Introduction to Description Logic and Query Rewriting

Transcription

1 Introduction to Description Logic and Query Rewriting Roman Kontchakov Department of Computer Science and Inf. Systems, Birkbeck College, London Alessandro Artale, Diego Calvanese, Carsten Lutz, Mariano Rodríguez-Muro, David Toman, Frank Wolter and Michael Zakharyaschev

2 The Web Ontology Language RW 2014, Athens,

3 Part 1 DL Basics RW 2014, Athens,

4 Description Logics by Example SubClassOf(ObjectIntersectionOf(Person, ObjectSomeValuesFrom(takesCourse, Course)), Student) Person takescourse.course Student SubObjectPropertyOf(mastersDegreeFrom, degreefrom) mastersdegreefrom degreefrom SubClassOf(ObjectSomeValuesFrom( ObjectInverseOf(takesCourse), owl:thing), Course) ClassAssertion(Student, john) takescourse. Course Student(john) ObjectPropertyAssertion(takesCourse, john, sw) takescourse(john, sw) RW 2014, Athens,

5 DL Syntax concepts (classes) C ::= A }{{} i concept name C }{{} ObjectComplementOf(C) }{{} owl:thing R.C }{{} ObjectSomeValuesFrom(R,C) }{{} owl:nothing C 1 C 2 }{{} ObjectIntersectionOf(C 1,C 2 ) R.C }{{} ObjectAllValuesFrom(R,C) C 1 C 2 }{{} ObjectUnionOf(C 1,C 2 ) roles (object properties) R ::= P }{{} i P i role name TBox T C 1 C 2 }{{} SubClassOf(C 1,C 2 ) and R 1 R 2 }{{} SubObjectPropertyOf(R 1,R 2 ) ABox A C(a) and R(a, b) knowledge base K = (T, A) (ontology) RW 2014, Athens,

6 interpretation I = ( }{{} I, I) domain DL Semantics I sw GraduateCourse FirstYearStudent john takescourse takescourse sp1 takescourse Student kate I (interpretation function) individuals a i elements a I i I concept names A i sets A I i I role names P i binary relations P I i I I RW 2014, Athens,

7 DL Semantics (2) j takescourse s (P ) I = {(v, u) (u, v) P I } takescourse I = I and I = j s l ( C) I = I \ C I (C 1 C 2 ) I = C I 1 CI 2 and (C 1 C 2 ) I = C I 1 CI 2 j k b RW 2014, Athens,

8 DL Semantics (3) ( R.C) I = { u there is v C I such that (u, v) R I } takescourse.undergraduatecourse takescourse.graduatecourse john takescourse sw GraduateCourse takescourse takescourse sp1 UndergraduateCourse Student kate ( R.C) I = { u v C I, for all v with (u, v) R } I R.C = R. C NB. for all is true when there are no v with (u, v) R I e.g., sp1 ( takescourse.undergraduatecourse) I sp1 ( takescourse. ) I RW 2014, Athens,

9 DL Semantics (4) I = C 1 C 2 C I 1 CI 2 j k b C 1 C 2 I = R 1 R 2 R I 1 RI 2 I = C(a) a I C I I = R(a, b) (a I, b I ) R I I is a model of (T, A) if I = α, for all inclusions α in T and assertions α in A RW 2014, Athens,

10 Open World Assumption T = { GraduateStudent Student GraduateStudent takescourse.graduatecourse } A = { GraduateStudent(john) } I 1 GraduateStudent Student GraduateCourse j s takescourse john GraduateStudent Student GraduateCourse a john GraduateStudent Student j john takescourse I 2 I 3 john I1 = j GraduateStudent I1 = {j} Student I1 = {j} GraduateCourse I1 = {s} takescourse I1 = {(j, s)} is a model john I2 = a GraduateStudent I2 = {a} Student I2 = {a} GraduateCourse I2 = {a} takescourse I2 = {(a, a)} is a model john I3 = j GraduateStudent I3 = {j} Student I3 = {j} GraduateCourse I3 = takescourse I3 = is not a model RW 2014, Athens,

11 Reasoning: Consistency a knowledge base K is satisfiable (or consistent) (in other words, K implies no contradictions) if there exists at least one model of K Example T : A: UndergraduateStudent takescourse.undergraduatecourse UndergraduateCourse GraduateCourse UndergraduateStudent(john) takescourse(john, sw) GraduateCourse(sw) (T, A) is inconsistent: John (as an undergraduate student) can take only undergraduate courses. We know, however, that he takes a graduate course, which cannot be an undergraduate one. RW 2014, Athens,

