint, it would be of interestto solve the open problems in Figures 1 and 2, and also consider other expressive DLssuch as DL-LiteHbool or ALCI. For example, our undecidability proof goes through forDL-LiteHbool , but the other cases remain open. From a practical viewpoint, our model-theoretic criteria for query entailment are a good starting point for developing algorithmsfor approximations of query entailment based on simulations. Our undecidability andcomplexity results also indicate that rUCQ-entailment is more amenable to practicalalgorithms than, say, CQ-entailment and can be used as an approximation of the latter.References1. Arenas, M., Botoeva, E., Calvanese, D., Ryzhikov, V.: Exchanging OWL 2 QL knowledgebases. In: Proc. of the 23rd Int. Joint Conf. on Artificial Intelligence (IJCAI 2013). AAAI Press(2013)2. Bienvenu, M., Rosati, R.: Query-based comparison of OBDA specifications. In: Proc. of the28th Int. Workshop on Description Logics (DL 2015). vol. 1350. CEUR-WS (2015)3. Botoeva, E., Kontchakov, R., Ryzhikov, V., Wolter, F., Zakharyaschev, M.: Query inseparabilityfor description logic knowledge bases. In: Proc. of the 14th Int. Conf. on the Principles ofKnowledge Representation and Reasoning (KR 2014) (2014)4. Ghilardi, S., Lutz, C., Wolter, F.: Did I damage my ontology? A case for conservative extensionsin description logics. In: Doherty, P., Mylopoulos, J., Welty, C. (eds.) Proc. of the 10th Int.Conf. on the Principles of Knowledge Representation and Reasoning (KR 2006). pp. 187–197(2006)5. Konev, B., Ludwig, M., Walther, D., Wolter, F.: The logical difference for the lightweightdescription logic EL. Journal of Artificial Intelligence Research (JAIR) 44, 633–708 (2012)6. Kontchakov, R., Pulina, L., Sattler, U., Schneider, T., Seimer, P., Wolter, F., Zakharyaschev, M.:Minimal module extraction from DL-Lite ontologies using QBF solvers. In: Proc. of the 21stInt. Joint Conf. on Artificial Intelligence (IJCAI 2009). pp. 836–840 (2009)7. Kontchakov, R., Wolter, F., Zakharyaschev, M.: Logic-based ontology comparison and moduleextraction, with an application to DL-Lite. Artificial Intelligence 174, 1093–1141 (2010)8. Lutz, C., Wolter, F.: Deciding inseparability and conservative extensions in the descriptionlogic EL. J. of Symbolic Computation 45(2), 194–228 (2010)9. Wang, K., Wang, Z., Topor, R.W., Pan, J.Z., Antoniou, G.: Eliminating concepts and roles fromontologies in expressive descriptive logics. Computational Intelligence 30(2), 205–232 (2014)

Is Query Inseparability for ALC Ontologies Decidable?

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Is Query Inseparability for ALC Ontologies Decidable?

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