Ontology-Based Data Access

Transcription

1 Ontology-Based Data Access Roman Kontchakov Dept. of Computer Science and Inf. Systems, Birkbeck, University of London acknowledgements: Alessandro Artale, Diego Calvanese, Carsten Lutz, Mariano Rodríguez-Muro, David Toman, Frank Wolter and Michael Zakharyaschev

2 Data Management: New Challenges Statoil (Norway) many databases, e.g., EPDS (Exploration and Production Data Store over 1500 tables historical exploration data (e.g., layers of rocks, porosity), production logs, maps, etc. business information such as license areas and companies direct data access by engineers (and geologists in particular) is often challenging SIGMOD 2015, Moscow,

3 Data Management: New Challenges Statoil (Norway) many databases, e.g., EPDS (Exploration and Production Data Store over 1500 tables historical exploration data (e.g., layers of rocks, porosity), production logs, maps, etc. business information such as license areas and companies direct data access by engineers (and geologists in particular) is often challenging Siemens Energy (Germany) power generation facilities (gas and steam turbines) 50 service centres linked to a common database each turbine 2000 sensors 150 tables 30 GB of data is generated daily (hundreds of terabytes in total) SIGMOD 2015, Moscow,

4 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 Director domain directed range ontology TBox EuropeanDirector Movie mappings r 2 (Director, Bio) r 1 (Title, Year, Director) data sources ABox since 1960, european directors since 1990 SIGMOD 2015, Moscow,

5 Issues in OBDA what is the right ontology language? there is a wide spectrum of languages that differ in expressive power and complexity of inference scalability to very large amounts of data is key SIGMOD 2015, Moscow,

6 Issues in OBDA what is the right ontology language? there is a wide spectrum of languages that differ in expressive power and complexity of inference scalability to very large amounts of data is key what is the query language? SIGMOD 2015, Moscow,

7 Issues in OBDA what is the right ontology language? there is a wide spectrum of languages that differ in expressive power and complexity of inference scalability to very large amounts of data is key what is the query language? how do we connect ontologies to data sources? multiple data sources and ontologies SIGMOD 2015, Moscow,

8 Issues in OBDA what is the right ontology language? there is a wide spectrum of languages that differ in expressive power and complexity of inference scalability to very large amounts of data is key what is the query language? how do we connect ontologies to data sources? multiple data sources and ontologies available tools? sound and complete reasoning practical scalability SIGMOD 2015, Moscow,

9 Part 1 Databases and Logic SIGMOD 2015, Moscow,

10 ID Databases: Specifying Schema an Entity-Relationship diagram ID Director directed Movie name ISA year title EuropeanDirector SIGMOD 2015, Moscow,

11 Databases: Specifying Schema ID an Entity-Relationship diagram ID Director directed Movie name ISA year title EuropeanDirector integrity constraints or dependencies (in the language of FO): d ( m directed(d, m) n Director(d, n)) m ( d directed(d, m) ty Movie(m, t, y)) dn (EuropeanDirector(d, n) Director(d, n)) (foreign keys, inclusion or tuple-generating dependencies, TGDs) SIGMOD 2015, Moscow,

12 Databases: Specifying Schema ID an Entity-Relationship diagram ID Director directed Movie name ISA year title EuropeanDirector integrity constraints or dependencies (in the language of FO): d ( m directed(d, m) n Director(d, n)) m ( d directed(d, m) ty Movie(m, t, y)) dn (EuropeanDirector(d, n) Director(d, n)) (foreign keys, inclusion or tuple-generating dependencies, TGDs) dn 1 n 2 (Director(d, n 1 ) Director(d, n 2 ) (n 1 = n 2 )) (keys, functional mt 1 t 2 y 1 y 2 (Movie(m, t 1, y 1 ) Movie(m, t 2, y 2 ) (t 1 = t 2 )) or equality-generating dependencies, EGDs) SIGMOD 2015, Moscow,

13 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 SIGMOD 2015, Moscow,

14 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: Director = { (0, peter ), (1, quentin ), (2, danny ) } EuropeanDirector = { (0, peter ), (2, danny ) } Movie = { (10, DC ), (11, TS ) } directed = { (0, 10), (2, 11) } query: q(n) = d Director(d, n) SIGMOD 2015, Moscow,

