OWL + Rules =..? David Carral 1 Matthias Knorr 2 Adila Krisnadhi 1. 10th Extended Semantic Web Conference (ESWC) 2013

Transcription

1 OWL + Rules =..? David Carral 1 Matthias Knorr 2 Adila Krisnadhi 1 1 Kno.e.sis Center, Wright State University, USA 2 CENTRIA, Universidade Nova de Lisboa, Portugal 10th Extended Semantic Web Conference (ESWC) 2013

2 Support David Carral and Adila Krisnadhi: At Kno.e.sis Center, supported by NSF grant III: Small: TROn - Tractable Reasoning with Ontologies Matthias Knorr: At Centria, supported by FCT grant ERRO. efficient reasoning with rules and ontologies

3 Slides Latest version available from

4 Introduction

5 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

6 Two Different Paradigms Ontologies: OWL Rules: RIF, SWRL Investigation towards a Unifying Logic

7 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

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

9 OWL 2 DL (SROIQ(D)+... ) Complex Classes Class intersection ( ) Mother Person Woman Class union ( ) Parent Mother Father Class complement ( ) ChildlessPerson Person Parent

10 OWL 2 DL (SROIQ(D)+... ) Complex Classes owl:thing ( ) and owl:nothing ( ) Existential quantification ( ) Man Woman Parent haschild.person Universal quantification ( ) haswife. Man NoDaughters haschild.male haswife.woman

11 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

12 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

13 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)) HasKey(: Person()(: hasssn))

14 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)

15 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

16 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

17 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)

18 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

19 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 tj 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)

20 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)

21 Can t we bring them together?

22 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?

23 Some historical bits (not complete...) : Description Logics (DLs) turn into the W3C OWL standard (logic programming still used for modeling ontologies); 2003: Description Logic Programs (DLP) [Grosof et al., WWW03] intersection of OWL 1 DL and Datalog; 2004: Semantic Web Rules Language (SWRL) OWL plus unbounded first-order rules, yet undecidable; 2004: dl-programs [Eiter et al., KR04] OWL + non-monotonic rules modularly with limited interaction; 2005: DL-safety [Motik et al., JWS05] DL-safe SWRL is decidable; 2006: DL+log [Rosati, KR06] weakly-dl-safe (non-monotonic) rules plus OWL, still separate semantics for each part;

24 Some historical bits (not complete...) 2007: Hybrid MKNF by [Motik and Rosati, IJCAI07] seamless integration of OWL and non-monotonic (DL-safe) rules; : standardization effort of OWL 2 by W3C; : Description Logic Rules and ELP [Krötzsch et al., ECAI08; ISWC08] and [Rudolph et al., JELIA08] significantly extended DLP; 2011: Well-founded Semantics for Hybrid MKNF [Knorr et al., AI11] tractable for polynomial DLs; 2011: Nominal schemas by [Krötzsch et al., WWW11] strongly integrate OWL 2 and DL-safe SWRL; 2012: Extending DL rules [Carral and Hitzler, ESWC12]; 2012: Nominal schemas with non-monotonic extensions [Knorr et al., ECAI12].

25 Rules readily expressible in OWL

26 Reasoning Needs z newsfrom rome. rome locadedin italy.

27 Reasoning Needs We want to conclude z newsfrom rome. rome locadedin italy. z newsfrom italy.

28 Reasoning Needs We want to conclude Rule: z newsfrom rome. rome locadedin italy. z newsfrom italy. newsfrom(x, y) locatedin(y, z) newsfrom(x, z)

29 Reasoning Needs We want to conclude Rule: In OWL: z newsfrom rome. rome locadedin italy. z newsfrom italy. newsfrom(x, y) locatedin(y, z) newsfrom(x, z) (using owl:propertychainaxiom) newsfrom locatedin newsfrom

30 Reasoning Needs e.g. knowledge base of authors and papers <paper> hasauthor <Author>

31 Reasoning Needs e.g. knowledge base of authors and papers <paper> hasauthor <Author> insufficient because author order is missing use of RDF-lists not satisfactory due to the lack of formal semantics.

32 Reasoning Needs e.g. knowledge base of authors and papers <paper> hasauthor <Author> insufficient because author order is missing use of RDF-lists not satisfactory due to the lack of formal semantics. better: <paper> hasauthornumbered :x ; :x authornumber nˆˆxsd:positiveinteger ; authorname <author>.