12 Reasoning: Entailment C 1 C 2 is entailed by K K = C 1 C 2 if I = C 1 C 2 for all models I of K (entailment for role inclusions and concept and role assertions is defined similarly) T : takescourse.undergraduatecourse UndergraduateStudent FirstYearStudent takescourse.undergraduatecourse. I 1 FirstYearStudent UndergraduateStudent j takescourse s UndergraduateCourse I 2 FirstYearStudent j takescourse s UndergraduateCourse takescourse l I 1 = T I 1 = FirstYearStudent UndergraduateStudent I 2 = T I 2 = FirstYearStudent UndergraduateStudent RW 2014, Athens,

13 Reasoning: Entailment (2) C 1 is subsumed by C 2 with respect to K if K = C 1 C 2 Proposition (T, A) = C 1 C 2 iff (T, A {C 1 (a), C 2 (a)}) is not satisfiable for a fresh a Proof iff = if and only if (, only if) Let (T, A) = C 1 C 2. Assume, for the sake of contradiction, that (T, A {C 1 (a), C 2 (a)}) is satisfiable. Then there is an interpretation I such that I = (T, A) (a model of (T, A)), a I C I 1 and ai / C I 2, contrary to (T, A) = C 1 C 2 (, if) Let (T, A {C 1 (a), C 2 (a)}) be not satisfiable. Assume, for the sake of contradiction, that (T, A) = C 1 C 2. Then there is a model of (T, A) and some u C I 1 such that u / CI 2. Since a is fresh, we can redefine a I by taking a I = u, which will contradict inconsistency of (T, A {C 1 (a), C 2 (a)}). a is an instance of C with respect to K if K = C(a) Proposition (T, A) = C(a) iff (T, A { C(a)}) is not satisfiable RW 2014, Athens,

14 Conjunctive Queries (CQs) conjunctive query q( x }{{} answer variables ) = y ϕ( x, y) }{{} existentially quantified variables ϕ( x, y) is a conjunction of atoms such as A }{{} concept name (z) and P (z 1, z 2 ) }{{} role name z, z 1 and z 2 are terms = an individual name or a variable from x or y Example q(x) = y, z ( friend(x, y) Female(y) loves(y, z) Male(z) ) SELECT?x WHERE {?x :friend?y.?y a :Female.?y :loves?z.?z a :Male. } RW 2014, Athens,

15 Certain Answers to CQs q( x) = y ϕ( x, y) is a CQ with x = (x 1,..., x n ) a = (a 1,..., a n ) is a tuple of individual names from A q( a) is the result of replacing each x i in y ϕ( x, y) with a i a is a certain answer to q( x) over (T, A) I = q( a) if, for any model I of (T, A), the sentence q( a) is true in I q without answer variables is Boolean a certain answer to q over (T, A) is yes if (T, A) = q and no otherwise RW 2014, Athens,

16 Andrea s Example T : Male Female, Male Female A: friend(john, susan), friend(john, andrea), Female(susan) loves(susan, andrea), loves(andrea, bill), Male(bill) q(x) = y, z ( friend(x, y) Female(y) loves(y, z) Male(z) ) friend john loves andrea susan Female Female loves bill Male friend A john loves andrea susan Male Female loves friend bill Male friend A NB: the same as finding instances of friend.(female loves.male) RW 2014, Athens,

17 From OWL to DL ObjectPropertyDomain(takesCourse, Student) takescourse. Student ObjectPropertyRange(takesCourse, Course) takescourse. Course or takescourse.course Proposition The following are equivalent (have the same models) C 1 P.C 2 and P.C 1 C 2 C 1 C 2 C and { C 1 C, C 2 C} C C 1 C 2 and { C C 1, C C 2 } C 1 C 2 and C 1 C 2 RW 2014, Athens,

18 From OWL to DL (2) EquivalentClasses(C 1, C 2,..., C n ) NB: A B stands for A B and B A C 1 C 2, C 2 C 3,..., C n 1 C n, C n C 1 DisjointClasses(C 1, C 2,..., C n ) C i C j, for all i, j with 1 i < j n DisjointUnion(C, C 1, C 2,..., C n ), C C 1 C n SymmetricObjectProperty(P ) P P RW 2014, Athens,

19 DL Zoo ALCHI AL attributive language C complement C I role inverses P H role inclusions R 1 R 2 S ALC + transitive roles N unqualified number restrictions q R. O nominals {a} Q qualified number restrictions q R.C F functionality constraints 2 R. R role chains and R.Self SHOIN OWL 1.0 SHIF OWL Lite SROIQ OWL 2 RW 2014, Athens,