15 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: Director = { (0, peter ), (1, quentin ), (2, danny ) } EuropeanDirector = { (0, peter ), (2, danny ) } Movie = { (10, DC ), (11, TS ) } directed = { (0, 10), (2, 11) } query: q(n) = d Director(d, n) answer: { peter, quentin, danny } NB: not having (2, danny ) in Director would violate the integrity constraint dn (EuropeanDirector(d, n) Director(d, n)) SIGMOD 2015, Moscow,

16 Databases: Query Languages SQL domain-independent FO queries: database predicates + logical connectives,, + quantifiers, SIGMOD 2015, Moscow,

17 Databases: Query Languages SQL domain-independent FO queries: database predicates + logical connectives,, + quantifiers, data D = FO interpretation I D (CWA) a is an answer to q( x) iff I D = q( a) SIGMOD 2015, Moscow,

18 Databases: Query Languages SQL domain-independent FO queries: database predicates + logical connectives,, + quantifiers, data D = FO interpretation I D (CWA) a is an answer to q( x) iff I D = q( a) Select-Project-Join (SPJ) = conjunctive queries (CQs): database predicates + + database engines are optimised for CQs Example: SELECT M.title, D.name FROM Movie M, Directed MD, Director D WHERE M.id = MD.movieId AND MD.directorId = D.id AND M.Year = 1982 SIGMOD 2015, Moscow,

19 Databases: Query Languages SQL domain-independent FO queries: database predicates + logical connectives,, + quantifiers, data D = FO interpretation I D (CWA) a is an answer to q( x) iff I D = q( a) Select-Project-Join (SPJ) = conjunctive queries (CQs): database predicates + + database engines are optimised for CQs Example: SELECT M.title, D.name FROM Movie M, Directed MD, Director D WHERE M.id = MD.movieId AND MD.directorId = D.id AND M.Year = 1982 Datalog notation: q( x) }{{} head P 1 ( z 1 ),..., P k ( z k ) }{{} body where each z i is a vector, which may contain answer variables x and existentially quantified variables y (implicit) Example: q(t, n) Movie(m, t, 1982), directed(m, d), director(d, n) SIGMOD 2015, Moscow,

20 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? SIGMOD 2015, Moscow,

21 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? SIGMOD 2015, Moscow,

22 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? SIGMOD 2015, Moscow,

23 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? SIGMOD 2015, Moscow,

24 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? SIGMOD 2015, Moscow,

25 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 SIGMOD 2015, Moscow,

26 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 SIGMOD 2015, Moscow,

27 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 SIGMOD 2015, Moscow,

28 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 SIGMOD 2015, Moscow,

29 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 SIGMOD 2015, Moscow,

30 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) SIGMOD 2015, Moscow,

31 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 SIGMOD 2015, Moscow,

32 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 SIGMOD 2015, Moscow,

33 Part 2 Basics of Ontology Languages SIGMOD 2015, Moscow,

34 DLs and OWL: Syntax concepts (classes, sets of elements) 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 ) SIGMOD 2015, Moscow,

35 DLs and OWL: Syntax concepts (classes, sets of elements) 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, binary relations) R ::= P }{{} i P i role name SIGMOD 2015, Moscow,

36 DLs and OWL: Syntax concepts (classes, sets of elements) 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, binary relations) 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) SIGMOD 2015, Moscow,

37 DLs and OWL: Syntax concepts (classes, sets of elements) 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, binary relations) 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) SIGMOD 2015, Moscow,

38 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 subsets A I i I role names P i binary relations P I i I I SIGMOD 2015, Moscow,

39 DL Semantics (2) (P ) I = {(v, u) (u, v) P I } j takescourse takescourse s SIGMOD 2015, Moscow,

40 DL Semantics (2) (P ) I = {(v, u) (u, v) P I } j takescourse takescourse s I = I and I = SIGMOD 2015, Moscow,

41 DL Semantics (2) (P ) I = {(v, u) (u, v) P I } j takescourse takescourse s I = I and I = ( C) I = I \ C I j s l SIGMOD 2015, Moscow,

42 DL Semantics (2) (P ) I = {(v, u) (u, v) P I } j takescourse takescourse s I = I and I = ( C) I = I \ C I j s l (C 1 C 2 ) I = C I 1 CI 2 j k b (C 1 C 2 ) I = C I 1 CI 2 SIGMOD 2015, Moscow,

