Querying Data through Ontologies

Transcription

1 Querying Data through Ontologies Instructor: Sebastian Link Thanks to Serge Abiteboul, Ioana Manolescu, Philippe Rigaux, Marie-Christine Rousset and Pierre Senellart Web Data Management and Distribution March 26, 2018 Sebastian Link Querying Data through Ontologies March 26, / 64

2 Introduction Outline 1 Introduction The Semantic Web Ontologies and Reasoning Illustration 2 Three ontology languages for the Web 3 Reasoning in Description Logics 4 Querying Data through Ontologies 5 Conclusion Sebastian Link Querying Data through Ontologies March 26, / 64

3 Introduction The Semantic Web The Semantic Web A Web in which the semantics of resources is described annotations give information about a page, explain an expression in a page, etc. Sebastian Link Querying Data through Ontologies March 26, / 64

4 Introduction The Semantic Web The Semantic Web A Web in which the semantics of resources is described annotations give information about a page, explain an expression in a page, etc. A resource is anything that can be referred to by a URI a web page, identified by a URL a fragment of an XML document, identified by an element node of the document, a web service, a thing, an object, a concept, a property, etc. Sebastian Link Querying Data through Ontologies March 26, / 64

5 Introduction The Semantic Web The Semantic Web A Web in which the semantics of resources is described annotations give information about a page, explain an expression in a page, etc. A resource is anything that can be referred to by a URI a web page, identified by a URL a fragment of an XML document, identified by an element node of the document, a web service, a thing, an object, a concept, a property, etc. Semantic annotations are logical assertions that associate resources with some terms in pre-defined ontologies Sebastian Link Querying Data through Ontologies March 26, / 64

6 Introduction Ontologies and Reasoning Ontologies Formal descriptions providing human users with a shared understanding of some given domain A controlled vocabulary Sebastian Link Querying Data through Ontologies March 26, / 64

7 Introduction Ontologies and Reasoning Ontologies Formal descriptions providing human users with a shared understanding of some given domain A controlled vocabulary Formally defined so that it can also be processed by machines Sebastian Link Querying Data through Ontologies March 26, / 64

8 Introduction Ontologies and Reasoning Ontologies Formal descriptions providing human users with a shared understanding of some given domain A controlled vocabulary Formally defined so that it can also be processed by machines Logical semantics empowers machines to reason Sebastian Link Querying Data through Ontologies March 26, / 64

9 Introduction Ontologies and Reasoning Ontologies Formal descriptions providing human users with a shared understanding of some given domain A controlled vocabulary Formally defined so that it can also be processed by machines Logical semantics empowers machines to reason Reasoning is the key for various important tasks in Web data management, in particular to answer queries (over possibly distributed data) Sebastian Link Querying Data through Ontologies March 26, / 64

10 Introduction Ontologies and Reasoning Ontologies Formal descriptions providing human users with a shared understanding of some given domain A controlled vocabulary Formally defined so that it can also be processed by machines Logical semantics empowers machines to reason Reasoning is the key for various important tasks in Web data management, in particular to answer queries (over possibly distributed data) to link objects from different data sources enabling their integration Sebastian Link Querying Data through Ontologies March 26, / 64

11 Introduction Ontologies and Reasoning Ontologies Formal descriptions providing human users with a shared understanding of some given domain A controlled vocabulary Formally defined so that it can also be processed by machines Logical semantics empowers machines to reason Reasoning is the key for various important tasks in Web data management, in particular to answer queries (over possibly distributed data) to link objects from different data sources enabling their integration to detect inconsistencies or redundancies Sebastian Link Querying Data through Ontologies March 26, / 64

12 Introduction Ontologies and Reasoning Ontologies Formal descriptions providing human users with a shared understanding of some given domain A controlled vocabulary Formally defined so that it can also be processed by machines Logical semantics empowers machines to reason Reasoning is the key for various important tasks in Web data management, in particular to answer queries (over possibly distributed data) to link objects from different data sources enabling their integration to detect inconsistencies or redundancies to refine queries with too many answers, or to relax queries with no answer Sebastian Link Querying Data through Ontologies March 26, / 64

13 Introduction Illustration Classes and class hierarchies Backbone of an ontology Sebastian Link Querying Data through Ontologies March 26, / 64

14 Introduction Illustration Classes and class hierarchies Backbone of an ontology AcademicStaff is a Class (A class will be interpreted as a set of objects) Sebastian Link Querying Data through Ontologies March 26, / 64

15 Introduction Illustration Classes and class hierarchies Backbone of an ontology AcademicStaff is a Class (A class will be interpreted as a set of objects) AcademicStaff isa Staff (isa is interpreted as set inclusion) Sebastian Link Querying Data through Ontologies March 26, / 64

16 Introduction Illustration Classes and class hierarchies Backbone of an ontology AcademicStaff is a Class (A class will be interpreted as a set of objects) AcademicStaff isa Staff (isa is interpreted as set inclusion) Faculty Staff Department Student Course AdministrativeStaff AcademicStaff CSDept MathsDept PhysicsDept PhDStudent MasterStudent UndergraduateStudent CSCourse Logic MathCourse Professor Researcher Lecturer Java AI DB Algebra Probabilities Sebastian Link Querying Data through Ontologies March 26, / 64

17 Introduction Illustration Relations Declaration of relations with their signature (Relations will be interpreted as binary relations between objects) Sebastian Link Querying Data through Ontologies March 26, / 64

18 Introduction Illustration Relations Declaration of relations with their signature (Relations will be interpreted as binary relations between objects) TeachesIn(AcademicStaff, Course) if one states that X TeachesIn Y, then X belongs to AcademicStaff and Y to Course, Sebastian Link Querying Data through Ontologies March 26, / 64

19 Introduction Illustration Relations Declaration of relations with their signature (Relations will be interpreted as binary relations between objects) TeachesIn(AcademicStaff, Course) if one states that X TeachesIn Y, then X belongs to AcademicStaff and Y to Course, TeachesTo(AcademicStaff, Student) Sebastian Link Querying Data through Ontologies March 26, / 64

20 Introduction Illustration Relations Declaration of relations with their signature (Relations will be interpreted as binary relations between objects) TeachesIn(AcademicStaff, Course) if one states that X TeachesIn Y, then X belongs to AcademicStaff and Y to Course, TeachesTo(AcademicStaff, Student) Leads(Staff, Department) Sebastian Link Querying Data through Ontologies March 26, / 64

