Knowledge Representation and Description Logic Part 3

Transcription

1 Knowledge Representation and Description Logic Part 3 Renata Wassermann renata@ime.usp.br Computer Science Department University of São Paulo September 2014 IAOA School Vitória Renata Wassermann Knowledge Representation and Description Logic Part 3 1 / 44

2 ALC C, D A C C D C D R.C R.C Renata Wassermann Knowledge Representation and Description Logic Part 3 2 / 44

3 Below ALC EL C, D A C D R.C Looks rather inexpressive, but: SNOMED, Galen, etc. Renata Wassermann Knowledge Representation and Description Logic Part 3 3 / 44

4 Below ALC DL-Lite B A R R 1 inverse of roles Renata Wassermann Knowledge Representation and Description Logic Part 3 4 / 44

5 Below ALC DL-Lite B A R R 1 inverse of roles (ispetof = haspet 1 ) Renata Wassermann Knowledge Representation and Description Logic Part 3 4 / 44

6 Below ALC DL-Lite B A R R 1 inverse of roles (ispetof = haspet 1 ) C, D B B C D Renata Wassermann Knowledge Representation and Description Logic Part 3 4 / 44

7 Above ALC SHIF S = ALC+ transitive roles H = role hierarchies I = inverse of roles F = functional roles Renata Wassermann Knowledge Representation and Description Logic Part 3 5 / 44

8 Above ALC SHIF S = ALC+ transitive roles (ancestor) H = role hierarchies I = inverse of roles F = functional roles Renata Wassermann Knowledge Representation and Description Logic Part 3 5 / 44

9 Above ALC SHIF S = ALC+ transitive roles (ancestor) H = role hierarchies (hasson haschild) I = inverse of roles F = functional roles Renata Wassermann Knowledge Representation and Description Logic Part 3 5 / 44

10 Above ALC SHIF S = ALC+ transitive roles (ancestor) H = role hierarchies (hasson haschild) I = inverse of roles (ispetof = haspet 1 ) F = functional roles Renata Wassermann Knowledge Representation and Description Logic Part 3 5 / 44

11 Above ALC SHIF S = ALC+ transitive roles (ancestor) H = role hierarchies (hasson haschild) I = inverse of roles (ispetof = haspet 1 ) F = functional roles (isfatherof) Renata Wassermann Knowledge Representation and Description Logic Part 3 5 / 44

12 Above ALC SHOIN S = ALC+ transitive roles H = role hierarchies O = nominals I = inverse of roles N = number restriction Renata Wassermann Knowledge Representation and Description Logic Part 3 6 / 44

13 Above ALC SHOIN S = ALC+ transitive roles H = role hierarchies O = nominals ({hogwarts}) I = inverse of roles N = number restriction Renata Wassermann Knowledge Representation and Description Logic Part 3 6 / 44

14 Above ALC SHOIN S = ALC+ transitive roles H = role hierarchies O = nominals ({hogwarts}) I = inverse of roles N = number restriction (> 3hasChild) Renata Wassermann Knowledge Representation and Description Logic Part 3 6 / 44

15 Above ALC SROIQ S = ALC+ transitive roles R = role chains + hierarchies O = nominals I = inverse of roles Q = qualified number restriction Renata Wassermann Knowledge Representation and Description Logic Part 3 7 / 44

16 Above ALC SROIQ S = ALC+ transitive roles R = role chains + hierarchies (hasparent hasbrother hasuncle) O = nominals I = inverse of roles Q = qualified number restriction Renata Wassermann Knowledge Representation and Description Logic Part 3 7 / 44

17 Above ALC SROIQ S = ALC+ transitive roles R = role chains + hierarchies (hasparent hasbrother hasuncle) O = nominals I = inverse of roles Q = qualified number restriction (> 3hasChild.Male) Renata Wassermann Knowledge Representation and Description Logic Part 3 7 / 44

18 OWL (1.0) Web Ontology Language W3C standard for ontologies since 2004 Based on SHIF and SHOIN XML syntax Renata Wassermann Knowledge Representation and Description Logic Part 3 8 / 44