43 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 or y (R(x, y) C(y)) SIGMOD 2015, Moscow,

44 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 or y (R(x, y) C(y)) ( R.C) I = { u v C I, for all v with (u, v) R I } R.C = R. C SIGMOD 2015, Moscow,

45 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 or y (R(x, y) C(y)) ( 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 R C or y (R(x, y) C(y)) SIGMOD 2015, Moscow,

46 DL Semantics (4) I = C 1 C 2 C I 1 CI 2 j k b C 1 C 2 SIGMOD 2015, Moscow,

47 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 SIGMOD 2015, Moscow,

48 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 SIGMOD 2015, Moscow,

49 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 SIGMOD 2015, Moscow,

50 Open World Assumption T = { GraduateStudent Student GraduateStudent takescourse.graduatecourse } A = { GraduateStudent(john) } SIGMOD 2015, Moscow,

51 Open World Assumption T = { GraduateStudent Student GraduateStudent takescourse.graduatecourse } A = { GraduateStudent(john) } I 1 GraduateStudent Student GraduateCourse j s takescourse john john I1 = j GraduateStudent I1 = {j} Student I1 = {j} GraduateCourse I1 = {s} takescourse I1 = {(j, s)} is a model of (T, A) SIGMOD 2015, Moscow,

52 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 takescourse I 2 john I1 = j GraduateStudent I1 = {j} Student I1 = {j} GraduateCourse I1 = {s} takescourse I1 = {(j, s)} is a model of (T, A) john I2 = a GraduateStudent I2 = {a} Student I2 = {a} GraduateCourse I2 = {a} takescourse I2 = {(a, a)} is a model of (T, A) SIGMOD 2015, Moscow,

53 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 of (T, A) john I2 = a GraduateStudent I2 = {a} Student I2 = {a} GraduateCourse I2 = {a} takescourse I2 = {(a, a)} is a model of (T, A) john I3 = j GraduateStudent I3 = {j} Student I3 = {j} GraduateCourse I3 = takescourse I3 = is not a model of (T, A) SIGMOD 2015, Moscow,

54 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 SIGMOD 2015, Moscow,

55 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) SIGMOD 2015, Moscow,

56 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. SIGMOD 2015, Moscow,

57 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) SIGMOD 2015, Moscow,

58 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. SIGMOD 2015, Moscow,

59 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 1 = T I 1 = FirstYearStudent UndergraduateStudent SIGMOD 2015, Moscow,

60 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 SIGMOD 2015, Moscow,

61 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 SIGMOD 2015, Moscow,

62 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 (T, A) = q( a) if, for any model I of (T, A), the sentence q( a) is true in I I = q( a) SIGMOD 2015, Moscow,

63 Andrea s Example (Schaerf, 1993) T : Male Female, Male Female A: friend(john, susan), friend(john, andrea), Female(susan) loves(susan, andrea), loves(andrea, bill), Male(bill) q = y, z ( friend(john, y) Female(y) loves(y, z) Male(z) ) SIGMOD 2015, Moscow,

64 Andrea s Example (Schaerf, 1993) T : Male Female, Male Female A: friend(john, susan), friend(john, andrea), Female(susan) loves(susan, andrea), loves(andrea, bill), Male(bill) q = y, z ( friend(john, y) Female(y) loves(y, z) Male(z) ) john loves andrea susan Female Female loves friend bill Male friend A SIGMOD 2015, Moscow,

65 Andrea s Example (Schaerf, 1993) T : Male Female, Male Female A: friend(john, susan), friend(john, andrea), Female(susan) loves(susan, andrea), loves(andrea, bill), Male(bill) q = y, z ( friend(john, 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 SIGMOD 2015, Moscow,

66 Andrea s Example (Schaerf, 1993) T : Male Female, Male Female A: friend(john, susan), friend(john, andrea), Female(susan) loves(susan, andrea), loves(andrea, bill), Male(bill) q = y, z ( friend(john, 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 checking whether john is an instance of friend.(female loves.male) SIGMOD 2015, Moscow,

67 DL Zoo ALCHI AL attributive language C complement C (ALC is multi-modal K m ) I role inverses P H role inclusions R 1 R 2 SIGMOD 2015, Moscow,