21 Introduction Illustration Instances Classes have instances Dupond is an instance of the class Professor it corresponds to the fact: Professor(Dupond) Sebastian Link Querying Data through Ontologies March 26, / 64

22 Introduction Illustration Instances Classes have instances Dupond is an instance of the class Professor it corresponds to the fact: Professor(Dupond) Relations also have instances (Dupond,CS101) is an instance of the relation TeachesIn it corresponds to the fact: TeachesIn(Dupond,CS101) Sebastian Link Querying Data through Ontologies March 26, / 64

23 Introduction Illustration Instances Classes have instances Dupond is an instance of the class Professor it corresponds to the fact: Professor(Dupond) Relations also have instances (Dupond,CS101) is an instance of the relation TeachesIn it corresponds to the fact: TeachesIn(Dupond,CS101) The instance statements can be seen as (and stored in) a database Sebastian Link Querying Data through Ontologies March 26, / 64

24 Introduction Illustration Ontology = schema + instance Schema The set of class and relation names The signatures of relations and also constraints Constraints are used for two purposes checking data consistency (like dependencies in databases) inferring new facts Sebastian Link Querying Data through Ontologies March 26, / 64

25 Introduction Illustration Ontology = schema + instance Schema The set of class and relation names The signatures of relations and also constraints Constraints are used for two purposes checking data consistency (like dependencies in databases) inferring new facts Instance The set of facts The set of base facts together with the inferred facts should satisfy the constraints Sebastian Link Querying Data through Ontologies March 26, / 64

26 Introduction Illustration Ontology = schema + instance Schema The set of class and relation names The signatures of relations and also constraints Constraints are used for two purposes checking data consistency (like dependencies in databases) inferring new facts Instance The set of facts The set of base facts together with the inferred facts should satisfy the constraints Ontology (i.e., Knowledge Base) = Schema + Instance Sebastian Link Querying Data through Ontologies March 26, / 64

27 Three ontology languages for the Web Outline 1 Introduction 2 Three ontology languages for the Web 3 Reasoning in Description Logics 4 Querying Data through Ontologies 5 Conclusion Sebastian Link Querying Data through Ontologies March 26, / 64

28 Three ontology languages for the Web Three ontology languages for the Web RDF: a very simple ontology language Sebastian Link Querying Data through Ontologies March 26, / 64

29 Three ontology languages for the Web Three ontology languages for the Web RDF: a very simple ontology language RDFS: Schema for RDF Can be used to define more expressive ontologies Sebastian Link Querying Data through Ontologies March 26, / 64

30 Three ontology languages for the Web Three ontology languages for the Web RDF: a very simple ontology language RDFS: Schema for RDF Can be used to define more expressive ontologies OWL: a much more expressive ontology language Sebastian Link Querying Data through Ontologies March 26, / 64

31 Three ontology languages for the Web Three ontology languages for the Web RDF: a very simple ontology language RDFS: Schema for RDF Can be used to define more expressive ontologies OWL: a much more expressive ontology language We present them in brief We will further introduce a family of ontology languages: Description logics Sebastian Link Querying Data through Ontologies March 26, / 64

32 Three ontology languages for the Web RDF: Resource Description Framework RDF facts are triplets :Dupond :Leads :InfoDept :Dupond :TeachesIn :UE111 :Dupond :TeachesTo :Pierre :Pierre :EnrolledIn :InfoDept :Pierre :RegisteredTo :UE111 :UE111 :OfferedBy :InfoDept Sebastian Link Querying Data through Ontologies March 26, / 64

33 Three ontology languages for the Web RDF: Resource Description Framework RDF facts are triplets :Dupond :Leads :InfoDept :Dupond :TeachesIn :UE111 :Dupond :TeachesTo :Pierre :Pierre :EnrolledIn :InfoDept :Pierre :RegisteredTo :UE111 :UE111 :OfferedBy :InfoDept Linked open data: publish open data sets on the Web By September 2011, more than 31 billions RDF triplets Sebastian Link Querying Data through Ontologies March 26, / 64

34 Three ontology languages for the Web Linked Open Data Cloud Sebastian Link Querying Data through Ontologies March 26, / 64

35 Three ontology languages for the Web RDF graph A set of RDF facts defines a set of relations between objects an RDF graph where the nodes are objects: :InfoDept :Leads :EnrolledIn :Dupond :TeachesTo :Pierre :OfferedBy :RegisteredTo :TeachesIn UE111 Sebastian Link Querying Data through Ontologies March 26, / 64

36 Three ontology languages for the Web RDF semantics A triplet s P o is interpreted in first-order logic (FOL) as a fact P(s, o) Sebastian Link Querying Data through Ontologies March 26, / 64

37 Three ontology languages for the Web RDF semantics A triplet s P o is interpreted in first-order logic (FOL) as a fact P(s, o) Example: Leads(Dupond, InfoDept) TeachesIn(Dupond, UE111) TeachesTo(Dupond,Pierre) EnrolledIn(Pierre, InfoDept) RegisteredTo(Pierre, UE111) OfferedBy(UE111, InfoDept) Sebastian Link Querying Data through Ontologies March 26, / 64

38 RDFS: RDF Schema Three ontology languages for the Web Not detailed here: the schema in RDF is super simplistic Sebastian Link Querying Data through Ontologies March 26, / 64

39 RDFS: RDF Schema Three ontology languages for the Web Not detailed here: the schema in RDF is super simplistic An RDF Schema defines the schema of a more expressive ontology Sebastian Link Querying Data through Ontologies March 26, / 64

40 Three ontology languages for the Web RDF Schema Do net get confused: RDFS can use RDF as syntax, i.e., RDFS statements can be themselves expressed as RDF triplets using some specific properties and objects used as RDFS keywords with a particular meaning. Sebastian Link Querying Data through Ontologies March 26, / 64

41 Three ontology languages for the Web RDF Schema Do net get confused: RDFS can use RDF as syntax, i.e., RDFS statements can be themselves expressed as RDF triplets using some specific properties and objects used as RDFS keywords with a particular meaning. Declaration of classes and subclass relationships Staff rdf:type rdfs:class Java rdfs:subclassof CSCourse Sebastian Link Querying Data through Ontologies March 26, / 64