20 Complexity of Reasoning The satisfiability problem is ExpTime-complete for ALCHI KBs and N2ExpTime-complete for SROIQ KBs Concept and role subsumption and instance checking are ExpTime- and con2exptime-complete for, respectively, ALCHI and SROIQ KBs CQ entailment over ALCHI KBs is 2ExpTime-complete CQ entailment over SROIQ is not even known to be decidable DL Complexity Navigator: practical reasoners for OWL 2 DL: FaCT++, HermiT, Pellet RW 2014, Athens,

21 Part 2 OWL 2 Profiles OWL 2 DL OWL 2 QL RDFS OWL 2 EL OWL 2 RL RW 2014, Athens,

22 (In)tractable Reasoning Garey & Johnson, 1979 Computers and Intractability: A Guide to the Theory of NP-Completeness RW 2014, Athens,

23 (In)tractable Reasoning Garey & Johnson, 1979 Computers and Intractability: A Guide to the Theory of NP-Completeness RW 2014, Athens,

24 (In)tractable Reasoning Garey & Johnson, 1979 Computers and Intractability: A Guide to the Theory of NP-Completeness RW 2014, Athens,

25 OWL 2 Profiles: No Disjunction graph 3-colourability problem is NP-complete given an (undirected) graph G = (V, E), decide whether each of its vertices can be painted in one of the three colours in such a way that no pair of adjacent vertices has the same colour A G : edge(v 1, v 2 ), for each {v 1, v 2 } E T : C 1 C 2 C 3 C i C j, 1 i < j 3, C i edge.c i, 1 i 3 models of (T, A G ) 3-colouring of G (T, A G ) is satisfiable G is 3-colourable the satisfiability problem for KBs in any DL able to express this TBox is NP-hard (intractable) RW 2014, Athens,

26 OWL 2 RL Description Logic Programs (Grosof, Horrocks, Volz, Decker 03) and pd (ter Horst 05) supported by Oracle Database 11g, OWLIM, BaseVISor, ELLY, Jena and RDFox RL TBox: B A, R 1 R 2 and B only concept names on the right B ::= A R. R.B B 1 B 2 simple ABox: A(a) and P (a, b) NB: this is a simplified version see the equivalences RW 2014, Athens,

27 RL and Datalog UndergraduateStudent Student x ( UndergraduateStudent(x) Student(x) ) takescourse.undergraduatecourse UndergraduateStudent x y ( takescourse(x, y) UndergraduateCourse(y) UndergraduateStudent(x) ) standard translation ST x (A) = A(x) and ST x,y (P ) = P (x, y) ST x,y (P ) = P (y, x) ST x (B 1 B 2 ) = ST x (B 1 ) ST x (B 2 ) ST x ( R. ) = y ST x,y (R) ST x ( R.B) = y ( ST x,y (R) ST y (B) ) (B A) = x ( ST x (B) ST x (A) ) (R 1 R 2 ) = x y ( ST x,y (R 1 ) ST x,y (R 2 ) ) (B ) = x ( ST x (B) ) y ( γ 1 ( y) γ k ( y) γ 0 ( y) ) y ( γ 1 ( y) γ k ( y) ) Horn clauses (datalog programs) RW 2014, Athens,

28 T : Chase: Example takescourse.undergraduatecourse UndergraduateStudent UndergraduateStudent Student A: takescourse(john, sp1), UndergraduateCourse(sp1) step 0: I 0 A I 0 = {john, sp1} takescourse I 0 = {(john, sp1)} UndergraduateCourse I 0 = {sp1} UndergraduateStudent I 0 = Student I 0 = I 0 = T step 1: I 1 = I 0 and I1 expands I0 with UndergraduateStudent I 1 = {john} I 1 = T step 2: I 2 = I 1 and I1 expands I1 with Student I 2 = {john} I 2 = T stop: I 2 is a model of (T, A) RW 2014, Athens,

29 Chase for RL the standard model I A of a simple ABox A is I A a I A A I A P I A = ind(a) = a, for a ind(a) = {a A(a) A}, for concept name A = {(a, b) P (a, b) A}, for role name P chase rules: (forward chaining) (c) if a B I k, B A T but a / A I k, then add a to A I k+1 (r) if (a, b) R I k 1, R 1 R 2 T but (a, b) / R I k 2, then add (a, b) to R I k+1 2 (b) if a B I k and B T, then the process terminates the domains of the I k are finite and all coincide with ind(a), the process terminates after a finite number of steps if (b) is not applicable then the result is a model of (T, A), the canonical model C T,A of (T, A) otherwise, (T, A) is inconsistent RW 2014, Athens,