68 DL Zoo ALCHI AL attributive language C complement C (ALC is multi-modal K m ) 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 SHOIN OWL 1.0 SIGMOD 2015, Moscow,

69 DL Zoo ALCHI AL attributive language C complement C (ALC is multi-modal K m ) 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. SHOIN OWL 1.0 SHIF OWL Lite SIGMOD 2015, Moscow,

70 DL Zoo ALCHI AL attributive language C complement C (ALC is multi-modal K m ) 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 SIGMOD 2015, Moscow,

71 Complexity of Reasoning The satisfiability problem is ExpTime-complete for ALCHI KBs and N2ExpTime-complete for SROIQ KBs SIGMOD 2015, Moscow,

72 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 SIGMOD 2015, Moscow,

73 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 SIGMOD 2015, Moscow,

74 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: SIGMOD 2015, Moscow,

75 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 SIGMOD 2015, Moscow,

76 Part 3 Conjunctive Query Rewriting SIGMOD 2015, Moscow,

77 Query Rewriting Approach (Calvanese et al. 2008): use off-the-shelf RDBMS conjunctive query q TBox T ABox A + union of conjunctive queries q ABox A AC 0 SIGMOD 2015, Moscow,

78 Query Rewriting Approach (Calvanese et al. 2008): use off-the-shelf RDBMS conjunctive query q TBox T ABox A + union of conjunctive queries q 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) SIGMOD 2015, Moscow,

79 Query Rewriting Approach (Calvanese et al. 2008): use off-the-shelf RDBMS conjunctive query q TBox T ABox A + union of conjunctive queries q 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 SIGMOD 2015, Moscow,

80 W3C Standard OWL 2 QL OWL 2 QL is a profile of OWL 2 designed with the aim of OBDA (and based on the DL-Lite family of DLs) roles R ::= P i P i basic concepts B ::= A i R concepts C ::= B R.B a TBox T is a finite set of axioms of the form ( R is an abbreviation for R. ) B C, R 1 R 2, B 1 B 2, R 1 R 2 (plus reflexivity/irreflexivity assertions for roles) an ABox A s a finite set of atoms the form A k (a i ) and P k (a i, a j ) (plus inequality constraints a i a j for i j) SIGMOD 2015, Moscow,

81 W3C Standard OWL 2 QL OWL 2 QL is a profile of OWL 2 designed with the aim of OBDA (and based on the DL-Lite family of DLs) roles R ::= P i P i basic concepts B ::= A i R concepts C ::= B R.B a TBox T is a finite set of axioms of the form ( R is an abbreviation for R. ) B C, R 1 R 2, B 1 B 2, R 1 R 2 (plus reflexivity/irreflexivity assertions for roles) an ABox A s a finite set of atoms the form A k (a i ) and P k (a i, a j ) (plus inequality constraints a i a j for i j) NB. axioms B R.B are syntactic sugar SIGMOD 2015, Moscow,

82 W3C Standard OWL 2 QL OWL 2 QL is a profile of OWL 2 designed with the aim of OBDA (and based on the DL-Lite family of DLs) roles R ::= P i P i basic concepts B ::= A i R concepts C ::= B R.B a TBox T is a finite set of axioms of the form ( R is an abbreviation for R. ) B C, R 1 R 2, B 1 B 2, R 1 R 2 (plus reflexivity/irreflexivity assertions for roles) an ABox A s a finite set of atoms the form A k (a i ) and P k (a i, a j ) (plus inequality constraints a i a j for i j) NB. axioms B R.B are syntactic sugar B R R.B, R B B, R B R SIGMOD 2015, Moscow,

83 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 } SIGMOD 2015, Moscow,

84 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 SIGMOD 2015, Moscow,

85 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 SIGMOD 2015, Moscow,

86 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 SIGMOD 2015, Moscow,

87 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 SIGMOD 2015, Moscow,

88 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 SIGMOD 2015, Moscow,

89 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 SIGMOD 2015, Moscow,

90 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 SIGMOD 2015, Moscow,

91 Can We Use A B C in QL? graph 3-colouring problem is NP-complete: given a graph G = (V, E), decide whether its vertices can be painted in one of three colours so that no adjacent vertices have the same colour represent G as the ABox A G = { R(v 1, v 2 ) {v 1, v 2 } E } 3-colouring is encoded by the TBox T with the axioms C 1 C 2 C 3, C i C j, C i R.C i B, 1 i 3 SIGMOD 2015, Moscow,