42 Three ontology languages for the Web RDF Schema Do net get confused: RDFS can use RDF as syntax, i.e., RDFS statements can be themselves expressed as RDF triplets using some specific properties and objects used as RDFS keywords with a particular meaning. Declaration of classes and subclass relationships Staff rdf:type rdfs:class Java rdfs:subclassof CSCourse Declaration of instances (beyond the pure schema) Dupond rdf:type AcademicStaff Sebastian Link Querying Data through Ontologies March 26, / 64

43 RDF Schema - continued Three ontology languages for the Web Declaration of relations (properties in RDFS terminology) RegisteredTo rdf:type rdfs:property Sebastian Link Querying Data through Ontologies March 26, / 64

44 RDF Schema - continued Three ontology languages for the Web Declaration of relations (properties in RDFS terminology) RegisteredTo rdf:type rdfs:property Declaration of subproperty relationships LateRegisteredTo rdfs:subpropertyof RegisteredTo Sebastian Link Querying Data through Ontologies March 26, / 64

45 RDF Schema - continued Three ontology languages for the Web Declaration of relations (properties in RDFS terminology) RegisteredTo rdf:type rdfs:property Declaration of subproperty relationships LateRegisteredTo rdfs:subpropertyof RegisteredTo Declaration of domain and range restrictions for predicates TeachesIn rdfs:domain AcademicStaff TeachesIn rdfs:range Course TeachesIn( AcademicStaff, Course) Sebastian Link Querying Data through Ontologies March 26, / 64

46 RDFS logical semantics Three ontology languages for the Web RDF and RDFS statements FOL translation DL notation i rdf:type C C(i) i : C or C(i) Ignore for now DL column This is just a notation We will come back to it to discuss Description logics

47 RDFS logical semantics Three ontology languages for the Web RDF and RDFS statements FOL translation DL notation i rdf:type C C(i) i : C or C(i) i P j P(i,j) i P j or P(i,j) Ignore for now DL column This is just a notation We will come back to it to discuss Description logics

48 RDFS logical semantics Three ontology languages for the Web RDF and RDFS statements FOL translation DL notation i rdf:type C C(i) i : C or C(i) i P j P(i,j) i P j or P(i,j) C rdfs:subclassof D X (C(X) D(X)) C D Ignore for now DL column This is just a notation We will come back to it to discuss Description logics

49 RDFS logical semantics Three ontology languages for the Web RDF and RDFS statements FOL translation DL notation i rdf:type C C(i) i : C or C(i) i P j P(i,j) i P j or P(i,j) C rdfs:subclassof D X (C(X) D(X)) C D P rdfs:subpropertyof R X Y (P(X,Y ) R(X,Y )) P R Ignore for now DL column This is just a notation We will come back to it to discuss Description logics

50 RDFS logical semantics Three ontology languages for the Web RDF and RDFS statements FOL translation DL notation i rdf:type C C(i) i : C or C(i) i P j P(i,j) i P j or P(i,j) C rdfs:subclassof D X (C(X) D(X)) C D P rdfs:subpropertyof R X Y (P(X,Y ) R(X,Y )) P R P rdfs:domain C X Y (P(X,Y ) C(X)) P C Ignore for now DL column This is just a notation We will come back to it to discuss Description logics

51 RDFS logical semantics Three ontology languages for the Web RDF and RDFS statements FOL translation DL notation i rdf:type C C(i) i : C or C(i) i P j P(i,j) i P j or P(i,j) C rdfs:subclassof D X (C(X) D(X)) C D P rdfs:subpropertyof R X Y (P(X,Y ) R(X,Y )) P R P rdfs:domain C X Y (P(X,Y ) C(X)) P C P rdfs:range D X Y (P(X,Y ) D(Y )) P D Ignore for now DL column This is just a notation We will come back to it to discuss Description logics Sebastian Link Querying Data through Ontologies March 26, / 64

52 Three ontology languages for the Web OWL: Web Ontology Language OWL extends RDFS with the possibility to express additional constraints Sebastian Link Querying Data through Ontologies March 26, / 64

53 Three ontology languages for the Web OWL: Web Ontology Language OWL extends RDFS with the possibility to express additional constraints Main OWL constructs Disjointness between classes Constraints of functionality and symmetry on predicates Intentional class definitions Class union and intersection Sebastian Link Querying Data through Ontologies March 26, / 64

54 Three ontology languages for the Web OWL: Web Ontology Language OWL extends RDFS with the possibility to express additional constraints Main OWL constructs Disjointness between classes Constraints of functionality and symmetry on predicates Intentional class definitions Class union and intersection We will see these are all expressible in Description logics Sebastian Link Querying Data through Ontologies March 26, / 64

55 Three ontology languages for the Web OWL constructs Ignore again the DL column Disjointness between classes: OWL notation FOL translation DL notation C owl:disjointwith D X (C(X) D(X)) C D Sebastian Link Querying Data through Ontologies March 26, / 64

56 Three ontology languages for the Web OWL constructs Ignore again the DL column Disjointness between classes: OWL notation FOL translation DL notation C owl:disjointwith D X (C(X) D(X)) C D Constraints of functionality and symmetry on predicates: OWL notation FOL translation DL notation P rdf:type X Y Z (funct P) owl:functionalproperty (P(X,Y ) P(X,Z) Y = Z) or P ( 1P)

57 Three ontology languages for the Web OWL constructs Ignore again the DL column Disjointness between classes: OWL notation FOL translation DL notation C owl:disjointwith D X (C(X) D(X)) C D Constraints of functionality and symmetry on predicates: OWL notation FOL translation DL notation P rdf:type X Y Z (funct P) owl:functionalproperty (P(X,Y ) P(X,Z) Y = Z) or P ( 1P) P rdf:type X Y Z (funct P ) owl:inversefunctionalproperty (P(X,Y ) P(Z,Y ) X = Z) or P ( 1P )

58 Three ontology languages for the Web OWL constructs Ignore again the DL column Disjointness between classes: OWL notation FOL translation DL notation C owl:disjointwith D X (C(X) D(X)) C D Constraints of functionality and symmetry on predicates: OWL notation FOL translation DL notation P rdf:type X Y Z (funct P) owl:functionalproperty (P(X,Y ) P(X,Z) Y = Z) or P ( 1P) P rdf:type X Y Z (funct P ) owl:inversefunctionalproperty (P(X,Y ) P(Z,Y ) X = Z) or P ( 1P ) P owl:inverseof Q X Y (P(X,Y ) Q(Y,X)) P Q