33 Reasoning Needs <paper> hasauthornumbered :x ; :x authornumber nˆˆxsd:positiveinteger ; authorname <author>.

34 Reasoning Needs <paper> hasauthornumbered :x ; :x authornumber nˆˆxsd:positiveinteger ; authorname <author>. In OWL: Paper hasauthornumbered.numberedauthor NumberedAuthor = 1 authornumber. < xsd:positiveinteger > NumberedAuthor = 1 authorname.name

35 Reasoning Needs <paper> hasauthornumbered :x ; :x authornumber nˆˆxsd:positiveinteger ; authorname <author>. In OWL: Paper hasauthornumbered.numberedauthor NumberedAuthor = 1 authornumber. < xsd:positiveinteger > NumberedAuthor = 1 authorname.name hasauthornumbered authorname hasauthor

36 Reasoning Needs Property hasfirstauthor:

37 Reasoning Needs Property hasfirstauthor: Paper(x) hasauthornumbered(x, y) authornumber(y, 1) authorname(y, z) hasfirstauthor(x, z)

38 Reasoning Needs Property hasfirstauthor: Paper(x) hasauthornumbered(x, y) authornumber(y, 1) in OWL: authorname(y, z) hasfirstauthor(x, z) Paper reflexivepaper.self authornumber.{1} FirstAuthor FirstAuthor reflexivefirstauthor.self reflexivepaper hasauthornumbered reflexivefirstauthor authorname hasfirstauthor

39 Graphical Example

40 Graphical Example Paper reflexivepaper.self

41 Graphical Example authornumber.{1} FirstAuthor

42 Graphical Example FirstAuthor reflexivefirstauthor.self

43 Graphical Example reflexivepaper hasauthornumbered reflexivefirstauthor authorname hasfirstauthor

44 Reasoning as first-class citizen Why would we want to have knowledge/rules such as newfrom(x, y) locatedin(y, z) newsfrom(x, z) if we can also just do this with some software code?

45 Reasoning as first-class citizen Why would we want to have knowledge/rules such as newfrom(x, y) locatedin(y, z) newsfrom(x, z) if we can also just do this with some software code? It declaratively describes what you do.

46 Reasoning as first-class citizen Why would we want to have knowledge/rules such as newfrom(x, y) locatedin(y, z) newsfrom(x, z) if we can also just do this with some software code? It declaratively describes what you do. It separates knowledge (as knowledge base) from programming.

47 Reasoning as first-class citizen Why would we want to have knowledge/rules such as newfrom(x, y) locatedin(y, z) newsfrom(x, z) if we can also just do this with some software code? It declaratively describes what you do. It separates knowledge (as knowledge base) from programming. It makes knowledge shareable.

48 Reasoning as first-class citizen Why would we want to have knowledge/rules such as newfrom(x, y) locatedin(y, z) newsfrom(x, z) if we can also just do this with some software code? It declaratively describes what you do. It separates knowledge (as knowledge base) from programming. It makes knowledge shareable. It makes knowledge easier to maintain.

49 Translating OWL Axioms Which OWL axioms can be encoded as rules? Let s see some examples

50 Translating OWL Axioms Which OWL axioms can be encoded as rules?

51 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)

52 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)

53 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)

54 Rules in OWL Which OWL axioms can be encoded as rules?

55 Rules in OWL Which OWL axioms can be encoded as rules? A B C becomes A(x) B(x) C(x)

56 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

57 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

58 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)

59 Rules in OWL Which OWL axioms can be encoded as rules? {a} {b} becomes a = b

60 Rules in OWL Which OWL axioms can be encoded as rules? {a} {b} becomes a = b A B becomes A(x) B(x) f

61 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)

62 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?

63 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: DLP plus some more (OWL 2) e.g., R S T becomes R(x, y) S(y, z) T (x, z) This essentially results in OWL RL

64 Which rules can be translated into OWL axioms? Rolification Examples Formal definition: Rule Graphs

65 Rolification Elephant(x) Mouse(y) biggerthan(x, y)

66 Rolification Elephant(x) Mouse(y) biggerthan(x, y) Rolification of a concept A: A R A.Self

67 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

68 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

69 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

70 Rolification worksat(x, y) University(y) supervises(x, z) PhDStudent(z) professorof(x, z) R worksat.university supervises R PhDStudent professorof

71 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

72 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

73 So how can we pinpoint this? C(x) R(x, a) S(x, y) D(y) T (y, a) V (x, y)

74 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

75 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

76 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)

77 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)

78 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

79 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)

80 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)