19 OWL (1.0) Three flavours: OWL Full - Allows for any combination of OWL operators, extends RDF (undecidable) Renata Wassermann Knowledge Representation and Description Logic Part 3 9 / 44

20 OWL (1.0) Three flavours: OWL Full - Allows for any combination of OWL operators, extends RDF (undecidable) OWL DL - Restricts constructors, decidable, corresponds to SHOIN. Renata Wassermann Knowledge Representation and Description Logic Part 3 9 / 44

21 OWL (1.0) Three flavours: OWL Full - Allows for any combination of OWL operators, extends RDF (undecidable) OWL DL - Restricts constructors, decidable, corresponds to SHOIN. OWL Lite - Excludes disjoint classes, nominals, cardinality only 0/1. Decidable, corresponds to SHIF. Renata Wassermann Knowledge Representation and Description Logic Part 3 9 / 44

22 Classes owl:disjointwith owl:equivalentclass owl:thing owl:nothing Renata Wassermann Knowledge Representation and Description Logic Part 3 10 / 44

23 Boolean combinations owl:complementof owl:unionof owl:intersectionof Renata Wassermann Knowledge Representation and Description Logic Part 3 11 / 44

24 Properties Two kinds of properties: Object - teaches, advises Datatype - name, age Renata Wassermann Knowledge Representation and Description Logic Part 3 12 / 44

25 Properties Two kinds of properties: Object - teaches, advises Datatype - name, age <owl:datatypeproperty rdf:id="age"> <rdfs:range rdf:resource= " </owl:datatypeproperty> <owl:objectproperty rdf:id="taughtby"> <owl:domain rdf:resource="course"/> <owl:range rdf:resource="academicstaff"/> <rdfs:subpropertyof rdf:resource="involves"/> </owl:objectproperty> Renata Wassermann Knowledge Representation and Description Logic Part 3 13 / 44

26 Properties Restrictions: owl:allvaluesfrom universal quantification owl:hasvalue specific value owl:somevaluesfrom existential quantification Cardinality owl:mincardinality and owl:maxcardinality Renata Wassermann Knowledge Representation and Description Logic Part 3 14 / 44

27 Special Properties owl:transitiveproperty ex.: ancestral owl:symmetricproperty ex.: hassamegradeas owl:functionalproperty ex.: age, father owl:inversefunctionalproperty (two different objects cannot have the same value) ex.: marriedto Renata Wassermann Knowledge Representation and Description Logic Part 3 15 / 44

28 Example OWL-RDF <owl:class rdf:id="associateprofessor"> <rdfs:subclassof rdf:resource="#academicstaffmember"/ </owl:class> <owl:class rdf:about="associateprofessor"> <owl:disjointwith rdf:resource="#professor"/> <owl:disjointwith rdf:resource="#assistantprofessor"/ </owl:class> <owl:class rdf:id="faculty"> <owl:equivalentclass rdf:resource="#academicstaffmemb </owl:class> Renata Wassermann Knowledge Representation and Description Logic Part 3 16 / 44

29 Example OWL-RDF <owl:objectproperty rdf:id="istaughtby"> <rdfs:domain rdf:resource="#course"/> <rdfs:range rdf:resource="#academicstaffmember"/> <rdfs:subpropertyof rdf:resource="#involves"/> </owl:objectproperty> <owl:objectproperty rdf:id="teaches"> <rdfs:range rdf:resource="#course"/> <rdfs:domain rdf:resource="#academicstaffmember"/> <owl:inverseof rdf:resource="#istaughtby"/> </owl:objectproperty> Renata Wassermann Knowledge Representation and Description Logic Part 3 17 / 44