59 Three ontology languages for the Web OWL constructs Ignore again the DL column Disjointness between classes: OWL notation FOL translation DL notation C owl:disjointwith D X (C(X) D(X)) C D Constraints of functionality and symmetry on predicates: OWL notation FOL translation DL notation P rdf:type X Y Z (funct P) owl:functionalproperty (P(X,Y ) P(X,Z) Y = Z) or P ( 1P) P rdf:type X Y Z (funct P ) owl:inversefunctionalproperty (P(X,Y ) P(Z,Y ) X = Z) or P ( 1P ) P owl:inverseof Q X Y (P(X,Y ) Q(Y,X)) P Q X Y (P(X,Y ) P(Y,X)) P rdf:type P P owl:symmetricproperty Sebastian Link Querying Data through Ontologies March 26, / 64

60 Three ontology languages for the Web Definition of intentional classes in OWL Goal: express complex constraints such as departments can be led only by professors only professors or lecturers may teach to undergraduate students professors teach in at least 2 and in at most 3 courses Sebastian Link Querying Data through Ontologies March 26, / 64

61 Three ontology languages for the Web Definition of intentional classes in OWL Goal: express complex constraints such as departments can be led only by professors only professors or lecturers may teach to undergraduate students professors teach in at least 2 and in at most 3 courses The keyword owl:restriction is used in association with a blank node class, and some specific restriction properties: owl:somevaluesfrom owl:allvaluesfrom owl:mincardinality owl:maxcardinality Sebastian Link Querying Data through Ontologies March 26, / 64

62 Three ontology languages for the Web OWL Semantics OWL notation FOL translation DL notation _a owl:onproperty P _a owl:allvaluesfrom C Y (P(X, Y ) C(Y )) P.C

63 Three ontology languages for the Web OWL Semantics OWL notation FOL translation DL notation _a owl:onproperty P _a owl:allvaluesfrom C Y (P(X, Y ) C(Y )) P.C _a owl:onproperty P _a owl:somevaluesfrom C Y (P(X, Y ) C(Y )) P.C

64 Three ontology languages for the Web OWL Semantics OWL notation FOL translation DL notation _a owl:onproperty P _a owl:allvaluesfrom C Y (P(X, Y ) C(Y )) P.C _a owl:onproperty P _a owl:somevaluesfrom C Y (P(X, Y ) C(Y )) P.C _a owl:onproperty P _a owl:mincardinality n Y 1... Y n (P(X,Y 1 )... P(X,Y n ) ( n P) i,j [1..n],i =j(y i = Y j ))

65 Three ontology languages for the Web OWL Semantics OWL notation FOL translation DL notation _a owl:onproperty P _a owl:allvaluesfrom C Y (P(X, Y ) C(Y )) P.C _a owl:onproperty P _a owl:somevaluesfrom C Y (P(X, Y ) C(Y )) P.C _a owl:onproperty P _a owl:mincardinality n Y 1... Y n (P(X,Y 1 )... P(X,Y n ) ( n P) i,j [1..n],i =j(y i = Y j )) _a owl:maxcardinality n Y 1... Y n Y n+1 (P(X,Y 1 )... P(X,Y n ) ( n P) P(X,Y n+1 ) i,j [1..n+1],i =j(y i = Y j )) Sebastian Link Querying Data through Ontologies March 26, / 64

66 Three ontology languages for the Web Unnamed new classes by example Departments can be led only by professors Sebastian Link Querying Data through Ontologies March 26, / 64

67 Three ontology languages for the Web Unnamed new classes by example Departments can be led only by professors Recall Leads(Staff,Department), and declare inverse: LedBy owl : inverseof Lead Sebastian Link Querying Data through Ontologies March 26, / 64

68 Three ontology languages for the Web Unnamed new classes by example Departments can be led only by professors Recall Leads(Staff,Department), and declare inverse: LedBy owl : inverseof Lead Define the set of objects that are led by professors _a rdfs:subclassof owl:restriction _a owl:onproperty LedBy _a owl:allvaluesfrom Professor Sebastian Link Querying Data through Ontologies March 26, / 64

69 Three ontology languages for the Web Unnamed new classes by example Departments can be led only by professors Recall Leads(Staff,Department), and declare inverse: LedBy owl : inverseof Lead Define the set of objects that are led by professors _a rdfs:subclassof owl:restriction _a owl:onproperty LedBy _a owl:allvaluesfrom Professor Now specify that all departments are led by professors Department rdfs:subclassof _a Sebastian Link Querying Data through Ontologies March 26, / 64

70 Three ontology languages for the Web Union and intersection of classes by example Only professors or lecturers may teach to undergraduate students Sebastian Link Querying Data through Ontologies March 26, / 64

71 Three ontology languages for the Web Union and intersection of classes by example Only professors or lecturers may teach to undergraduate students _a rdfs:subclassof owl:restriction _a owl:onproperty TeachesTo _a owl:somevaluesfrom Undergrad _b owl:unionof (Professor, Lecturer) _a rdfs:subclassof _b Sebastian Link Querying Data through Ontologies March 26, / 64

72 Three ontology languages for the Web Union and intersection of classes by example Only professors or lecturers may teach to undergraduate students _a rdfs:subclassof owl:restriction _a owl:onproperty TeachesTo _a owl:somevaluesfrom Undergrad _b owl:unionof (Professor, Lecturer) _a rdfs:subclassof _b This corresponds to an inclusion axiom in Description Logic: TeachesTo.UndergraduateStudent Professor Lecturer Sebastian Link Querying Data through Ontologies March 26, / 64

73 Three ontology languages for the Web Union and intersection of classes by example Only professors or lecturers may teach to undergraduate students _a rdfs:subclassof owl:restriction _a owl:onproperty TeachesTo _a owl:somevaluesfrom Undergrad _b owl:unionof (Professor, Lecturer) _a rdfs:subclassof _b This corresponds to an inclusion axiom in Description Logic: TeachesTo.UndergraduateStudent Professor Lecturer owl:equivalentclass corresponds to double inclusion: MathStudent Student RegisteredTo.MathCourse Sebastian Link Querying Data through Ontologies March 26, / 64

74 Three ontology languages for the Web Cardinality constraints by example Professors teach in at least 2 and in at most 3 courses Sebastian Link Querying Data through Ontologies March 26, / 64

75 Three ontology languages for the Web Cardinality constraints by example Professors teach in at least 2 and in at most 3 courses Define the set of objects who teach in 2-3 courses _a rdfs:subclassof owl:restriction _a owl:onproperty TeachesIn _a owl:mincardinality 2 _a owl:maxcardinality 3 Sebastian Link Querying Data through Ontologies March 26, / 64

