FOUNDATIONS OF SEMANTIC WEB TECHNOLOGIES

Transcription

1 FOUNDATIONS OF SEMANTIC WEB TECHNOLOGIES OWL and Rules Sebastian Rudolph Dresden, 25 June 2014

2 Content Overview & XML Introduction into RDF RDFS Syntax & Intuition Tutorial 1 RDFS Semantics RDFS Rule-based Reasoning Tutorial 2 SPARQL Syntax & Intuition SPARQL Semantics SPARQL Algebra Tutorial 3 OWL Syntax & Intuition OWL & Description Logics OWL 2 Tutorial 4 11 APR DS5 11 APR DS6 16 APR DS6 23 APR DS6 25 APR DS5 25 APR DS6 30 APR DS6 02 MAY DS5 02 MAY DS6 09 MAY DS5 09 MAY DS6 14 MAY DS6 16 MAY DS5 16 MAY DS6 23 MAY DS5 Tableau I Tableau II Tutorial 5 Hypertableau I Hypertableau II Tutorial 6 SPARQL 1.1 SPARQL Entailment Tutorial 7 OWL & Rules Ontology Editing Ontology Engineering Tutorial 8 Linked Data & Applications Q&A Session Q&A Session 23 MAY DS6 30 MAY DS5 30 MAY DS6 4 JUN DS6 6 JUN DS5 6 JUN DS6 18 JUN DS6 20 JUN DS5 20 JUN DS6 25 JUN DS6 27 JUL DS5 27 JUL DS6 2 JUL DS6 4 JUL DS5 9 JUL DS6 11 JUL DS5 Foundations of Semantic Web Technologies slide 2 of 96

3 Semantic Web Stack / Layer Cake Each layer builds on the layers below Standardization in progress and driven by W3C Hypertext Web technologies and some Semantic Web technologies already standardized Foundations of Semantic Web Technologies slide 3 of 96

4 Two Different Paradigms Ontologies: OWL Rules: RIF, SWRL Investigation towards a Unifying Logic Foundations of Semantic Web Technologies slide 4 of 96

5 OWL 2 DL (SROIQ(D)+... ) Basic Language Constructs Classes (concepts) unary predicates Person, Woman, Mother, Uncle Properties (roles) binary predicates haschild, hasparent, haswife Individuals constants Mary, John, Bill Foundations of Semantic Web Technologies slide 5 of 96

6 OWL 2 DL (SROIQ(D)+... ) Axioms Assertions of named individuals to (complex) classes and properties Woman(Mary) hasmother(bill,mary) (Sub)class and property hierarchies ( ) Woman Person hasmother hasparent Equivalent classes ( shortcut for and ) Person Human Foundations of Semantic Web Technologies slide 6 of 96

7 OWL 2 DL (SROIQ(D)+... ) Complex Classes Class intersection ( ) Mother Person Woman Class union ( ) Parent Mother Father Class complement ( ) ChildlessPerson Person Parent Foundations of Semantic Web Technologies slide 7 of 96

8 OWL 2 DL (SROIQ(D)+... ) Complex Classes owl:thing ( ) and owl:nothing ( ) Existential quantification ( ) Man Woman Parent haschild.person haswife. Man Universal quantification ( ) NoDaughters haschild.male haswife.woman Foundations of Semantic Web Technologies slide 8 of 96

9 OWL 2 DL (SROIQ(D)+... ) Complex Classes Qualified cardinality constraints ( and ) 2hasChild.Parent(John) 4hasChild. (John) Nominals/enumerations of individuals (owl:oneof) {William} {Bill} SiblingsOfJohn {Mary, Tom} Self NarcisticPerson loves.self Foundations of Semantic Web Technologies slide 9 of 96

10 OWL 2 DL (SROIQ(D)+... ) Property Characteristics Inverse haschild instead of hasparent Symmetric hasspouse Asymmetric haschild Disjoint hasparent and haschild Reflexive hasrelative Irreflexive haschild Functional hasspouse Inverse functional hasspouse Transitive hasdescendent Foundations of Semantic Web Technologies slide 10 of 96

11 OWL 2 DL (SROIQ(D)+... ) And also... Property chains hasparent hasparent hasgrandparent Top property (U universal role) Bottom property (not part of SROIQ(D)) Datatypes (with facets) (D) Keys (not part of SROIQ(D)) Foundations of Semantic Web Technologies slide 11 of 96