30 Example OWL-RDF <owl:class rdf:about="#firstyearcourse"> <rdfs:subclassof> <owl:restriction> <owl:onproperty rdf:resource="#istaughtby"/> <owl:allvaluesfrom rdf:resource="#professor"/> </owl:restriction> </rdfs:subclassof> </owl:class> <owl:class rdf:about="#course"> <rdfs:subclassof> <owl:restriction> <owl:onproperty rdf:resource="#istaughtby"/> <owl:mincardinality Renata Wassermann Knowledge Representation and Description Logic Part 3 18 / 44

31 Example OWL-RDF <owl:class rdf:id="peopleatuni"> <owl:unionof rdf:parsetype="collection"> <owl:class rdf:about="#staffmember"/> <owl:class rdf:about="#student"/> </owl:unionof> </owl:class> <academicstaffmember rdf:id="949352"> <uni:age rdf:datatype="&xsd;integer">39<uni:age> </academicstaffmember> <course rdf:about="cit1111"> <istaughtby rdf:resource="949318"/> <istaughtby rdf:resource="949352"/> </course> Renata Wassermann Knowledge Representation and Description Logic Part 3 19 / 44

32 Example OWL-Turtle :academicstaffmember rdf:type owl:class ; owl:equivalentclass :faculty. :assistantprofessor rdf:type owl:class ; rdfs:subclassof :academicstaffmember ; owl:disjointwith :associateprofessor. :associateprofessor rdf:type owl:class ; rdfs:subclassof :academicstaffmember ; owl:disjointwith :professor. Renata Wassermann Knowledge Representation and Description Logic Part 3 20 / 44

33 Example OWL-Turtle :course rdf:type owl:class ; rdfs:subclassof [ rdf:type owl:restriction ; owl:onproperty :istaughtby ; owl:mincardinality "1"^^xsd:nonNegati ]. :faculty rdf:type owl:class. :firstyearcourse rdf:type owl:class ; rdfs:subclassof :course, [ rdf:type owl:restriction ; owl:onproperty :istaughtby ; owl:allvaluesfrom :professor ]. Renata Wassermann Knowledge Representation and Description Logic Part 3 21 / 44

34 OWL (2.0) W3C standard for ontologies since 2009 Even OWL-Lite is intractable OWL-DL 2 is based on SROIQ Three new profiles for Lite fragment Renata Wassermann Knowledge Representation and Description Logic Part 3 22 / 44

35 OWL 2.0 EL: polynomial time reasoning for schema and data Useful for ontologies with large conceptual part Renata Wassermann Knowledge Representation and Description Logic Part 3 23 / 44

36 OWL 2.0 EL: polynomial time reasoning for schema and data Useful for ontologies with large conceptual part QL: fast (logspace) query answering using RDBMs via SQL Useful for large datasets already stored in RDBs Renata Wassermann Knowledge Representation and Description Logic Part 3 23 / 44

37 OWL 2.0 EL: polynomial time reasoning for schema and data Useful for ontologies with large conceptual part QL: fast (logspace) query answering using RDBMs via SQL Useful for large datasets already stored in RDBs RL: fast (polynomial) query answering using rule-extended DBs Useful for large datasets stored as RDF triples Renata Wassermann Knowledge Representation and Description Logic Part 3 23 / 44

38 Motivation Study the dynamics of ontologies, specially OWL-like DL ontologies. AGM Belief Revision deals with the problem of adding/removing information in a consistent way. AGM is most commonly applied to propositional classical logic and cannot be directly used with DLs. How can we adapt AGM so that it can deal with interesting DLs? Renata Wassermann Knowledge Representation and Description Logic Part 3 24 / 44

39 In this work Show reasons why AGM fails to apply to DLs. Adapt Contraction (easy). Adapt Revision (less easy). Renata Wassermann Knowledge Representation and Description Logic Part 3 25 / 44

40 AGM AGM Belief Revision Three operations defined to deal with knowledge base dynamics: Expansion - adding knowledge (possibly inconsistent) Renata Wassermann Knowledge Representation and Description Logic Part 3 26 / 44

41 AGM AGM Belief Revision Three operations defined to deal with knowledge base dynamics: Expansion - adding knowledge (possibly inconsistent) Contraction - removing knowledge Renata Wassermann Knowledge Representation and Description Logic Part 3 26 / 44