81 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

82 Extending DL Rules Extended Description Logic Rules [ Carral and Hitzler, ESWC12 ] Conjunction over complex roles? Some cyclic rules

83 Extending DL: SROIQ( ) hasfather(x, y) hasbrother(y, z) hasteacher(x, z) TaughtByUncle(x)

84 Extending DLs: SROIQ( ) hasfather(x, y) hasbrother(y, z) hasteacher(x, z) TaughtByUncle(x)

85 Extending DLs: SROIQ( ) hasfather(x, y) hasbrother(y, z) hasteacher(x, z) TaughtByUncle(x) Equivalent Translation: hasfather(x, y) hasbrother(y, z) hasuncle(x, z) hasuncle(x, z) hasteacher(x, z) TaughtbyUncle(x) hasfather hasbrother hasuncle

86 Extending DLs: SROIQ( ) Middle rule: hasuncle(x, z) hasteacher(x, z) TaughtbyUncle(x) Equivalent Translation: hasuncle(x, z) hasteacher(x, z) hasuncleandteacher(x, z) hasuncleandteacher(x, z) TaughtbyUncle(x) hasuncle hasteacher hasuncle hasuncleandteacher. TaughtByUncle

87 Extending DLs: SROIQ( ) hasfather(x, y) hasbrother(y, z) hasteacher(x, z) TaughtByUncle(x) hasfather hasbrother hasuncle hasuncle hasteacher hasuncle hasteacheranduncle. TaughtByUncle

88 Conclusions and Future Work Definition of DL Rules Automatic Transformation: implementation Extending DL Rules:

89 Nominal Schemas

90 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.

91 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)

92 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)

93 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

94 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)

95 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.

96 DL-safe Rules How about simply adding rules as-is to the ontology?

97 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!

98 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?

99 DL-safety Decidability guaranteed if rules only operate on named individuals Named individuals are finite.

100 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.

101 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.

102 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.

103 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 u S R T z R S T x A

104 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)

105 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.

106 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

107 Nominal schemas: syntax and semantics Nominal schemas: a variable nominal construct in the form of {x} where x is a variable.

108 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.

109 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.

110 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.

111 What can we express with nominal schemas? Let SROELV(, ) = SROEL(, ) + nominal schemas.

112 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

113 Complexity bounds Reasoning for SROIQV = SROIQ + nominal schemas, is theoretically no worse than SROIQ [Krötzsch et al., WWW11]

114 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)

115 An Efficient Implementation for Nominal Schemas

116 Naive grounding Naive reasoning directly from the semantics:

117 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.

118 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.

119 Naive grounding example hparent. married.{z} hparent.{z} C

120 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

121 Naive grounding example R 1.( R 4.{z} R 5.({w} R 6.{z})) R 2.{z} R 3.{w} C

122 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

123 Defining Optimizations Delayed grounding: [ Cong et al. at JIST 2012 ] Ordered Resolution: [ Adila et al. at RR 2012 ]

124 Towards an Efficient Algorithm for DL Extended with Nominal Schemas Algorithm extension presented at [ Markus Krotzsch at Jelia 2010 ] Define a mapping from a normalized OWL EL knowledge base to a Datalog program. Make use of an existing Datalog engine to derive all inferences.

125 OWL EL Normal Form Transformation

126 Input Transformation

127 Input Transformation: some mappings A C subclass(a, C)

128 Input Transformation: some mappings A C subclass(a, C) R S subrole(r, T )

129 Input Transformation: some mappings A C subclass(a, C) R S subrole(r, T ) A B R subprod(a, B, R)

130 Set of Rules

131 Set of Rules: some mappings subclass(y, z) inst(x, y) inst(x, z)

132 Set of Rules: some mappings subclass(y, z) inst(x, y) inst(x, z) subrole(v, w) triple(x, v, x ) inst(v, w)

133 Set of Rules: some mappings subclass(y, z) inst(x, y) inst(x, z) subrole(v, w) triple(x, v, x ) inst(v, w) subprod(y 1, y 2, w) inst(x, y 1 ) inst(z, y 2 ) triple(x, w, x )

134 Extending the Algorithm: Nominal Schemas Normalization and Input Transformation. Set of Rules. Set of facts + Set of Rules + Set of Rules derived from axioms with nominal schemas.