92 Can We Use A B C in QL? graph 3-colouring problem is NP-complete: given a graph G = (V, E), decide whether its vertices can be painted in one of three colours so that no adjacent vertices have the same colour represent G as the ABox A G = { R(v 1, v 2 ) {v 1, v 2 } E } 3-colouring is encoded by the TBox T with the axioms C 1 C 2 C 3, C i C j, C i R.C i B, 1 i 3 consider CQ q B(y) (T, A G ) = q iff G is 3-colourable T and q do not depend on G (T, A G ) = q? is conp-hard for data complexity q and T are not FO-rewritable SIGMOD 2015, Moscow,

93 OWL 2 QL as TGDs (aka Datalog ± aka existential rules) concept inclusion tuple-generating dependency PhDStudent Student x ( PhDStudent(x) Student(x) ) Student HasTutor x ( Student(x) y HasTutor(x, y) ) SIGMOD 2015, Moscow,

94 OWL 2 QL as TGDs (aka Datalog ± aka existential rules) concept inclusion tuple-generating dependency PhDStudent Student x ( PhDStudent(x) Student(x) ) Student HasTutor x ( Student(x) y HasTutor(x, y) ) TGDs: x y ( ϕ( x, y) z ψ( x, z) ) ϕ and ψ are conjunctions of predicate atoms SIGMOD 2015, Moscow,

95 OWL 2 QL as TGDs (aka Datalog ± aka existential rules) concept inclusion tuple-generating dependency PhDStudent Student x ( PhDStudent(x) Student(x) ) Student HasTutor x ( Student(x) y HasTutor(x, y) ) TGDs: x y ( ϕ( x, y) z ψ( x, z) ) ϕ and ψ are conjunctions of predicate atoms linear TGDs: ϕ and ψ are atoms (all OWL 2 QL axioms are linear) Calì, Gottlob & Pieris, 2010: sets of sticky TGDs are FO-rewritable in particular, linear TGDs SIGMOD 2015, Moscow,

96 OWL 2 QL as TGDs (aka Datalog ± aka existential rules) concept inclusion tuple-generating dependency PhDStudent Student x ( PhDStudent(x) Student(x) ) Student HasTutor x ( Student(x) y HasTutor(x, y) ) TGDs: x y ( ϕ( x, y) z ψ( x, z) ) ϕ and ψ are conjunctions of predicate atoms linear TGDs: ϕ and ψ are atoms (all OWL 2 QL axioms are linear) Calì, Gottlob & Pieris, 2010: sets of sticky TGDs are FO-rewritable in particular, linear TGDs NB: these TGDs are used with the open world assumption (for enriching data) TGDs in DBs are used with the closed world assumption (integrity constraints) SIGMOD 2015, Moscow,

97 Practical Query Answering in OWL 2 QL systems QuOnto (Rome, 2005) REQUIEM (Oxford, 2009) / Stardog (Washington, DC, 2011) Presto (Rome, 2010) IQAROS (Athens, 2011) Rapid (Athens-Oxford, 2011) Nyaya (Milan-Oxford, 2010) for TGDs Clipper (Vienna, 2012) for Horn-SHIQ kyrie (Madrid, 2013) Pure (Montpellier, 2013) for TGDs Quest/ontop (Bolzano, 2011) SIGMOD 2015, Moscow,

98 Practical Query Answering in OWL 2 QL systems QuOnto (Rome, 2005) REQUIEM (Oxford, 2009) / Stardog (Washington, DC, 2011) Presto (Rome, 2010) IQAROS (Athens, 2011) Rapid (Athens-Oxford, 2011) Nyaya (Milan-Oxford, 2010) for TGDs Clipper (Vienna, 2012) for Horn-SHIQ kyrie (Madrid, 2013) Pure (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) SIGMOD 2015, Moscow,

99 Does a Rewriting Have to be Exponential? TBox mother parent and father parent query grandparent(x, z) parent(x, y) parent(y, z) SIGMOD 2015, Moscow,

100 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)... SIGMOD 2015, Moscow,

101 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)) SIGMOD 2015, Moscow,

102 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 SIGMOD 2015, Moscow,

103 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) SIGMOD 2015, Moscow,

104 Case 1: Flat QL TBoxes a TBox T is flat if it does not contain generating axioms B R.B RDF Schema SIGMOD 2015, Moscow,