76 Three ontology languages for the Web Cardinality constraints by example Professors teach in at least 2 and in at most 3 courses Define the set of objects who teach in 2-3 courses _a rdfs:subclassof owl:restriction _a owl:onproperty TeachesIn _a owl:mincardinality 2 _a owl:maxcardinality 3 Now specify that professors belong to this restriction class Professor rdfs:subclassof _a Sebastian Link Querying Data through Ontologies March 26, / 64

77 Reasoning in Description Logics Outline 1 Introduction 2 Three ontology languages for the Web 3 Reasoning in Description Logics ALC Polynomial DLs 4 Querying Data through Ontologies 5 Conclusion Sebastian Link Querying Data through Ontologies March 26, / 64

78 Reasoning in Description Logics Description Logics Philosophy: isolate decidable fragments of first-order logic allowing reasoning on complex logical axioms over unary and binary predicates These fragments are called Description Logics Sebastian Link Querying Data through Ontologies March 26, / 64

79 Reasoning in Description Logics Description Logics Philosophy: isolate decidable fragments of first-order logic allowing reasoning on complex logical axioms over unary and binary predicates These fragments are called Description Logics The DL jargon: the classes are called concepts the properties are called roles the ontology (the knowledge base) = Tbox + Abox the schema is called the Tbox the instance is called the Abox Sebastian Link Querying Data through Ontologies March 26, / 64

80 Reasoning in Description Logics The DL family Few constructs: atomic concepts and roles, inverse of roles, unqualified restriction on roles, restricted negation Revisit RDFS checking out the DL column If you don t like the syntax: neither do I Sebastian Link Querying Data through Ontologies March 26, / 64

81 Semantics of main conctructs Reasoning in Description Logics I(C 1 C 2 ) = I(C 1 ) I(C 2 ) Sebastian Link Querying Data through Ontologies March 26, / 64

82 Semantics of main conctructs Reasoning in Description Logics I(C 1 C 2 ) = I(C 1 ) I(C 2 ) I( R.C) = {o 1 o 2 [(o 1,o 2 ) I(R) o 2 I(C)]} Sebastian Link Querying Data through Ontologies March 26, / 64

83 Semantics of main conctructs Reasoning in Description Logics I(C 1 C 2 ) = I(C 1 ) I(C 2 ) I( R.C) = {o 1 o 2 [(o 1,o 2 ) I(R) o 2 I(C)]} I( R.C) = {o 1 o 2.[(o 1,o 2 ) I(R) o 2 I(C)]} Sebastian Link Querying Data through Ontologies March 26, / 64

84 Semantics of main conctructs Reasoning in Description Logics I(C 1 C 2 ) = I(C 1 ) I(C 2 ) I( R.C) = {o 1 o 2 [(o 1,o 2 ) I(R) o 2 I(C)]} I( R.C) = {o 1 o 2.[(o 1,o 2 ) I(R) o 2 I(C)]} I( C) = I \ I(C) Sebastian Link Querying Data through Ontologies March 26, / 64

85 Semantics of main conctructs Reasoning in Description Logics I(C 1 C 2 ) = I(C 1 ) I(C 2 ) I( R.C) = {o 1 o 2 [(o 1,o 2 ) I(R) o 2 I(C)]} I( R.C) = {o 1 o 2.[(o 1,o 2 ) I(R) o 2 I(C)]} I( C) = I \ I(C) I(R ) = {(o 2,o 1 ) (o 1,o 2 ) I(R)} Sebastian Link Querying Data through Ontologies March 26, / 64

86 Reasoning in Description Logics Defining a particular description logic Choose how to construct complex concepts and roles starting from atomic concepts and roles Professor Lecturer (those who are either professor or lecturer) Sebastian Link Querying Data through Ontologies March 26, / 64

87 Reasoning in Description Logics Defining a particular description logic Choose how to construct complex concepts and roles starting from atomic concepts and roles Professor Lecturer (those who are either professor or lecturer) Choose the constraints you want to consider Sebastian Link Querying Data through Ontologies March 26, / 64

88 Reasoning in Description Logics Defining a particular description logic Choose how to construct complex concepts and roles starting from atomic concepts and roles Professor Lecturer (those who are either professor or lecturer) Choose the constraints you want to consider The complexity of the logic depends on these choices Sebastian Link Querying Data through Ontologies March 26, / 64

89 Reasoning in Description Logics Reasoning problems studied in DL Satisfiability checking: Given a DL knowledge base K = T,A, is K satisfiable? Sebastian Link Querying Data through Ontologies March 26, / 64

90 Reasoning in Description Logics Reasoning problems studied in DL Satisfiability checking: Given a DL knowledge base K = T,A, is K satisfiable? Subsumption checking: Given a Tbox T and two concept expressions C and D, does T = C D? Sebastian Link Querying Data through Ontologies March 26, / 64

91 Reasoning in Description Logics Reasoning problems studied in DL Satisfiability checking: Given a DL knowledge base K = T,A, is K satisfiable? Subsumption checking: Given a Tbox T and two concept expressions C and D, does T = C D? Instance checking: Given a DL knowledge base K = T,A, an individual e and a concept expression C, does K = C(e)? Sebastian Link Querying Data through Ontologies March 26, / 64

92 Reasoning in Description Logics Reasoning problems studied in DL Satisfiability checking: Given a DL knowledge base K = T,A, is K satisfiable? Subsumption checking: Given a Tbox T and two concept expressions C and D, does T = C D? Instance checking: Given a DL knowledge base K = T,A, an individual e and a concept expression C, does K = C(e)? Query answering: Given a DL knowledge base K = T,A, and a concept expression C, finds the set of individuals e such that K = C(e)? Sebastian Link Querying Data through Ontologies March 26, / 64

93 Reasoning in Description Logics Remarks For DLs with full negation: instance checking and subsumption checking can be reduced to (un)satisfiability checking T = C D T,{(C D)(a)} is unsatisfiable. T,A = C(e) T,A { C(e)} is unsatisfiable. Sebastian Link Querying Data through Ontologies March 26, / 64