12 OWL 2 Profiles fragments of OWL 2 for better computational properties OWL 2 EL: Corresponds to SROEL(D) /EL ++ Allows,,,, nominals, property chains and hierarchies, and datatypes Also allows: reflexive and transitive properties, keys Used in large biomedicine ontologies, such as SNOMED CT, or GALEN (containing complex structural descriptions) Foundations of Semantic Web Technologies slide 12 of 96

13 OWL 2 Profiles OWL 2 QL: Corresponds to DL-Lite Left of only allows: limited to Right of only allows:,, Allows property inclusions but not property chains Closely related to database technology, query answering can be realized by rewriting queries OWL 2 RL: Corresponds to DLP, a rule fragment of OWL 2 DL Well-suited for enriching RDF data Details follow later Foundations of Semantic Web Technologies slide 13 of 96

14 What we can(not) do with OWL Describe schema level knowledge: class hierarchy, properties about classes, relationships between classes, etc. Consistency and class subsumption checking Classifying individuals to classes Assert the existence of unknown individuals (i.e., those that must exist but cannot be named) Cannot specify arbitrary relationships between instances/individuals; due to the inherent tree structure of DLs Cannot express n-ary relationships between individuals with n > 2; DL extensions with n-ary predicates exist, but not part of OWL Foundations of Semantic Web Technologies slide 14 of 96

15 Rules Prominent alternative to OWL modeling: Rule-based expert systems Logic Programming/Prolog F-Logic [Kifer et al., 1995] W3C Rule Interchange Format RIF (standard since 2010) Often argued to be more intuitive for modeling: worksat(x,y),university(y), supervises(x,z),phdstudent(z) ProfessorOf(x, z) Foundations of Semantic Web Technologies slide 15 of 96

16 Rules Rules can be divided into 3 categories: First-order rules: logical implication F G Closely related to RIF-BLD (Basic Logic Dialect) Open world, declarative (first-order) semantics, monotonic Logic Programming/PROLOG rules: Close to first-order rules but with optional procedural aspects and possible built-ins Covered by RIF-FLD (Framework for Logic Dialects) Closed world, (semi-)declarative, non-monotonic Production rules: IF condition THEN action Roughly corresponds to RIF-PRD (Production Rule Dialect) Semantics varies, sometimes defined as ad hoc computational mechanisms Foundations of Semantic Web Technologies slide 16 of 96

17 First-order rules (Horn clauses) body head {}}{{}}{ A 1 A 2 A k H Each A i and H is a first-order atomic formula P(t 1,..., t m) with P a predicate symbol with arity m Each t j is a term: a variable or an expression f(s 1,..., s k ) where f is a function symbol of arity k and s 1,..., s k are terms No quantifiers; no negation Datalog rules: first-order rules without function symbols First used for deductive databases Complexity: PTime data complexity (ExpTime combined) Suitable for large datasets (and relatively small/fixed rule set) Foundations of Semantic Web Technologies slide 17 of 96

18 What we can(not) do with Rules Specify and infer arbitrary relationships between individuals, including n-ary relationships with n > 2 Many people find rules more natural for modeling Non-monotonic extensions are very well-studied, more than that of OWL (ASP solvers, etc.) Rules are usually only applied to known constants Cannot express the existence of unknown/unnamed individuals (unlike OWL) Foundations of Semantic Web Technologies slide 18 of 96

19 Can t we bring them together? Foundations of Semantic Web Technologies slide 19 of 96

20 Our Agenda... This tutorial provides a condensed exposition of the recent efforts to answer the previous main question by focusing on the following issues: What kind of rules are readily expressible in OWL? What is DL-safety notion for rules? Why does it allow one to combine rules and OWL ontologies without losing decidability? Can we integrate DL-safe rules seamlessly within OWL framework by some small syntactic extension to OWL? Can we add non-monotonic flavor to such integration between DLs and rules? Foundations of Semantic Web Technologies slide 20 of 96

21 Rules readily expressible in OWL Foundations of Semantic Web Technologies slide 21 of 96

22 Reasoning Needs z newsfrom rome. rome locadedin italy. Foundations of Semantic Web Technologies slide 22 of 96

23 Reasoning Needs We want to conclude z newsfrom rome. rome locadedin italy. z newsfrom italy. Foundations of Semantic Web Technologies slide 23 of 96