105 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 ) SIGMOD 2015, Moscow,

106 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 SIGMOD 2015, Moscow,

107 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) SIGMOD 2015, Moscow,

108 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) SIGMOD 2015, Moscow,

109 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 SIGMOD 2015, Moscow,

110 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) SIGMOD 2015, Moscow,

111 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 SIGMOD 2015, Moscow,

112 Case 2: Rewriting the Labelled Nulls RA(x) Professor h 1 Professor z Project ismanagedby involves h 1 involves y workson RA involves h 1 workson x q(x) SIGMOD 2015, Moscow,

113 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 z involves y workson RA involves h 1 workson x q(x) SIGMOD 2015, Moscow,

114 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 (x and z are the roots of the tree witness) SIGMOD 2015, Moscow,

115 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) ) SIGMOD 2015, Moscow,

116 Tree-Witness Rewriting TBox T : A R, R T, B R, R S R T BR A P SIGMOD 2015, Moscow,

117 Tree-Witness Rewriting TBox T : A R, R T, B R, R S BR T R R T T y y R P R y A x x x P P SIGMOD 2015, Moscow,

118 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 y A x R P x x z R(x, z) P BR P P x R(x, z) (x = x ) SIGMOD 2015, Moscow,

119 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 x R(x, z) (x = x ) SIGMOD 2015, Moscow,

120 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 ) (Kikot, K & Zakharyaschev, 2012) SIGMOD 2015, Moscow,

121 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 ) SIGMOD 2015, Moscow,

122 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 ) SIGMOD 2015, Moscow,

123 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 ) (Kikot, K, Podolskii & Zakharyaschev, 2012) lower bounds from circuit complexity exponential non-recursive datalog (and positive existential) rewritings superpolynomial first-order rewritings (unless NP P/poly) SIGMOD 2015, Moscow,

124 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) ) SIGMOD 2015, Moscow,

125 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) 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! SIGMOD 2015, Moscow,

126 Part 4 Practical OBDA with Ontop SIGMOD 2015, Moscow,

127 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 & 5, SPARQL end-point, OWL API and Sesame libraries SIGMOD 2015, Moscow,

128 OBDA with Databases rewriting unfolding SPARQL query q + FO-query q + SQL TBox T R2RML mappings M virtual ABox A + data D ABox virtualisation SIGMOD 2015, Moscow,

129 OBDA with Databases rewriting unfolding SPARQL query q + FO-query q + SQL TBox T R2RML mappings M virtual ABox A + data D ABox virtualisation Why SQL rewritings are large: (1) a large number of tree witnesses (2) large concept/role hierarchies in OWL 2 QL ontology T (3) multiple definitions of the ontology terms in R2RML mappings M SIGMOD 2015, Moscow,

130 OBDA with Databases rewriting unfolding SPARQL query q + FO-query q + SQL TBox T R2RML mappings M virtual ABox A + data D ABox virtualisation Why SQL rewritings are large: (1) a large number of tree witnesses (2) large concept/role hierarchies in OWL 2 QL ontology T very few for real-world CQs/ontologies (3) multiple definitions of the ontology terms in R2RML mappings M SIGMOD 2015, Moscow,

131 OBDA with Databases rewriting unfolding SPARQL query q + FO-query q + SQL TBox T R2RML mappings M virtual ABox A + data D ABox virtualisation Why SQL rewritings are large: (1) a large number of tree witnesses (2) large concept/role hierarchies in OWL 2 QL ontology T very few for real-world CQs/ontologies (3) multiple definitions of the ontology terms in R2RML mappings M many inclusions in T follow from Σ and M SIGMOD 2015, Moscow,

132 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 ) SIGMOD 2015, Moscow,

133 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,... SIGMOD 2015, Moscow,

134 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,... Mappings (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 ) SIGMOD 2015, Moscow,

135 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 SIGMOD 2015, Moscow,

136 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 SIGMOD 2015, Moscow,

137 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 ) SIGMOD 2015, Moscow,

138 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 SIGMOD 2015, Moscow,

139 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) ) SIGMOD 2015, Moscow,

140 Optimising T-mappings using foreign keys (inclusion dependencies) SIGMOD 2015, Moscow,