42 AGM AGM Belief Revision Three operations defined to deal with knowledge base dynamics: Expansion - adding knowledge (possibly inconsistent) Contraction - removing knowledge Revision - adding knowledge consistently Renata Wassermann Knowledge Representation and Description Logic Part 3 26 / 44

43 AGM AGM Belief Revision Three operations defined to deal with knowledge base dynamics: Expansion - adding knowledge (possibly inconsistent) Contraction - removing knowledge Revision - adding knowledge consistently Revision usually defined in terms of contraction: K α = (K α) + α Renata Wassermann Knowledge Representation and Description Logic Part 3 26 / 44

44 AGM AGM Theory For contraction and revision: Rationality Postulates Renata Wassermann Knowledge Representation and Description Logic Part 3 27 / 44

45 AGM AGM Theory For contraction and revision: Rationality Postulates Construction Renata Wassermann Knowledge Representation and Description Logic Part 3 27 / 44

46 AGM AGM Theory For contraction and revision: Rationality Postulates Construction Representation Theorem (postulates construction) Renata Wassermann Knowledge Representation and Description Logic Part 3 27 / 44

47 AGM AGM Theory For contraction and revision: Rationality Postulates Construction Representation Theorem (postulates construction) AGM Assumptions: Tarskian, Compact, Deduction Theorem, Supraclassical. Renata Wassermann Knowledge Representation and Description Logic Part 3 27 / 44

48 AGM contraction (closure) K α = Cn(K α) (success) If α / Cn( ) then α / K α (inclusion) K α K (vacuity) If α / K then K α = K (recovery) K K α + α (extensionality) If Cn(α) = Cn(β) then K α = K β Renata Wassermann Knowledge Representation and Description Logic Part 3 28 / 44

49 Constructions Partial-meet contraction - remainder sets Definition The remainder set B A is defined as follows. X B A if and only if: 1 X B 2 X A 3 If X X B then X A The remainder set B A is the set of maximal subsets of B that do not imply A. Renata Wassermann Knowledge Representation and Description Logic Part 3 29 / 44

50 Constructions Partial-meet contraction - selection function A selection function then chooses some elements of B A. Definition γ is a selection function for B iff for every set A: 1 If B A then γ(b A) B A. 2 Otherwise γ(b A) = {B}. Multiple partial meet contraction is then defined as: B γ A = γ(b A) Renata Wassermann Knowledge Representation and Description Logic Part 3 30 / 44

51 Constructions Kernel contraction - kernels Definition (kernel) The kernel B A is defined as follows. X B A if and only if: 1 X B 2 X A 3 If X X then X A Renata Wassermann Knowledge Representation and Description Logic Part 3 31 / 44

52 Constructions Kernel contraction - incision function Definition An incision function for B is a function σ such that for every A: 1 σ(b A) (B A) 2 If X B A, then X σ(b A). Multiple kernel contraction is defined as: B σ A = B \ σ(b A) Renata Wassermann Knowledge Representation and Description Logic Part 3 32 / 44

53 Constructions Applying to DL AGM cannot be applied to every logic. In particular it can not be applied to SHIF and SHOIN. [Flouris 2006] Solution: substitute recovery by relevance (relevance) If β K \ K α, then there is K s. t. K α K K and α Cn(K ), but α Cn(K {β}). Good property: AGM assumptions + 5 postulates recovery and relevance are equivalent. Renata Wassermann Knowledge Representation and Description Logic Part 3 33 / 44

54 Constructions Results - contraction Representation Theorem [RW06] If the underlying logic is tarskian and compact, partial meet contraction is equivalent to the AGM postulates with relevance instead of recovery. Renata Wassermann Knowledge Representation and Description Logic Part 3 34 / 44

55 Constructions Results - contraction Representation Theorem [RW06] If the underlying logic is tarskian and compact, partial meet contraction is equivalent to the AGM postulates with relevance instead of recovery. Renata Wassermann Knowledge Representation and Description Logic Part 3 34 / 44