24 Reasoning Needs We want to conclude z newsfrom rome. rome locadedin italy. z newsfrom italy. Rule: newsfrom(x, y) locatedin(y, z) newsfrom(x, z) Foundations of Semantic Web Technologies slide 24 of 96

25 Reasoning Needs We want to conclude z newsfrom rome. rome locadedin italy. z newsfrom italy. Rule: In OWL: newsfrom(x, y) locatedin(y, z) newsfrom(x, z) newsfrom locatedin newsfrom (using owl:propertychainaxiom) Foundations of Semantic Web Technologies slide 25 of 96

26 Translating OWL Axioms Which OWL axioms can be encoded as rules? Let s see some examples Foundations of Semantic Web Technologies slide 26 of 96

27 Translating OWL Axioms Which OWL axioms can be encoded as rules? Foundations of Semantic Web Technologies slide 27 of 96

28 Translating OWL Axioms Which OWL axioms can be encoded as rules? A B becomes A(x) B(x) R S becomes R(x, y) S(x, y) Foundations of Semantic Web Technologies slide 28 of 96

29 Translating OWL Axioms Which OWL axioms can be encoded as rules? A B becomes A(x) B(x) R S becomes R(x, y) S(x, y) A R. S.B C becomes A(x) R(x, y) S(y, z) B(z) C(x) Foundations of Semantic Web Technologies slide 29 of 96

30 Translating OWL Axioms Which OWL axioms can be encoded as rules? A B becomes A(x) B(x) R S becomes R(x, y) S(x, y) A R. S.B C becomes A(x) R(x, y) S(y, z) B(z) C(x) A R.B becomes A(x) R(x, y) B(y) Foundations of Semantic Web Technologies slide 30 of 96

31 Rules in OWL Which OWL axioms can be encoded as rules? Foundations of Semantic Web Technologies slide 31 of 96

32 Rules in OWL Which OWL axioms can be encoded as rules? A B C becomes A(x) B(x) C(x) Foundations of Semantic Web Technologies slide 32 of 96

33 Rules in OWL Which OWL axioms can be encoded as rules? A B C becomes A(x) B(x) C(x) A B becomes A(x) B(x) f Foundations of Semantic Web Technologies slide 33 of 96

34 Rules in OWL Which OWL axioms can be encoded as rules? A B C becomes A(x) B(x) C(x) A B becomes A(x) B(x) f 1R. becomes R(x, y) R(x, z) y = z Foundations of Semantic Web Technologies slide 34 of 96

35 Rules in OWL Which OWL axioms can be encoded as rules? A B C becomes A(x) B(x) C(x) A B becomes A(x) B(x) f 1R. becomes R(x, y) R(x, z) y = z A R.{b} C becomes A(x) R(x, b) C(x) Foundations of Semantic Web Technologies slide 35 of 96

36 Rules in OWL Which OWL axioms can be encoded as rules? {a} {b} becomes a = b Foundations of Semantic Web Technologies slide 36 of 96

37 Rules in OWL Which OWL axioms can be encoded as rules? {a} {b} becomes a = b A B becomes A(x) B(x) f Foundations of Semantic Web Technologies slide 37 of 96

38 Rules in OWL Which OWL axioms can be encoded as rules? {a} {b} becomes a = b A B becomes A(x) B(x) f A B C becomes A(x) B(x) and A(x) C(x) A B C becomes A(x) C(x) and B(x) C(x) Foundations of Semantic Web Technologies slide 38 of 96

39 Rules in OWL A DL axiom α can be translated into rules if, after translating α into a first-order predicate logic expression α, and after normalizing this expression into a set of clauses M, each formula in M is a Horn clause (i.e., a rule). Issue: How complicated a translation is allowed? Foundations of Semantic Web Technologies slide 39 of 96

40 Rules in OWL A DL axiom α can be translated into rules if, after translating α into a first-order predicate logic expression α, and after normalizing this expression into a set of clauses M, each formula in M is a Horn clause (i.e., a rule). Issue: How complicated a translation is allowed? Naive translation: e.g., R S T becomes R(x, y) S(y, z) T(x, z) This essentially results in OWL RL Foundations of Semantic Web Technologies slide 40 of 96

41 Which rules can be translated into OWL axioms? Rolification Examples Formal definition: Rule Graphs Foundations of Semantic Web Technologies slide 41 of 96

42 Rolification Elephant(x) Mouse(y) biggerthan(x, y) Foundations of Semantic Web Technologies slide 42 of 96

43 Rolification Elephant(x) Mouse(y) biggerthan(x, y) Rolification of a concept A: A R A.Self Foundations of Semantic Web Technologies slide 43 of 96