30 Canonical Model is Universal a homomorphism h from an interpretation I 1 to an interpretation I 2 is a map from I 1 to I 2 such that h(a I 1 ) = a I 2, for each individual name a h(u) A I 2, for any u A I 1 and any concept name A (h(u), h(v)) P I 2, for any (u, v) P I 1 and any role name P Lemma C T,A is universal C T,A can be homomorphically mapped into any other model of (T, A) I C T,A I RW 2014, Athens,

31 Complexity of Reasoning in RL Theorem Let (T, A) be a consistent RL KB. Then C T,A = (T, A). In addition, (T, A) = B A C T,A = B A (T, A) = R 1 R 2 C T,A = R 1 R 2 (T, A) = q( a) C T,A = q( a) Theorem The problems of KB consistency, concept and role subsumption and instance checking are P-complete in RL The problem of CQ entailment in RL is NP-complete Proof Construct the canonical model (in a polynomial number of steps). Check the condition in the canonical model. RW 2014, Athens,

32 OWL 2 EL SNOMED CT (Spackman 00, Baader, Brandt & Lutz 05) Pericardium Tissue containedin.heart Pericarditis Inflammation haslocation.pericardium Inflammation Disease actson.tissue Disease haslocation.containedin.heart HeartDisease NeedsTreatment EL TBox: C 1 C 2 and P 1 P 2 C ::= A P. P.C C 1 C 2 no inverses simple ABox: A(a) and P (a, b) RW 2014, Athens,

33 EL and Existential Rules Person parent.person x ( Person(x) y (parent(x, y) Person(y)) ) the standard translation can be extended to EL y ( γ 1 ( y) γ k ( y) x γ 0 ( x, y) ) existential rules chase rules: (Johnson & Klug 84) (c ) if d C I k and C A T, then add d to A I k+1 (r ) if (d, d ) P I k 1 and P 1 P 2 T, then we add (d, d ) to P I k+1 2 (e) if d C I k and C P.D T, where D is a concept name or, then take a fresh labelled null, d, and add d to D I k+1 and (d, d ) to P I k+1 NB: oblivious chase no check whether there is such an element RW 2014, Athens,

34 EL Chase: Example T : Person parent.person A: Person(john) step 0: I 0 = {john}, Person I 0 = {john}, parent I 0 = I 0 = I A step 1: I 1 = {john, d 1 }, Person I 1 = {john, d 1 }, parent I 1 = {(john, d 1 )} step 2: I 2 = {john, d 1, d 2 }, Person I 2 = {john, d 1, d 2 }, A john d 1 d 2 parent parent parent parent I 2 = {(john, d 1 ), (d 1, d 2 )} 1. this canonical model is infinite 2. there is no finite universal model 3. there are many universal models (each is infinite) RW 2014, Athens,

35 Normal Form in EL normal form of concept inclusions A 1 A 2 A, P.D A or A P.D P is a role name, A, A 1 and A 2 are concept names and D is either a concept name or Example: P.A B R. P.A introduce abbreviations: C is P.A and D is R.C result: C B D, P.A C, C P.A, D R.C, R.C D RW 2014, Athens,

36 Reasoning in EL do not construct the infinite chase, re-use the labelled nulls for each positive S.D, take a fixed witness w S.D (e-c) if d C I k and C P.D T, where D is a concept name or, then add w S.D to D I k+1 and (d, d ) to P I k+1 A john parent w parent.person parent the generating model G T,A of (T, A) NB: alternatively, take the canonical model and identify all labelled nulls introduced for the same S.D RW 2014, Athens,

37 Reasoning in EL (2) Theorem Let (T, A) be an EL KB. Then G T,A = (T, A). In addition, (T, A) = C 1 C 2 G T,A = C 1 C 2 (T, A) = P 1 P 2 G T,A = P 1 P 2 (T, A) = C(a) G T,A = C(a) (T, A) = P (a, b) G T,A = P (a, b) Theorem The problems of concept and role subsumption and instance checking are P-complete in EL NB: G T,A does not give correct answers to queries q = x parent(x, x) RW 2014, Athens,

38 OWL 2 QL Employee 1..N worksfor 0..N Project (Calvanese et al. 05) ISA Manager worksfor. Employee worksfor. Project Manager Employee Project worksfor. QL TBox: simple ABox: B C and R 1 R 2 B ::= A R. C ::= A R. R.C A(a) and P (a, b) no R.C on the left RW 2014, Athens,

39 Chase for QL normal form of concept inclusions A A, R A or A R.D R is a role, A and A are concept names and D is either a concept name or chase rules: (c ) if d B I k and B A T, then add d to A I k+1 (r ) if (d, d ) R I k 1 and R 1 R 2 T, then add (d, d ) to R I k+1 2 (e) if d B I k and B R.D T, where D is a concept name or, then take a fresh labelled null, d, and add d to D I k+1 and (d, d ) to R I k+1 RW 2014, Athens,