135 Mapping for Axioms Containing Nominal Schemas hasparent(x, y) married(y, z) hasparent(x, z) C(x)

136 Mapping for Axioms Containing Nominal Schemas hasparent(x, y) married(y, z) hasparent(x, z) C(x) hasparent. married.{z} hasparent.{z} C

137 Mapping for Axioms Containing Nominal Schemas hasparent(x, y) married(y, z) hasparent(x, z) C(x) hasparent. married.{z} hasparent.{z} C triple(x, hasparent, y) triple(y, married, z) triple(x, hasparent, z) nom(z) inst(x, C)

138 Execution Example hasfather(mike, Joe) hasparent(mike, Mary) married(joe, Mary) hasfather hasparent hasparent(x, y) married(y, z) hasparent(x, z) C(x)

139 Execution Example triple(mike, hasfather, Joe) triple(mike, hasparent, Mary) triple(joe, married, Mary) subrole(hasfather, hasparent) subrole(v, w) triple(x, v, x ) triple(x, w, x ) triple(x, hasparent, y) triple(y, married, z) triple(x, hasparent, z) nom(z) inst(x, C)

140 Experimental Results

141 Conclusions DL Logic Rules Allows to express simple rules Push the DL fragments Nominal Schemas Convoluted Rules Efficient implementations

142 Non-monotonic rule extension to OWL

143 Why Non-monotonic Extensions? Open World Assumption (OWA) in general preferable on the Web Without clinical test, no assumptions can be made on outcome But with complete knowledge, Closed World Assumption (CWA) is better Patient s medication is fully known Requirement for local closure of certain information

144 Why Non-monotonic Extensions? HeartLeft HeartRight Person HeartLeft HeartRight Person has.spinalcolumn has.spinalcolumn Vertebrate Person(Bob) Vertebrate(Bob), Person Vertebrate, and x.spinalcolumn(x) derivable

145 Why Non-monotonic Extensions? Person HeartLeft HeartRight HeartLeft HeartRight Person has.spinalcolumn has.spinalcolumn Vertebrate Person(Bob) SSN OK(x) hasssn(x, y) f Person(x), notssn OK(x) HeartLeft(x) Vertebrate(x), notheartright(x) HeartLeft(Bob); hasssn(bob, y) for ground y required Model defaults and exceptions, and integrity constraints

146 Why focus on Rules? Non-monotonic extensions have been studied directly for DLs Extensions for, e.g., default logic, epistemic logic, and circumscription But non-trivial and few results in terms of implementations (apart from ad-hoc solutions in, e.g., recommender systems) Rules easily extended to non-monotonic features Well-studied field Logic Programming including fast reasoners Leverage the Knowledge and Reasoners available

147 Ways to Combine Ontologies and Rules Non-trivial matter because of Non-monotonicity Rules on top of ontologies First define concepts then define rules on top Rules deal with ontologies as external code Separate rule and ontology predicates Tight (full) integration Allow for defining predicates both in the ontology and the rule layer Rules may use concepts defined in the ontology Ontology can use predicates of the rules Modular combination Trades some expressiveness for an easier to implement interface integration

148 Difficulties of Tight Integration Rule languages (LP) use CWA Ontology languages (DL) use OWA What if a predicate is defined using both DL and LP? Should its negation be assumed by default? Or should it be kept open? How exactly can one define what is CWA or OWA in this context?

149 Hybrid MKNF Knowledge Bases Seamless integration, expressive, yet competitive w.r.t. computational complexity Introduced in [Motik and Rosati, IJCAI07] (extended in [Motik and Rosati, JACM10]) Based on Logics of Minimal Knowledge and Negation as Failure: first-order logics with equality and modal operators K and not [Lifschitz, IJCAI91] Consist of a (decidable) DL knowledge base O and a finite set of rules, P, of the form KH 1... KH l KA 1,..., KA n, notb 1,..., notb m Combined decidability ensured by DL-safety (restriction of application of rules to known individuals)

150 MKNF KB Example PortCity(Barcelona) OnSea(Barcelona, Mediterranean) PortCity(Hamburg) NonSeaSideCity(Hamburg) RainyCity(Manchester) Has(Manchester, AquaticsCenter) Recreational(AquaticsCenter) SeaSideCity Has.Beach Beach Recreational Has.Recreational RecreationalCity KSeaSideCity(x) KPortCity(x), notnonseasidecity(x) KinterestingCity(x) KRecreationalCity(x), notrainycity(x) KhasOnSea(x) KOnSea(x, y) Kfalse KSeaSideCity(x), nothasonsea(x) KsummerDestination(x) KinterestingCity(x), KOnSea(x, y)