44 Rolification Elephant(x) Mouse(y) biggerthan(x, y) Rolification of a concept A: A R A.Self Elephant R Elephant.Self Mouse R Mouse.Self R Elephant U R Mouse biggerthan Foundations of Semantic Web Technologies slide 44 of 96

45 Rolification A(x) R(x, y) S(x, y) becomes R A R S A(y) R(x, y) S(x, y) becomes R R A S A(x) B(y) R(x, y) S(x, y) becomes R A R R B S Foundations of Semantic Web Technologies slide 45 of 96

46 Rolification A(x) R(x, y) S(x, y) becomes R A R S A(y) R(x, y) S(x, y) becomes R R A S A(x) B(y) R(x, y) S(x, y) becomes R A R R B S Woman(x) marriedto(x, y) Man(y) hashusband(x, y) R Woman marriedto R Man hashusband careful - role regularity needs to be preserved hashusband marriedto Foundations of Semantic Web Technologies slide 46 of 96

47 Rolification worksat(x, y) University(y) supervises(x, z) PhDStudent(z) professorof(x, z) R worksat.university supervises R PhDStudent professorof Foundations of Semantic Web Technologies slide 47 of 96

48 Rules in OWL 2 Man(x) hasbrother(x, y) haschild(y, z) Uncle(x) Man hasbrother. haschild. Uncle NutAllergic(x) NutProdcut(y) dislikes(x, y) NutAllergic nutallergic.self NutProduct nutproduct.self nutallergic U nutproduct dislikes dislikes(x, z) Dish(y) contains(y, z) dislikes(x, y) Dish dish.self dislikes contains dish dislikes Foundations of Semantic Web Technologies slide 48 of 96

49 So how can we pinpoint this? Tree-shaped bodies (variables) First argument of the conclusion is the root C(x) R(x, a) S(x, y) D(y) T(y, a) E(x) C R.{a} S.(D T.{a}) E Foundations of Semantic Web Technologies slide 49 of 96

50 So how can we pinpoint this? C(x) R(x, a) S(x, y) D(y) T(y, a) V(x, y) Foundations of Semantic Web Technologies slide 50 of 96

51 So how can we pinpoint this? C(x) R(x, a) S(x, y) D(y) T(y, a) V(x, y) C R.{a} R 1.Self D T.{a} R 2.Self R 1 S R 2 V Foundations of Semantic Web Technologies slide 51 of 96

52 So how can we pinpoint this? Rule graph: C(x) R(x, a) S(x, y) D(y) T(y, a) P(x, y) Graph analysis: determine whether a rule is expressible within a given profile Automatic Transformation Foundations of Semantic Web Technologies slide 52 of 96

53 DLs Rules: EL ++ R 1 (x, y) C 1 (y) R 2 (y, w) R 3 (y, z) C 2 (z) R 4 (x, x) C 3 (x) Foundations of Semantic Web Technologies slide 53 of 96

54 DLs Rules: EL ++ R 1 (x, y) C 1 (y) R 2 (y, w) R 3 (y, z) C 2 (z) R 4 (x, x) C 3 (x) Foundations of Semantic Web Technologies slide 54 of 96

55 DLs Rules: EL ++ R 1 (x, y) C 1 (y) R 2 (y, w) R 3 (y, z) C 2 (z) R 4 (x, x) C 3 (x) R 1.(C 1 R 2. R 3.C 2 ) R 4.Self C 3 Foundations of Semantic Web Technologies slide 55 of 96

56 DLs Rules: SROIQ R 1 (y, x) C 1 (y) R 2 (w, y) R 3 (y, z) C 2 (z) R 4 (x, x) C 3 (x) Foundations of Semantic Web Technologies slide 56 of 96

57 DLs Rules: SROIQ R 1 (y, x) C 1 (y) R 2 (w, y) R 3 (y, z) C 2 (z) R 4 (x, x) C 3 (x) Foundations of Semantic Web Technologies slide 57 of 96

58 DLs Rules: SROIQ R 1 (y, x) C 1 (y) R 2 (w, y) R 3 (y, z) C 2 (z) R 4 (x, x) C 3 (x) R 1.(C 1 R 2. R 3.C 2 ) R 4.Self C 3 Foundations of Semantic Web Technologies slide 58 of 96