56 Constructions Results - contraction Representation Theorem [RW06] If the underlying logic is tarskian and compact, partial meet contraction is equivalent to the AGM postulates with relevance instead of recovery. Can we do the same for revision??? Renata Wassermann Knowledge Representation and Description Logic Part 3 34 / 44

57 AGM Revision (closure) K α = Cn(K α) (success) α K a (inclusion) K α K + α (vacuity) If K + α is consistent then K α = K + α (consistency) If α is consistent then K α is consistent. (extensionality) If Cn(α) = Cn(β) then K α = K β Renata Wassermann Knowledge Representation and Description Logic Part 3 35 / 44

58 Applying to DL Problem: no negation no Levi identity. Solution: Direct constructions. Renata Wassermann Knowledge Representation and Description Logic Part 3 36 / 44

59 Applying to DL Problem: no negation no Levi identity. Solution: Direct constructions. Definition X K α iff X maximal subset of K such that X {α} is consistent. Renata Wassermann Knowledge Representation and Description Logic Part 3 36 / 44

60 Applying to DL Problem: no negation no Levi identity. Solution: Direct constructions. Definition X K α iff X maximal subset of K such that X {α} is consistent. Definition (Revision without negation) K γ α = γ(k α) + α where γ selects at least one element of K α. Renata Wassermann Knowledge Representation and Description Logic Part 3 36 / 44

61 Properties 1 Inconsistent explosion: Whenever K is inconsistent, then for all formulas α, α Cn(K) 2 Distributivity: For all sets of formulas X, Y and W, Cn(X (Cn(Y ) Cn(W ))) = Cn(X Y ) Cn(X W ) Renata Wassermann Knowledge Representation and Description Logic Part 3 37 / 44

62 Properties 1 Inconsistent explosion: Whenever K is inconsistent, then for all formulas α, α Cn(K) 2 Distributivity: For all sets of formulas X, Y and W, Cn(X (Cn(Y ) Cn(W ))) = Cn(X Y ) Cn(X W ) Representation Theorem [RW09] If the logic is monotonic and compact and satisfies Inconsistent explosion and Distributivity, then * is a revision without negation iff it satisfies closure, success, inclusion, consistency, relevance and uniformity. (uniformity) If for all K K, K {α} is inconsistent iff K {β} is inconsistent then K K α = K K β Renata Wassermann Knowledge Representation and Description Logic Part 3 37 / 44

63 Which Logics Satisfy Distributivity? Classical logic does. But what about DLs? ALC does not. ALC with empty ABox does. not many more... Renata Wassermann Knowledge Representation and Description Logic Part 3 38 / 44

64 New characterisation Representation Theorem [RW14] If the logic is monotonic and compact and satisfies Inconsistent explosion and Distributivity, then * is a revision without negation iff it satisfies closure, success, strong inclusion, consistency, relevance and uniformity. (strong inclusion) K α (K K α) + α In classical logics this postulate is equivalent to inclusion. Renata Wassermann Knowledge Representation and Description Logic Part 3 39 / 44

65 Implementing Using Protégé, OWL API, Pellet/HermiT Flexibility of choosing operation and properties (postulates) ReviewAndContractProtegePlugin Renata Wassermann Knowledge Representation and Description Logic Part 3 40 / 44

66 Protégé Figure: Pizza ontology modeled using the Protégé tutorial Renata Wassermann Knowledge Representation and Description Logic Part 3 41 / 44

67 Figure: Initial screen of the plug-in with functionality for multiple contraction Renata Wassermann Knowledge Representation and Description Logic Part 3 42 / 44 The Protégé Revision Plugin

68 Figure: Initial screen of the plug-in and example of how to add sentences Renata Wassermann Knowledge Representation and Description Logic Part 3 43 / 44 The Protégé Revision Plugin

69 Figure: Ontology after applying package Kernel contraction Renata Wassermann Knowledge Representation and Description Logic Part 3 44 / 44 The Protégé Revision Plugin