151 Properties of Hybrid MKNF Generalizes/captures (sometimes not entirely) quite a number of different approaches Faithful w.r.t. Stable Models for empty O and w.r.t. OWL for empty P Data complexity of instance checking in MKNF: rules DL = DL P DL conp definite P P conp stratified P P p 2 normal conp conp Π p 2 disjunctive Π p 2 Π p 2 Π p 2

152 Problems of two-valued Hybrid MKNF Models have to be guessed and checked Unrestricted rules increase computational complexity Queries for particular information require computation of the entire model Limited robustness, e.g., w.r.t. merging of KBs (Ku notu) [Knorr et al., AI11] provides alternative based on well-founded semantics for non-disjunctive logic programs

153 Stable Models vs. Well-Founded Model in LP p notq q notp a notb b has two stable models {p, b} and {q, b}, while the unique well-founded model assigns t to b, f to a, and u to both p and q. For p notp q notq a notb b the well-founded model is the same, but there are no stable models!

154 Stable Models vs. Well-Founded Model in LP Stable Models/Answer Sets More expressive language More derivable information Fast ASP solvers available Well-founded Model Lower computational complexity always exists top-down derivations possible Similar for combinations of rules and ontologies

155 Properties of Well-Founded MKNF Sound w.r.t. two-valued MKNF semantics Faithful w.r.t. first-order semantics for empty P and w.r.t. the Well-Founded Semantics for empty O given complexity C for instance checking in O we obtain a data complexity P C ; for C =P, polynomial data complexity Top-down procedure SLG(O) [Alferes et al., ACM TOCL13] combining a DL reasoner and XSB Prolog, special procedures defined for OWL 2 QL and a large fragment of OWL 2 EL

156 Non-monotonic DL Extension with MKNF ALCK N F [Donini et al., ACM TOCL02] ALC with MKNF logic-style modal operators K minimal knowledge and A autoepistemic assumption (corresponds to not) K can be used to derive new information, A to verify if information is already known Different expressiveness compared to Hybrid MKNF

157 Non-monotonic Features of ALCK N F from [Donini et al., ACM TOCL02] Defaults: KI K(employee belongsto.programmingdept) Amanager K(engineer mathematican) Integrity Constraints: Kemployee (Amale Afemale) Kemployee ASSN.Avalid

158 Non-monotonic Features of ALCK N F Concept and Role Closure UScitizen(Paula) UScitizen(Carl) Manages(Ann, Marc) UScitizen(Marc) adding ( KManages.KUScitizen)(Ann) closes the role adding KManages.A UScitizen(Ann) closes the concept

159 Can We find a joint formalism for both MKNF extensions? Contribute towards a unifying logic Reconcile OWL and Datalog together with CWA extensions (on both sides) Usage of one (DL-style) syntax in opposite to common hybrid languages Coverage of many different previous approaches

160 SROIQV(B s, )K N F OWL 2 DL (SROIQ) with concept products ( - [Krötzsch, SSW10]) and Boolean constructors over simple roles (B s - [Rudolph et al., JELIA08]) Nominal schemas (V - [Krötzsch et al., WWW11]) - variable nominals that can only bind to known individuals MKNF logic-style modal operators K minimal knowledge and A autoepistemic assumption (K N F - from ALCK N F [Donini et al., ACM TOCL02])

161 Syntax signature Σ = N I, N C, N R, N V Definition The set of SROIQV(B s, )K N F concepts C and (simple/non-simple) SROIQV(B s, )K N F roles R (R s /R n ) are defined by the following grammar. R s ::=N s R (Ns R ) U N C N C R s R s R s R s R s KR s AR s R n ::=N n R (Nn R ) U N C N C KR n AR n R ::=R s R n C ::= N C {N I } {N V } C C C C C R.C R.C R s.self k R s.c k R s.c KC AC

162 Semantics Principal Notions Based on interpretations I = ( I, I) plus variable assignments (for nominal variables) mapping each variable to the interpretation of one element in N I Variant of Standard Name Assumption applied: essentially I is a bijective function on N I while still allowing that elements of N I may be identified ( only one ) An MKNF structure is a triple (I, M, N ) where I is an interpretation, M and N are sets of interpretations, and I and all interpretations in M and N are defined over. For any such (I, M, N ) and assignment Z, the function (I,M,N ),Z is defined.