40 Unravelling of the Generating Structure for each R.D that occurs positively in T, take a witness w R.D generating relation T,A : a T,A w R.D if a ind(a), I A = B(a) and T = B R.D w S.B T,A w R.D if T = S R.D or T = B R.D T,A -path σ is any aw R1.D 1 w Rn.D n a ind(a) and, if n > 0 then a T,A w R1.D 1 and w Ri.D i T,A w Ri+1.D i+1, for i < n RW 2014, Athens,

41 Canonical Model: Example T : RA workson.project Project ismanagedby.prof workson involves ismanagedby involves A: RA(chris), workson(chris, dyn), Project(dyn), Lecturer(dave), workson(dave, dyn) generating relation w 1 = w workson.project and w 2 = w ismanagedby.prof chris T,A w 1, dyn T,A w 2, w 1 T,A w 2 T,A -paths chris, chris w 1, chris w 1 w 2, dyn, dyn w 2 and dave RW 2014, Athens,

42 Canonical Model: Example (2) T : RA workson.project Project ismanagedby.prof workson involves ismanagedby involves A: RA(chris), workson(chris, dyn), Project(dyn), Lecturer(dave), workson(dave, dyn) ismanagedby Prof chris w1 w 2 involves w 1 = w workson.project w 2 = w ismanagedby.prof Prof dyn w2 A Project chris w 1 workson involves chris involves RA workson ismanagedby involves dyn Project involves workson dave Lecturer RW 2014, Athens,

43 Comparing Profiles RL EL QL inverse roles P + + R.C on the right + + R.C on the left + + RW 2014, Athens,

44 Part 3 OBDA and Query Rewriting RW 2014, Athens,

45 Ontology-Based Data Access Aim: to achieve logical transparency in accessing data hide from the user where and how data is stored present only a conceptual view of the data query the data sources through the conceptual model using RDBMSs subclass Employee domain worksfor range ontology intensional level, conceptual structure of the domain, TBox Manager Project (GAV) mappings data sources extensional level, instances of conceptual elements, ABox RW 2014, Athens,

46 Databases: Data and the Closed World Assumption data is completely specified (closed world assumption) and is typically large what is specified is true, everything else is false data: Employee = {john, mary, nick} Manager = {john, nick} Project = {A, B} worksfor = {(john, A), (mary, B)} query: q(x) Employee(x) answer: {john, mary, nick} NB: not having nick in Employee would violate the integrity constraint x (Manager(x) Employee(x)) RW 2014, Athens,

47 Why do Databases Work? query answering problem (as a recognition problem): given a finite data D, a query q( x) and a tuple a, decide whether I D = q( a) I D makes the facts in D true (and only them) what is the complexity of CQ answering? naive algorithm: guess values for all existential variables and then evaluate the query in polynomial time in NP can it be done better? RW 2014, Athens,

48 Why Do Databases Work? (2) no, by reduction of the graph 3-colourability problem, which is NP-complete: given an undirected graph G = (V, E), decide whether it possible to colour it (using r, g, b) so that no edge has the same colour at both ends? D = {A(r, g), A(g, b), A(b, r), A(g, r), A(r, b), A(b, g)} q G = v 1,..., v n A(v i, v j ) (v i,v j ) E g A b A A r D = q G iff G is 3-colourable in fact, the query answering algorithm runs in O( D q ) data is large, query is short data complexity: only data D are counted as input (q is constant) (Vardi, 1982): query answering is in AC 0 for data complexity RW 2014, Athens,

49 Circuits and AC 0 a circuit is an acyclic graph of AND-, OR- and NOT-gates (with n inputs and a single output, sink) g 3 p 1 p 2 database instances D can be encoded on inputs (one input for each possible ground atom) FO-query is a circuit:, and are AND-, OR- and NOT-gates, respectively and are AND- and OR-gates with unbounded fan-in AC 0 = circuits of constant depth with AND- and OR-nodes of unbounded fan-in constant time by a polynomial number of processors (high degree of parallelism) the depth of this circuit does not depend on D Vardi s theorem NB: AC 0 is a proper subclass of LOGSPACE P (PARITY does not belong to AC 0 ) given a word w, decide whether its length is even RW 2014, Athens,

50 Query Rewriting Approach use off-the-shelf RDBMS (Calvanese et al. 05) conjunctive query q + FO query q TBox T ABox A ABox A AC 0 given a CQ q( x) over T, rewrite q( x) into an FO query q ( x) such that for all A and a, T, A = q( a) iff A = q ( a) FO-rewritability: only possible in DL with query answering in FO (=AC 0 ) for data complexity: OWL 2 QL RW 2014, Athens,