59 Nominal Schemas Foundations of Semantic Web Technologies slide 59 of 96

60 What we learned about rules expressible in OWL Rules with tree-shaped body can be expressed in DL, role conjunction allows DLs to express some rules with non-tree-shaped body, but many rules are not covered. Foundations of Semantic Web Technologies slide 60 of 96

61 Clique of 4 R 1 (x, y) R 2 (x, z) R 3 (x, w) R 4 (y, z) R 5 (y, w) R 6 (w, z) C(x) Foundations of Semantic Web Technologies slide 61 of 96

62 Clique of 4 R 1 (x, y) R 2 (x, z) R 3 (x, w) R 4 (y, z) R 5 (y, w) R 6 (w, z) C(x) Foundations of Semantic Web Technologies slide 62 of 96

63 Nominal Schemas A Better Uncle for OWL [ Krötzsch et al; WWW 2011 ] hasparent(x, y) married(y, z) hasparent(x, z) C(x) hasparent. married.{z} hasparent.{z} C Foundations of Semantic Web Technologies slide 63 of 96

64 Nominal Schemas A Better Uncle for OWL [ Krötzsch et al; WWW 2011 ] hasparent(x, y) married(y, z) hasparent(x, z) C(x) hasparent. married.{z} hasparent.{z} C {z} only binds to known/named individuals! Covers DL-safe datalog (arbitrary arity of predicates) Foundations of Semantic Web Technologies slide 64 of 96

65 Complex Rules to OWL Theorem Any rule R containing m different free variables, where m > 3, can be directly expressed in DL using n nominal schemas s.t. n m 2. Foundations of Semantic Web Technologies slide 65 of 96

66 DL-safe Rules How about simply adding rules as-is to the ontology? Foundations of Semantic Web Technologies slide 66 of 96

67 DL-safe Rules How about simply adding rules as-is to the ontology? Problem: although the DL-part and rule-part are both decidable, the combination is undecidable! Foundations of Semantic Web Technologies slide 67 of 96

68 DL-safe Rules How about simply adding rules as-is to the ontology? Problem: although the DL-part and rule-part are both decidable, the combination is undecidable! Work around: weaken the rule semantics? Foundations of Semantic Web Technologies slide 68 of 96

69 DL-safety Decidability guaranteed if rules only operate on named individuals Named individuals are finite. Foundations of Semantic Web Technologies slide 69 of 96

70 DL-safety Decidability guaranteed if rules only operate on named individuals Named individuals are finite. DL-safety: variables in the rules refer to only named individuals in the OWL ontology. Foundations of Semantic Web Technologies slide 70 of 96

71 DL-safety Decidability guaranteed if rules only operate on named individuals Named individuals are finite. DL-safety: variables in the rules refer to only named individuals in the OWL ontology. In the rule: R(u, x) A(x) S(u, y) T(x, z) T(y, z) R(u, z) S(x, y) B(u) the variables u, x, y, z refer only to named individuals. Foundations of Semantic Web Technologies slide 71 of 96

72 DL-safety Decidability guaranteed if rules only operate on named individuals Named individuals are finite. DL-safety: variables in the rules refer to only named individuals in the OWL ontology. In the rule: R(u, x) A(x) S(u, y) T(x, z) T(y, z) R(u, z) S(x, y) B(u) the variables u, x, y, z refer only to named individuals. Approach taken by DL-safe SWRL. Foundations of Semantic Web Technologies slide 72 of 96

73 Revisiting DL-safety: can it be relaxed? R(u, x) A(x) S(u, y) T(x, z) T(y, z) R(u, z) S(x, y) B(u) Rule body forms complicated graph: y S T u R z R S T x A Foundations of Semantic Web Technologies slide 73 of 96

74 Revisiting DL-safety: can it be relaxed? R(u, x) A(x) S(u, y) T(x, z) T(y, z) R(u, z) S(x, y) B(u) If y and z refer to named individuals, say a, b, it represents: R(u, x) A(x) S(u, a) T(x, a) T(a, a) R(u, a) S(x, a) B(u) R(u, x) A(x) S(u, a) T(x, b) T(a, b) R(u, b) S(x, a) B(u) R(u, x) A(x) S(u, b) T(x, a) T(b, a) R(u, a) S(x, b) B(u) R(u, x) A(x) S(u, b) T(x, b) T(b, b) R(u, b) S(x, b) B(u) Foundations of Semantic Web Technologies slide 74 of 96