163 Function (I,M,N ),Z (parts of it) Syntax a x Semantics a I Z(x) C \ C (I,M,N ),Z {t} {a a t (I,M,N ),Z } KC AC KR AR C D J M C (J,M,N ),Z J N C (J,M,N ),Z J M R(J,M,N ),Z J N R(J,M,N ),Z C (I,M,N ),Z D (I,M,N ),Z

164 (Monotonic) Semantics Definition (I, M, N ) satisfies axiom α, written (I, M, N ) = α, if (I, M, N ), Z = α for all variable assignments Z. A (non-empty) set of interpretations M satisfies α, written M = α, if (I, M, M) = α holds for all I M. M satisfies a SROIQV(B s, )K N F knowledge base KB, written M = KB, if M = α for all axioms α KB.

165 (Non-monotonic) Semantics Definition Given a SROIQV(B s, )K N F knowledge base KB, a (non-empty) set of interpretations M is an MKNF model of KB if (1) M = KB, and (2) for each M with M M, (I, M, M) = KB for some I M.

166 Example C Persons whose parents are married HasParent(mary, john) (1) ( HasParent. Married.{john})(mary) (2) HasParent.{z} HasParent. Married.{z} C (3) We can substitute (3) by K HasParent.{z} K HasParent. Married.{z} KC

167 Example C Persons whose parents are married HasParent(mary, john) (1) ( HasParent. Married.{john})(mary) (2) HasParent.{z} HasParent. Married.{z} C (3) We can also substitute (3) by K HasParent.{z} K HasParent. Married.{z} AC

168 Example C Persons whose parents are married HasParent(mary, john) (1) ( HasParent. Married.{john})(mary) (2) HasParent.{z} HasParent. Married.{z} C (3) We can also substitute (3) by HasParent.{z} HasParent. AMarried.{z} C Now C are Persons that are known to be not married

169 Decidability First, reduce reasoning in SROIQV(B s, )K N F to reasoning in SROIQ(B s )K N F by grounding and by simulating concept products Then, follow approach for ALCK N K : each model of a knowledge base in SROIQ(B s )K N F is cast into a SROIQ(B s ) KB. Consequently, reasoning in SROIQ(B s )K N F is reduced to a number of reasoning tasks in the non-modal SROIQ(B s ) For simplicity, appearance of modal operators restricted to simple KBs as in ALCK N F (finitely many, finite representations of models)

170 (Monotonic) Coverage SROIQ (a.k.a. OWL 2 DL); The tractable profiles OWL 2 EL, OWL 2 RL, OWL 2 QL; RIF-Core, i.e., n-ary Datalog, interpreted as DL-safe Rules (general case new result in [Knorr et al., ECAI12]); DL-safe SWRL [Motik et al., JWS05], AL-log [Donini et al., JIIS98], and CARIN [Levy and Rousset, AI98].

171 (Non-monotonic) Coverage ALCK N F [Donini et al., ACM TOCL02]; includes notions of concept and role closure present in this formalism; Closed Reiter defaults covered through the coverage of ALCK N F ; includes coverage of DLs extended with default rules [Baader and Hollunder, JAR95]; Hybrid MKNF [Motik and Rosati, JACM10]; Answer Set Programming, i.e., disjunctive Datalog with classical negation and non-monotonic negation under the answer set semantics; follows from the coverage of Hybrid MKNF.

172 N-nary Datalog N R = N P,2 {U} S, where S is a special set of roles: If P N P,>2 has arity k, then P 1,..., P k S are unique binary predicates associated with P; Translation: dl(p(t 1,..., t k )) := U.( P 1.{t 1 }... P k.{t k }); Family of interpretations of J for interpretation I of Datalog RB: (a) To each (d 1,..., d k ) P I, assign a unique element e in (i.e., we define a total, injective function from the set of tuples to ). (d) For each P N P,>2, if (d 1,..., d k ) P I, then (e, d i ) P J i, where e is the element assigned to (d 1,..., d k ) in point (a).

173 Hybrid MKNF Seamless integration of DL ontology O and rules of the form KH 1 KH l KA 1,..., KA n, notb 1,..., notb m Based on the n-nary Datalog embedding, additionally: dl(kh 1 KH l KA 1,..., KA n, notb 1,..., notb m ) := Kdl(A 1 )... Kdl(A n ) Adl(B 1 )... Adl(B m ) Kdl(H 1 )... Kdl(H l )

174 Example KC(x) KHasParent(x, y), KHasParent(x, z), K(y z), can be translated into notmarried(y, z). K U.({x} HasParent.{y}) K U.({x} HasParent.{z}) K U.({y}.{z}) A U.({y} Married.{z}) K U.({x} C)

175 Thank you!

176 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, 2010.

177 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, 1-16.

178 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,