51 Can We Use R.A B in QL? reachability problem for directed graphs is NLogSpace-complete: given a directed graph G = (V, E) and s, t V, decide whether there is a directed path from s to t in G ABox: A G,t = { edge(v 1, v 2 ) (v 1, v 2 ) E } {ReachableFromTarget(t)} TBox: T = { edge.reachablefromtarget ReachableFromTarget } t CQ: q ReachableFromTarget(s) (T, A G,t ) = q iff there is a path from s to t in G T and q do not depend on G, s, t (T, A G,t ) = q? is NLogSpace-hard for data complexity q and T are not FO-rewritable RW 2014, Athens,

52 Practical Query Answering in OWL 2 QL systems QuOnto (Rome, 2005) REQUIEM (Oxford, 2009) Presto (Rome, 2010) IQAROS (Athens, 2011) Rapid (Athens-Oxford, 2011) Nyaya (Milan-Oxford, 2010) for TGDs Clipper (Vienna, 2012) for Horn-SHIQ (Montpellier, 2013) for TGDs Quest/ontop (Bolzano, 2011) not so smoothly: the size of implemented rewritings q is O ( ( q T ) q ) (can t say query is small or fixed any longer) RW 2014, Athens,

53 Does a Rewriting Have to be Exponential? TBox mother parent and father parent query grandparent(x, z) parent(x, y) parent(y, z) UCQ-rewritings (unions of CQs) are exponential in the worst case grandparent(x, z) parent(x, y) parent(y, z) grandparent(x, z) father(x, y) father(y, z) grandparent(x, z) mother(x, y) father(y, z)... PE-rewritings (positive existential queries select-project-join-union) grandparent(x, z) (parent(x, y) father(x, y) mother(x, y)) (parent(y, z) father(y, z) mother(y, z)) NDL-rewriting (non-recursive Datalog SQL with views) grandparent(x, z) ext-parent(x, y) ext-parent(y, z) ext-parent(x, y) parent(x, y) ext-parent(x, y) father(x, y) ext-parent(x, y) mother(x, y) + structure sharing FO-rewriting (first-order queries SQL) RW 2014, Athens,

54 Case 1: Flat QL TBoxes a TBox T is flat if it does not contain generating axioms B R.B RDF Schema q( x) and a flat T q ext ( x) by replacing A(u) P (u, v) A (u) v R(u, v) T =A A T = R A R(u, v) T =R P for any CQ q( x) and any flat OWL 2 QL TBox T, q ext ( x) is a PE-rewriting of q and T of size O( q T ) easy in theory, not so in practice RW 2014, Athens,

55 Who Works with Professors? TBox: workson involves ismanagedby involves in English: find those who work with professors query: q(x) workson(x, y) involves(y, z) Professor(z) workson(z, y) ismanagedby(y, z) involves(y, z) RW 2014, Athens,

56 Rewriting over H-complete ABoxes an ABox A is H-complete with respect to T if A(a) A whenever A (a) A and T = A A A(a) A whenever R(a, b) A and T = R A P (a, b) A whenever R(a, b) A and T = R P an FO-query q ( x) is an FO-rewriting of q( x) and T over H-complete ABoxes if, for any H-complete (w.r.t. T ) ABox A and any a, (T, A) = q( a) iff A = q ( a) (thus we ignore the axioms considered in the flat rewriting) RW 2014, Athens,

57 Case 2: Who Works with Professors (2)? T : RA workson.project Project ismanagedby.prof workson involves ismanagedby involves A: RA(chris), workson(chris, dyn), Project(dyn), Lecturer(dave), workson(dave, dyn) ismanagedby Prof chris w1 w 2 involves w 1 = w workson.project w 2 = w ismanagedby.prof Prof dyn w2 A Project chris w 1 workson involves chris involves RA workson ismanagedby involves dyn Project involves workson dave Lecturer RW 2014, Athens,

58 Case 2: Rewriting the Labelled Nulls Project RA(x) Professor ismanagedby involves h 1 Professor ismanagedby involves Project h 1 h 2 h 2 workson(x, y) Project(y) Professor Professor z involves RA(x) Professor(z) (x = z) h 3 y h 3 ismanagedby involves Project workson RA involves h 1 workson x q(x) h 3 workson involves RA Professor PE-rewriting (over H-complete ABoxes): (x and z are the roots of the tree witness) q (x) RA(x) ( workson(x, y) Project(y) ) ( RA(x) Professor(z) (x = z) ) ( workson(x, y) involves(y, z) Professor(z) ) RW 2014, Athens,