94 Reasoning in Description Logics Remarks For DLs with full negation: instance checking and subsumption checking can be reduced to (un)satisfiability checking T = C D T,{(C D)(a)} is unsatisfiable. T,A = C(e) T,A { C(e)} is unsatisfiable. For DLs without negation: instance checking can be reduced to subsumption checking by computing the most specific concept satisfied by an individual in the Abox (denoted msc(a, e)) T,A = C(e) T = msc(a,e) C Sebastian Link Querying Data through Ontologies March 26, / 64

95 Reasoning in Description Logics Remarks For DLs with full negation: instance checking and subsumption checking can be reduced to (un)satisfiability checking T = C D T,{(C D)(a)} is unsatisfiable. T,A = C(e) T,A { C(e)} is unsatisfiable. For DLs without negation: instance checking can be reduced to subsumption checking by computing the most specific concept satisfied by an individual in the Abox (denoted msc(a, e)) T,A = C(e) T = msc(a,e) C the most specific concept of e in A (denoted by msc(a,e)) is the concept expression D in the given DL such that for every concept C in the given DL, A = C(e) implies D C Sebastian Link Querying Data through Ontologies March 26, / 64

96 Reasoning in Description Logics ALC ALC: the prototypical DL (standard) An ALC Abox is made of a set of facts of the form C(a) and R(a,b) where a and b are individuals, R is an atomic role and C is a possibly complex concept Sebastian Link Querying Data through Ontologies March 26, / 64

97 Reasoning in Description Logics ALC ALC: the prototypical DL (standard) An ALC Abox is made of a set of facts of the form C(a) and R(a,b) where a and b are individuals, R is an atomic role and C is a possibly complex concept ALC constructs: conjunction C 1 C 2, existential restriction R.C Y (R(X,Y ) C(Y )) negation C. Sebastian Link Querying Data through Ontologies March 26, / 64

98 Reasoning in Description Logics ALC ALC: the prototypical DL (standard) An ALC Abox is made of a set of facts of the form C(a) and R(a,b) where a and b are individuals, R is an atomic role and C is a possibly complex concept ALC constructs: conjunction C 1 C 2, existential restriction R.C Y (R(X,Y ) C(Y )) negation C. As a result, ALC also contains de facto: disjunctions C 1 C 2 ( ( C 1 C 2 )), value restrictions ( R.C ( R. C)), ( A A) and ( A A). Sebastian Link Querying Data through Ontologies March 26, / 64

99 Reasoning in Description Logics ALC ALC - continued An ALC Tbox may contain inclusion constraints between concepts and roles MathCourse Course LateRegisteredTo RegisteredTo Sebastian Link Querying Data through Ontologies March 26, / 64

100 Reasoning in Description Logics ALC ALC - continued An ALC Tbox may contain inclusion constraints between concepts and roles MathCourse Course LateRegisteredTo RegisteredTo An ALC Tbox may contain General Concept Inclusions (GCIs): TeachesTo.UndergraduateStudent Professor Lecturer Sebastian Link Querying Data through Ontologies March 26, / 64

101 Reasoning in Description Logics ALC Tableau method Reasoning is based on tableau calculus - a classical method in logic for checking satisfiability Extensively used in Description logics for implementing reasoners Sebastian Link Querying Data through Ontologies March 26, / 64

102 Reasoning in Description Logics ALC Tableau method Reasoning is based on tableau calculus - a classical method in logic for checking satisfiability Extensively used in Description logics for implementing reasoners Technique Get rid of the Tbox by recursively unfolding the concept definitions Transform the resulting Abox so that negations apply only to atomic concepts Try to construct a model or raise a contradiction Sebastian Link Querying Data through Ontologies March 26, / 64

103 Reasoning in Description Logics ALC Tableau method Reasoning is based on tableau calculus - a classical method in logic for checking satisfiability Extensively used in Description logics for implementing reasoners Technique Get rid of the Tbox by recursively unfolding the concept definitions Transform the resulting Abox so that negations apply only to atomic concepts Try to construct a model or raise a contradiction We illustrate the technique with a simple example without GCIs In general, much more involved Sebastian Link Querying Data through Ontologies March 26, / 64

104 Reasoning in Description Logics ALC Tableau method For satisfiability checking of a DL knowledge base T,A T = {C1 A B, C 2 R.A, C 3 R.B, C 4 R. C 1 } A = {C2 (a),c 3 (a),c 4 (a)} Sebastian Link Querying Data through Ontologies March 26, / 64

105 Reasoning in Description Logics ALC Tableau method For satisfiability checking of a DL knowledge base T,A T = {C1 A B, C 2 R.A, C 3 R.B, C 4 R. C 1 } A = {C2 (a),c 3 (a),c 4 (a)} Get rid of the Tbox, by recursively unfolding the concept definitions: A = {( R.A)(a),( R.B)(a),( R. (A B))(a)} T,A Sebastian Link Querying Data through Ontologies March 26, / 64

106 Reasoning in Description Logics ALC Tableau method For satisfiability checking of a DL knowledge base T,A T = {C1 A B, C 2 R.A, C 3 R.B, C 4 R. C 1 } A = {C2 (a),c 3 (a),c 4 (a)} Get rid of the Tbox, by recursively unfolding the concept definitions: A = {( R.A)(a),( R.B)(a),( R. (A B))(a)} T,A Transform the concepts expressions in A into negation normal form A = {( R.A)(a),( R.B)(a),( R.( A B))(a)} Sebastian Link Querying Data through Ontologies March 26, / 64

107 Reasoning in Description Logics ALC Tableau method For satisfiability checking of a DL knowledge base T,A T = {C1 A B, C 2 R.A, C 3 R.B, C 4 R. C 1 } A = {C2 (a),c 3 (a),c 4 (a)} Get rid of the Tbox, by recursively unfolding the concept definitions: A = {( R.A)(a),( R.B)(a),( R. (A B))(a)} T,A Transform the concepts expressions in A into negation normal form A = {( R.A)(a),( R.B)(a),( R.( A B))(a)} Apply tableau rules to extend the resulting Abox until no rule applies anymore: From an extended Abox which is complete (no rule applies) and clash-free (no obvious contradiction), a so-called canonical interpretation can be built, which is a model of the initial Abox. Sebastian Link Querying Data through Ontologies March 26, / 64

108 Reasoning in Description Logics ALC Tableau rules for ALC The -rule: Condition: A contains (C D)(a) but not both C(a) and D(a) Action: add A = A {C(a),D(a)} Sebastian Link Querying Data through Ontologies March 26, / 64