75 Revisiting DL-safety: can it be relaxed? R(u, x) A(x) S(u, y) T(x, z) T(y, z) R(u, z) S(x, y) B(u) R(u, x) A(x) S(u, a) T(x, a) T(a, a) R(u, a) S(x, a) B(u) R(u, x) A(x) S(u, a) T(x, b) T(a, b) R(u, b) S(x, a) B(u) R(u, x) A(x) S(u, b) T(x, a) T(b, a) R(u, a) S(x, b) B(u) R(u, x) A(x) S(u, b) T(x, b) T(b, b) R(u, b) S(x, b) B(u) Only y and z need to be DL-safe. Expressible in OWL EL. Foundations of Semantic Web Technologies slide 75 of 96

76 Nominal schemas: DL-safety only to some variables In the rule R(u, x) A(x) S(u, y) T(x, z) T(y, z) R(u, z) S(x, y) B(u) only y and z need to be DL-safe to be expressible in OWL EL. Expressing it in OWL EL need multiple axioms: R.{a} R.(A S.{a} T.{a}) S.({a} T.{a}) B R.{b} R.(A S.{a} T.{b}) S.({a} T.{b}) B R.{a} R.(A S.{b} T.{a}) S.({b} T.{a}) B R.{b} R.(A S.{b} T.{b}) S.({b} T.{b}) B With nominal schemas, the above 4 axioms can be condensed: R.{z} R.(A S.{y} T.{z}) S.({y} T.{z}) B Foundations of Semantic Web Technologies slide 76 of 96

77 Nominal schemas: syntax and semantics Nominal schemas: a variable nominal construct in the form of {x} where x is a variable. Foundations of Semantic Web Technologies slide 77 of 96

78 Nominal schemas: syntax and semantics Nominal schemas: a variable nominal construct in the form of {x} where x is a variable. Semantically, each occurrence of a nominal schema represents all possible nominals in the ontology. Foundations of Semantic Web Technologies slide 78 of 96

79 Nominal schemas: syntax and semantics Nominal schemas: a variable nominal construct in the form of {x} where x is a variable. Semantically, each occurrence of a nominal schema represents all possible nominals in the ontology. If an axiom α contains n different nominal schemas (each may occur more than once) while the ontology has m different named individuals, then α represents m n different axioms, each is obtained by substituting nominal schemas with named individuals. Foundations of Semantic Web Technologies slide 79 of 96

80 Nominal schemas: syntax and semantics Nominal schemas: a variable nominal construct in the form of {x} where x is a variable. Semantically, each occurrence of a nominal schema represents all possible nominals in the ontology. If an axiom α contains n different nominal schemas (each may occur more than once) while the ontology has m different named individuals, then α represents m n different axioms, each is obtained by substituting nominal schemas with named individuals. All those m n axioms are called groundings of α. Foundations of Semantic Web Technologies slide 80 of 96

81 What can we express with nominal schemas? Let SROELV(, ) = SROEL(, ) + nominal schemas. Foundations of Semantic Web Technologies slide 81 of 96

82 What can we express with nominal schemas? Let SROELV(, ) = SROEL(, ) + nominal schemas. SROELV(, ) covers: DL-safe Datalog rules with predicates of arbitrary arity is also in SROELV. OWL 2 EL without datatypes DL-safe OWL 2 RL without datatypes but, preserves only ABox entailments (the main inference task for OWL 2 RL) most of OWL 2 QL Foundations of Semantic Web Technologies slide 82 of 96

83 Complexity bounds Reasoning for SROIQV = SROIQ + nominal schemas, is theoretically no worse than SROIQ [Krötzsch et al., WWW11] Foundations of Semantic Web Technologies slide 83 of 96

84 Complexity bounds Reasoning for SROIQV = SROIQ + nominal schemas, is theoretically no worse than SROIQ [Krötzsch et al., WWW11] Reasoning for SROELV is: still polynomial (like SROEL) if the number of occurrences of different nominal schemas in an axiom is bounded by a fixed constant; in general, it is exponential (c.f., combined complexity of Datalog is ExpTime) Foundations of Semantic Web Technologies slide 84 of 96