59 Tree-Witness Rewriting TBox T : A R, R T, B R, R S z R(z, y) (y = y ) T z R(z, x) R T R T T y y R B(y ) (y = y ) y A x x x z R(x, z) R P BR BR A(x) (x = x ) P P P P q tw ( x) = Θ independent set of tree witnesses x R(x, z) (x = x ) ( y S( z) S( z) q\q Θ t Θ tw t ) RW 2014, Athens,

60 Rewritings as Boolean functions hypergraph H: vertices = atoms of the query hyperedges = tree witnesses T (y, z) T (y, z) R(x, y) R(x, y ) R(x, y ) S(x, z ) S(x, z ) hypergraph function of H = (V, E): f H = X E X independent ( v V \V X p v e X p e ) lower bounds from circuit complexity: exponential non-recursive datalog (and positive existential) rewritings superpolynomial first-order rewritings (unless NP P/poly) RW 2014, Athens,

61 Short Rewritings in Theory if q t1 q t2 = or q t1 q t2 or q t2 q t1, for each pair t 1 and t }{{ 2, then } q tw ( x) = (S( z) t : S( z) q t tw t compatible S( z) q is a rewriting (over H-complete ABoxes) ) EL: add (recursive) datalog rules that complete ABox = polynomial datalog rewriting (tree witnesses in EL are compatible) QL: replace S( z) with S ( z) T =S S = polynomial positive existential rewriting provided that the number of tree witnesses is polynomial and they are compatible not the case in general! RW 2014, Athens,

62 Part 4 Practical OBDA with Ontop RW 2014, Athens,

63 OBDA system Ontop implemented at the Free University of Bozen-Bolzano (Mariano Rodríguez-Muro, Martin Rezk, Guohui Xiao) open-source available as a plugin for Protégé 4, SPARQL end-point, OWLAPI and Sesame libraries RW 2014, Athens,

64 OBDA with Databases rewriting unfolding CQ q + FO-query q + SQL TBox T GAV mapping M ABox A + data D ABox virtualisation Why SQL rewritings are large: (1) a large number of tree witnesses very few for real-world CQs/ontologies (2) large concept/role hierarchies in T (3) multiple definitions of the ontology terms in M many inclusions in T follow from Σ and M RW 2014, Athens,

65 Ontop Example IMDb (simplified): database title movie ID title production year 728 Django Unchained dependencies castinfo person ID movie ID person role n n m ( p, r castinfo(p, m, r) t, y title(m, t, y) ) (FK) ( m t 1 t 2 y title(m, t1, y) y title(m, t 2, y) (t 1 = t 2 ) ) (PK 1 ) ( m y 1 y 2 t title(m, t, y1 ) t title(m, t, y 2 ) (y 1 = y 2 ) ) (PK 2 ) Movie Ontology MO mo:movie mo:title, mo:movie mo:year, mo:movie mo:cast, mo:cast mo:person,... Mapping (created by the Ontop development team) mo:movie(m), mo:title(m, t), mo:year(m, y) title(m, t, y) (M 1 ) mo:cast(m, p), mo:person(p) castinfo(p, m, r) (M 2 ) RW 2014, Athens,

66 ABox A Ontop: T-mappings forward chaining + H-complete ABox A virtualisation + flat TBox T mapping M + T -mapping M T + virtualisation composition data D T mo:movie mo:title, mo:movie mo:cast, mo:movie mo:year, mo:cast mo:person M mo:movie(m), mo:title(m, t), mo:year(m, y) title(m, t, y) (M 1) mo:cast(m, p), mo:person(p) castinfo(p, m, r) (M 2 ) mo:movie(m) title(m, t, y) M T by (M 1 ) mo:movie(m) castinfo(p, m, r) by (M 2 ) + mo:cast mo:movie redundant by (FK) m ( p, r castinfo(p, m, r) t, y title(m, t, y) ) RW 2014, Athens,

67 Optimising T-mappings using foreign keys (inclusion dependencies) using disjunction T mo:actor mo:artist, mo:artist mo:person, mo:director mo:person, mo:editor mo:person,... M mo:actor(p) castinfo(p, m, r), (r = 1) (M 1 )... mo:editor(p) castinfo(p, m, r), (r = 6) (M 6 ) M T mo:person(p) castinfo(p, m, r), ((r = 1) (r = 6)) RW 2014, Athens,