109 Reasoning in Description Logics ALC Tableau rules for ALC The -rule: Condition: A contains (C D)(a) but not both C(a) and D(a) Action: add A = A {C(a),D(a)} The -rule: Condition: A contains (C D)(a) but neither C(a) nor D(a) Action: add A = A {C(a)} and A = A {D(a)} Sebastian Link Querying Data through Ontologies March 26, / 64

110 Reasoning in Description Logics ALC Tableau rules for ALC The -rule: Condition: A contains (C D)(a) but not both C(a) and D(a) Action: add A = A {C(a),D(a)} The -rule: Condition: A contains (C D)(a) but neither C(a) nor D(a) Action: add A = A {C(a)} and A = A {D(a)} The -rule: Condition: A contains ( R.C)(a) but there is no c such that {R(a,c),C(c)} A Action: add A = A {R(a,b),C(b)} where b is a new individual name Sebastian Link Querying Data through Ontologies March 26, / 64

111 Reasoning in Description Logics ALC Tableau rules for ALC The -rule: Condition: A contains (C D)(a) but not both C(a) and D(a) Action: add A = A {C(a),D(a)} The -rule: Condition: A contains (C D)(a) but neither C(a) nor D(a) Action: add A = A {C(a)} and A = A {D(a)} The -rule: Condition: A contains ( R.C)(a) but there is no c such that {R(a,c),C(c)} A Action: add A = A {R(a,b),C(b)} where b is a new individual name The -rule: Condition: A contains ( R.C)(a) and R(a, b) but not C(b) Action: add A = A {C(b)} Sebastian Link Querying Data through Ontologies March 26, / 64

112 Reasoning in Description Logics ALC Illustration on the example The result of the application of the tableau method to A = {( R.A)(a),( R.B)(a),( R.( A B))(a)} gives the following Aboxes: A 1 = A { } Sebastian Link Querying Data through Ontologies March 26, / 64

113 Reasoning in Description Logics ALC Illustration on the example The result of the application of the tableau method to A = {( R.A)(a),( R.B)(a),( R.( A B))(a)} gives the following Aboxes: A 1 = A { } Sebastian Link Querying Data through Ontologies March 26, / 64

114 Reasoning in Description Logics ALC Illustration on the example The result of the application of the tableau method to A = {( R.A)(a),( R.B)(a),( R.( A B))(a)} gives the following Aboxes: A 1 = A {R(a,b), A(b) } Sebastian Link Querying Data through Ontologies March 26, / 64

115 Reasoning in Description Logics ALC Illustration on the example The result of the application of the tableau method to A = {( R.A)(a),( R.B)(a),( R.( A B))(a)} gives the following Aboxes: A 1 = A {R(a,b), A(b) } Sebastian Link Querying Data through Ontologies March 26, / 64

116 Reasoning in Description Logics ALC Illustration on the example The result of the application of the tableau method to A = {( R.A)(a),( R.B)(a),( R.( A B))(a)} gives the following Aboxes: A 1 = A {R(a,b), A(b),B(b), } Sebastian Link Querying Data through Ontologies March 26, / 64

117 Reasoning in Description Logics ALC Illustration on the example The result of the application of the tableau method to A = {( R.A)(a),( R.B)(a),( R.( A B))(a)} gives the following Aboxes: A 1 = A {R(a,b), A(b),B(b), } Sebastian Link Querying Data through Ontologies March 26, / 64

118 Reasoning in Description Logics ALC Illustration on the example The result of the application of the tableau method to A = {( R.A)(a),( R.B)(a),( R.( A B))(a)} gives the following Aboxes: A 1 = A {R(a,b), A(b),B(b), A(b) } A 2 = A {R(a,b),A(b), B(b), B(b) } Sebastian Link Querying Data through Ontologies March 26, / 64

119 Reasoning in Description Logics ALC Illustration on the example The result of the application of the tableau method to A = {( R.A)(a),( R.B)(a),( R.( A B))(a)} gives the following Aboxes: A 1 = A {R(a,b), A(b),B(b), A(b) } A 2 = A {R(a,b),A(b), B(b), B(b) } They both contain a clash: A (and the equivalent original knowledge base) is correctly decided unsatisfiable by the algorithm Sebastian Link Querying Data through Ontologies March 26, / 64

120 Reasoning in Description Logics ALC Complexity The tableau method shows that the satisfiability of ALC knowledge bases is decidable but with a complexity that may be exponential because of the disjunction construct and the associated -rule. Sebastian Link Querying Data through Ontologies March 26, / 64

121 Reasoning in Description Logics ALC Complexity The tableau method shows that the satisfiability of ALC knowledge bases is decidable but with a complexity that may be exponential because of the disjunction construct and the associated -rule. Satisfiability checking in ALC (and thus also subsumption and instance checking) is in fact EXPTIME-complete and therefore not in PTIME Sebastian Link Querying Data through Ontologies March 26, / 64

122 Reasoning in Description Logics ALC Complexity The tableau method shows that the satisfiability of ALC knowledge bases is decidable but with a complexity that may be exponential because of the disjunction construct and the associated -rule. Satisfiability checking in ALC (and thus also subsumption and instance checking) is in fact EXPTIME-complete and therefore not in PTIME Additional constructs like those in the fragment OWL DL of OWL do not change the complexity class of reasoning (which remains EXPTIME-complete) Sebastian Link Querying Data through Ontologies March 26, / 64

123 Reasoning in Description Logics ALC Complexity The tableau method shows that the satisfiability of ALC knowledge bases is decidable but with a complexity that may be exponential because of the disjunction construct and the associated -rule. Satisfiability checking in ALC (and thus also subsumption and instance checking) is in fact EXPTIME-complete and therefore not in PTIME Additional constructs like those in the fragment OWL DL of OWL do not change the complexity class of reasoning (which remains EXPTIME-complete) OWL Full is undecidable Sebastian Link Querying Data through Ontologies March 26, / 64

124 Reasoning in Description Logics Polynomial DLs DLs for which reasoning is polynomial FL: conjunction C 1 C 2, value restrictions R.C and unqualified existential restriction R For Tboxes without GCIs, subsumption checking is polynomial For Tboxes with (even simple) GCIs,subsumption checking is co-np complete Sebastian Link Querying Data through Ontologies March 26, / 64