85 An Efficient Implementation for Nominal Schemas? Foundations of Semantic Web Technologies slide 85 of 96

86 Naive grounding Naive reasoning directly from the semantics: Foundations of Semantic Web Technologies slide 86 of 96

87 Naive grounding Naive reasoning directly from the semantics: Ground each axiom containing nominal schemas into exponentially many axioms without nominal schemas. The resulting ontology is in standard DL or OWL and equivalent to the original one. Use any existing reasoning algorithm for the corresponding standard DL on the resulting ontology. Foundations of Semantic Web Technologies slide 87 of 96

88 Naive grounding Naive reasoning directly from the semantics: Ground each axiom containing nominal schemas into exponentially many axioms without nominal schemas. The resulting ontology is in standard DL or OWL and equivalent to the original one. Use any existing reasoning algorithm for the corresponding standard DL on the resulting ontology. Practically inefficient. Foundations of Semantic Web Technologies slide 88 of 96

89 Naive grounding example hparent. married.{z} hparent.{z} C Foundations of Semantic Web Technologies slide 89 of 96

90 Naive grounding example hparent. married.{z} hparent.{z} C Full grounding: hparent. married.{a} hparent.{a} C hparent. married.{b} hparent.{b} C hparent. married.{c} hparent.{c} C where a, b, and c are the only individuals in the knowledge base Foundations of Semantic Web Technologies slide 90 of 96

91 Naive grounding example R 1.( R 4.{z} R 5.({w} R 6.{z})) R 2.{z} R 3.{w} C Foundations of Semantic Web Technologies slide 91 of 96

92 Naive grounding example R 1.( R 4.{z} R 5.({w} R 6.{z})) R 2.{z} R 3.{w} C Full grounding: R 1.( R 4.{a} R 5.({a} R 6.{a})) R 2.{a} R 3.{a} C R 1.( R 4.{a} R 5.({b} R 6.{a})) R 2.{a} R 3.{b} C R 1.( R 4.{a} R 5.({c} R 6.{a})) R 2.{a} R 3.{c} C R 1.( R 4.{a} R 5.({b} R 6.{a})) R 2.{a} R 3.{b} C R 1.( R 4.{b} R 5.({c} R 6.{b})) R 2.{a} R 3.{c} C where a, b, and c are the only individuals in the knowledge base Foundations of Semantic Web Technologies slide 92 of 96

93 Defining Optimizations Delayed grounding: [ Adila et al. at RR 2012 ] Ordered Resolution: [ Cong et al. at JIST 2012 ] Foundations of Semantic Web Technologies slide 93 of 96

94 References Rules expressible in OWL: Description logic programs: combining logic programs with description logic. Benjamin Grosof, Ian Horrocks, Raphael Volz, and Stefan Decker, WWW 2003, Description logic rules. Markus Krötzsch, Sebastian Rudolph, and Pascal Hitzler, ECAI 2008, ELP: tractable rules for OWL 2. Markus Krötzsch, Sebastian Rudolph, and Pascal Hitzler, ISWC 2008, Cheap boolean role constructors for description logics. Sebastian Rudolph, Markus Krötzsch, and Pascal Hitzler, JELIA 2008, Description logic rules. Markus Krötzsch, Studies on the Semantic Web, Vol. 08, Foundations of Semantic Web Technologies slide 94 of 96

95 References Rules expressible in OWL and with nominal schemas: A better uncle for OWL: nominal schemas for integrating rules and ontologies. Markus Krötzsch, Frederick Maier, Adila A. Krisnadhi, and Pascal Hitzler, WWW 2011, Extending description logic rules. David Carral Martínez and Pascal Hitzler, ESWC 2012, A tableau algorithm for description logics with nominal schemas. Adila Krisnadhi and Pascal Hitzler, RR2012, A resolution procedure for description logics with nominal schemas. Cong Wang and Pascal Hitzler, JIST 2012, Foundations of Semantic Web Technologies slide 95 of 96

96 References Nonmonotonic Extensions: Description logics of minimal knowledge and negation as failure. Francesco M. Donini, Daniele Nardi, and Riccardo Rosati, ACM Transactions on Computational Logic, 3(2), 2002, Reconciling Description Logics and Rules. Boris Motik and Riccardo Rosati, Journal of the ACM, 57(5), 2010, Local closed world reasoning with description logics under the well-founded semantics. Matthias Knorr, José J. Alferes, and Pascal Hitzler, Artificial Intelligence, 175(9-10), 2011, Reconciling OWL and non-monotonic rules for the Semantic Web. Matthias Knorr, Pascal Hitzler, and Frederick Maier, ECAI 2012, Foundations of Semantic Web Technologies slide 96 of 96