68 Unfolding with Semantic Query Optimisation Query q(t, y) mo:movie(m), mo:title(m, t), mo:year(m, y), (y > 2010) Rewriting q (t, y) mo:movie(m), mo:title(m, t), mo:year(m, y), (y > 2010) M mo:movie(m) title(m, t, y) (M 1 ) mo:title(m, t) title(m, t, y) (M 2 ) mo:year(m, y) title(m, t, y) (M 3 ) Unfolding q (t, y) title(m, t 0, y 0 ), title(m, t, y 1 ), title(m, t 2, y), (y > 2010) primary keys m t 1 t 2 ( y title(m, t1, y) y title(m, t 2, y) (t 1 = t 2 ) ) (PK 1 ) m y 1 y 2 ( t title(m, t, y1 ) t title(m, t, y 2 ) (y 1 = y 2 ) ) (PK 2 ) q (t, y) title(m, t, y), (y > 2010) Semantic Query Optimisation RW 2014, Athens,

69 Practical OBDA with Ontop tw-rewriting ➊ unfolding CQ q + UCQ q tw + SQL ontology T composition ➋ + SQO ➌ dependencies Σ SQO ➍ mapping M T -mapping + virtual ABox + H-completion ABox virtualisation H-complete ABox A ➊ tree-witness rewriting q tw over H-complete ABoxes + data D ABox virtualisation (no concept/role hierarchies) ➋ T -mapping = system mapping M + T makes virtual ABoxes H-complete ➌ T -mapping is simplified using SQO and SQL features constructed and optimised for T and Σ only once ➍ unfolding uses SQO to produce small and efficient SQL queries RW 2014, Athens,

70 Recommended Reading (1) A. Poggi, D. Lembo, D. Calvanese, G. De Giacomo, M. Lenzerini and R. Rosati. Linking Data to Ontologies. J. Data Semantics 10: (2008) A. Artale, D. Calvanese, R. Kontchakov and M. Zakharyaschev: The DL-Lite Family and Relations. J. Artif. Intell. Res. (JAIR) 36:1 69 (2009) A. Calì, G. Gottlob and A. Pieris. Query Rewriting under Non-Guarded Rules. In Proc. AMW 2010 M. Vardi. The Complexity of Relational Query Languages (Extended Abstract). In Proc. STOC 1982: D. Maier, A. O. Mendelzon, and Y. Sagiv. Testing implications of data dependencies. ACM Trans. Database Syst., 4(4): , 1979 D. S. Johnson and A. C. Klug. Testing containment of conjunctive queries under functional and inclusion dependencies. J. Comput. Syst. Sci., 28(1): , 1984 A. Deutsch, A. Nash, and J. B. Remmel. The chase revisited. Proc. PODS 2008: M. Rodriguez-Muro and D. Calvanese. Semantic Index: Scalable Query Answering without Forward Chaining or Exponential Rewritings Posters of ISWC 2011 M. Rodriguez-Muro and D. Calvanese. Dependencies: Making Ontology Based Data Access Work. Proc. AMW 2011 G. Gottlob and T. Schwentick. Rewriting Ontological Queries into Small Nonrecursive Datalog Programs. Proc. KR 2012 RW 2014, Athens,

71 Recommended Reading (2) S. Kikot, R. Kontchakov, V. Podolskii and M. Zakharyaschev: Exponential Lower Bounds and Separation for Query Rewriting. Proc. ICALP (2) 2012: F. Baader, S. Brandt and C. Lutz. Pushing the EL envelope. Proc. IJCAI 2005 B. N. Grosof, I. Horrocks, R. Volz and S. Decker. Description logic programs: Combining logic programs with description logic. Proc. WWW 2003 A. Calì, G. Gottlob and A. Pieris. Advanced Processing for Ontological Queries. PVLDB 3(1): (2010) D. Calvanese, G. De Giacomo, D. Lembo, M. Lenzerini and R. Rosati. DL-Lite: Tractable Description Logics for Ontologies. Proc. AAAI 2005: A. Chortaras, D. Trivela and G. Stamou. Optimized query rewriting for OWL 2 QL. Proc. CADE 2011 T. Eiter, Ortiz, M. Šimkus, T.-K. Tran and G. Xiao. Query rewriting for Horn-SHIQ plus rules. Proc. AAAI 2012 M. König, M. Leclère, M.-L. Mugnier and M. Thomazo: A sound and complete backward chaining algorithm for existential rules. Proc. RR 2012 M. Rodríguez-Muro, R. Kontchakov and M. Zakharyaschev. Ontology-Based Data Access: Ontop of Databases. Proc. ISWC 2013 R. Kontchakov, M. Rezk, M. Rodríguez-Muro, G. Xiao and M. Zakharyaschev. Answering SPARQL Queries under the OWL 2 QL Entailment Regime with Databases. Proc. ISWC 2014 RW 2014, Athens,