125 Reasoning in Description Logics Polynomial DLs DLs for which reasoning is polynomial FL: conjunction C 1 C 2, value restrictions R.C and unqualified existential restriction R For Tboxes without GCIs, subsumption checking is polynomial For Tboxes with (even simple) GCIs,subsumption checking is co-np complete EL: conjunctions C 1 C 2 and existential restrictions R.C Subsumption checking in E L is polynomial even for general Tboxes. Sebastian Link Querying Data through Ontologies March 26, / 64

126 Reasoning in Description Logics Polynomial DLs DLs for which reasoning is polynomial FL: conjunction C 1 C 2, value restrictions R.C and unqualified existential restriction R For Tboxes without GCIs, subsumption checking is polynomial For Tboxes with (even simple) GCIs,subsumption checking is co-np complete EL: conjunctions C 1 C 2 and existential restrictions R.C Subsumption checking in E L is polynomial even for general Tboxes. FLE: conjunction C 1 C 2, value restrictions R.C, and existential restrictions R.C Subsumption checking in F LE is NP-complete Sebastian Link Querying Data through Ontologies March 26, / 64

127 Reasoning in Description Logics Polynomial DLs DLs for which reasoning is polynomial FL: conjunction C 1 C 2, value restrictions R.C and unqualified existential restriction R For Tboxes without GCIs, subsumption checking is polynomial For Tboxes with (even simple) GCIs,subsumption checking is co-np complete EL: conjunctions C 1 C 2 and existential restrictions R.C Subsumption checking in E L is polynomial even for general Tboxes. FLE: conjunction C 1 C 2, value restrictions R.C, and existential restrictions R.C Subsumption checking in F LE is NP-complete Other reasoning problems for FL, EL and FLE: Satisfiability checking is trivial Instance checking can be reduced to subsumption checking via mscs Sebastian Link Querying Data through Ontologies March 26, / 64

128 Reasoning in Description Logics Polynomial DLs DLs for which reasoning is polynomial FL: conjunction C 1 C 2, value restrictions R.C and unqualified existential restriction R For Tboxes without GCIs, subsumption checking is polynomial For Tboxes with (even simple) GCIs,subsumption checking is co-np complete EL: conjunctions C 1 C 2 and existential restrictions R.C Subsumption checking in E L is polynomial even for general Tboxes. FLE: conjunction C 1 C 2, value restrictions R.C, and existential restrictions R.C Subsumption checking in F LE is NP-complete Other reasoning problems for FL, EL and FLE: Satisfiability checking is trivial Instance checking can be reduced to subsumption checking via mscs The DL-LITE family: a good trade-off, specially designed for guaranteeing query answering through ontologies to be polynomial in data complexity Sebastian Link Querying Data through Ontologies March 26, / 64

129 Querying Data through Ontologies Outline 1 Introduction 2 Three ontology languages for the Web 3 Reasoning in Description Logics 4 Querying Data through Ontologies Querying using RDFS Querying using DL-LITE Complexity 5 Conclusion Sebastian Link Querying Data through Ontologies March 26, / 64

130 Querying Data through Ontologies Querying using RDFS Querying using RDFS RDFS statements can be used to infer new triples Example Base fact ResponsibleOf (durand, ue111) Sebastian Link Querying Data through Ontologies March 26, / 64

131 Querying Data through Ontologies Querying using RDFS Querying using RDFS RDFS statements can be used to infer new triples Example Base fact ResponsibleOf (durand, ue111) Use the statement ResponsibleOf rdfs:domain Professor i.e., the logical rule: ResponsibleOf (X,Y ) Professor(X) With substitution {X/durand, Y/ue111} Infer fact Professor(durand) Sebastian Link Querying Data through Ontologies March 26, / 64

132 Querying Data through Ontologies Querying using RDFS Querying using RDFS RDFS statements can be used to infer new triples Example Base fact ResponsibleOf (durand, ue111) Use the statement ResponsibleOf rdfs:domain Professor i.e., the logical rule: ResponsibleOf (X,Y ) Professor(X) With substitution {X/durand, Y/ue111} Infer fact Professor(durand) Use the statement Professor rdfs:subclassof AcademicStaff i.e., the rule Professor(X) AcademicStaff (X) With substitution {X/durand} Infer fact AcademicStaff (durand) etc. Sebastian Link Querying Data through Ontologies March 26, / 64

133 Querying Data through Ontologies Querying using RDFS The saturation algorithm Keep inferring new facts until a fix-point is reached Note: Only polynomially many facts can be added PTIME Sebastian Link Querying Data through Ontologies March 26, / 64

134 Querying Data through Ontologies Querying using RDFS RDFS ontology expressed in different notations RDFS notation DL notation FOL notation AcademicStaff rdfs:subclassof Staff AcademicStaff Staff AcademicStaff (X) Staff (X) Professor rdfs:subclassof AcademicStaff Professor AcademicStaff Professor(X) AcademicStaff (X) Lecturer rdfs:subclassof AcademicStaff Lecturer AcademicStaff Lecturer(X) AcademicStaff (X) PhDStudent rdfs:subclassof Lecturer PhDStudent Lecturer PhDStudent(X) Lecturer(X) PhDStudent rdfs:subclassof Student PhDStudent Student PhDStudent(X) Student(X) TeachesIn rdfs:domain AcademicStaff TeachesIn AcademicStaff TeachesIn(X, Y ) AcademicStaff (X) TeachesIn rdfs:range Course TeachesIn Course TeachesIn(X,Y ) Course(Y ) ResponsibleOf rdfs:domain Professor ResponsibleOf Professor ResponsibleOf (X, Y ) Professor(X) ResponsibleOf rdfs:range Course ResponsibleOf Course ResponsibleOf (X,Y ) Course(Y ) TeachesTo rdfs:domain AcademicStaff TeachesTo AcademicStaff TeachesTo(X, Y ) AcademicStaff (X) TeachesTo rdfs:range Student TeachesTo Student TeachesTo(X,Y ) Student(Y ) Leads rdfs:domain AdminStaff Leads AdminStaff Leads(X, Y ) AdminStaff (X) Leads rdfs:range Dept Leads Dept Leads(X,Y ) Dept(Y ) RegisteredIn rdfs:domain Student RegisteredIn Student RegisteredIn(X, Y ) Student(X) RegisteredIn rdfs:range Course RegisteredIn Course RegisteredIn(X,Y ) Course(Y ) ResponsibleOf rdfs:subpropertyof TeachesIn ResponsibleOf TeachesIn ResponsibleOf (X, Y ) TeachesIn(X, Y ) Sebastian Link Querying Data through Ontologies March 26, / 64