OWL Three species of OWL OWL full is union of OWL syntax and RDF - PowerPoint PPT Presentation

OWL ◮ Three species of OWL ◮ OWL full is union of OWL syntax and RDF (Undecidable) ◮ OWL DL restricted to FOL fragment (decidable in NEXPTIME) ◮ OWL Lite is “easier to implement” subset of OWL DL (decidable in EXPTIME) ◮ Semantic layering ◮ OWL DL within Description Logic (DL) fragment ◮ OWL DL is based on SHOIN ( D n ) DL ◮ OWL Lite is based on SHIF ( D n ) DL ◮ OWL 2 is based on SROIQ ( D n ) DL

Description Logics (DLs) ◮ The logics behind OWL-DL and OWL-Lite, http://dl.kr.org/ . ◮ Concept/Class: names are equivalent to unary predicates ◮ In general, concepts equiv to formulae with one free variable ◮ Role or attribute: names are equivalent to binary predicates ◮ In general, roles equiv to formulae with two free variables ◮ Taxonomy: Concept and role hierarchies can be expressed ◮ Individual: names are equivalent to constants ◮ Operators: restricted so that: ◮ Language is decidable and, if possible, of low complexity ◮ No need for explicit use of variables ◮ Restricted form of ∃ and ∀ ◮ Features such as counting can be succinctly expressed

The DL Family ◮ A given DL is defined by set of concept and role forming operators ◮ Basic language: ALC ( A ttributive L anguage with C omplement) Syntax Semantics Example C , D → ⊤ | ⊤ ( x ) ⊥ | ⊥ ( x ) A | A ( x ) Human C ⊓ D | C ( x ) ∧ D ( x ) Human ⊓ Male C ⊔ D | C ( x ) ∨ D ( x ) Nice ⊔ Rich ¬ C | ¬ C ( x ) ¬ Meat ∃ R . C | ∃ y . R ( x , y ) ∧ C ( y ) ∃ has _ child . Blond ∀ R . C ∀ y . R ( x , y ) ⇒ C ( y ) ∀ has _ child . Human C ⊑ D ∀ x . C ( x ) ⇒ D ( x ) Happy _ Father ⊑ Man ⊓ ∃ has _ child . Female a : C C ( a ) John : Happy _ Father

Toy Example Sex = Male ⊔ Female Male ⊓ Female ⊑ ⊥ Person ⊑ Human ⊓ ∃ hasSex . Sex MalePerson = Person ⊓ ∃ hasSex . Male functional ( hasSex ) umberto : Person ⊓ ∃ hasSex . ¬ Female KB | = umberto : MalePerson

Note on DL Naming AL : − → ⊤ | ⊥ | A | C ⊓ D | ¬ A | ∃ R . ⊤ |∀ R . C C , D C : Concept negation, ¬ C . Thus, ALC = AL + C S : Used for ALC with transitive roles R + U : Concept disjunction, C 1 ⊔ C 2 E : Existential quantification, ∃ R . C H : Role inclusion axioms, R 1 ⊑ R 2 , e.g. is _ component _ of ⊑ is _ part _ of N : Number restrictions, ( ≥ n R ) and ( ≤ n R ) , e.g. ( ≥ 3 has _ Child ) (has at least 3 children) Q : Qualified number restrictions, ( ≥ n R . C ) and ( ≤ n R . C ) , e.g. ( ≤ 2 has _ Child . Adult ) (has at most 2 adult children) O : Nominals (singleton class), { a } , e.g. ∃ has _ child . { mary } . Note : a : C equiv to { a } ⊑ C and ( a , b ) : R equiv to { a } ⊑ ∃ R . { b } I : Inverse role, R − , e.g. isPartOf = hasPart − F : Functional role, f , e.g. functional ( hasAge ) R + : transitive role, e.g. transitive ( isPartOf ) For instance, SHIF = S + H + I + F = ALCR + HIF OWL-Lite SHOIN = S + H + O + I + N = ALCR + HOIN OWL-DL SROIQ = S + R + O + I + Q = ALCR + ROIN OWL 2

Semantics of Additional Constructs = R 1 ⊑ R 2 iff R 1 I ⊆ R 1 I H : Role inclusion axioms, I | N : Number restrictions, ( ≥ n R ) I = { x ∈ ∆ I : |{ y | � x , y � ∈ R I }| ≥ n } , ( ≤ n R ) I = { x ∈ ∆ I : |{ y | � x , y � ∈ R I }| ≤ n } Q : Qualified number restrictions, ( ≥ n R . C ) I = { x ∈ |{ y | � x , y � ∈ R I ∧ y ∈ C I }| ≥ n } , ( ≤ n R . C ) I = { x ∈ ∆ I : |{ y | � x , y � ∈ R I ∧ y ∈ C I }| ≤ n } O : Nominals (singleton class), { a } I = { a I } I : Inverse role, ( R − ) I = {� x , y � | � y , x � ∈ R I } = fun ( f ) iff ∀ z ∀ y ∀ z if � x , y � ∈ f I and � x , z � ∈ f I F : Functional role, I | the y = z R + : transitive role, ( R + ) I = {� x , y � | ∃ z such that � x , z � ∈ R I ∧ � z , y � ∈ R I }

Basics on Concrete Domains ◮ Concrete domains: reals, integers, strings, . . . ( tim , 14 ) :hasAge ( sf , “ SoftComputing ”) :hasAcronym ( source 1 , “ ComputerScience ”) :isAbout ( service 2 , “ InformationRetrievalTool ′′ ) :Matches Minor = Person ⊓ ∃ hasAge . ≤ 18 ◮ Semantics: a clean separation between “object” classes and concrete domains ◮ D = � ∆ D , Φ D � ◮ ∆ D is an interpretation domain ◮ Φ D is the set of concrete domain predicates d with a predefined arity n and fixed interpretation d D ⊆ ∆ n D ◮ Concrete properties: R I ⊆ ∆ I × ∆ D ◮ Notation: ( D ) . E.g., ALC ( D ) is ALC + concrete domains

◮ Example: assume I = (∆ I , · I ) such that ∆ I = { a , b , c , d , e , f , 1 , 2 , 4 , 5 , 8 } Person I = { a , b , c , d } ◮ Consider the following concrete domain with of some unary predicates ( n = 1) over reals ◮ ∆ D = R , ◮ Φ D = { = m , ≥ m , ≤ m , > m , < m | m ∈ R } ◮ the fixed interpretation of the predicates is (= m ) D = { m } ( > m ) D = { k | k > m } ( ≥ m ) D { k | k ≥ m } = ( < m ) D { k | k < m } = ( ≤ m ) D = { k | k ≤ m } Concrete properties: hasAge I ⊆ ∆ I × R ◮ hasAge I = {� a , 9 � , � c , 20 � , � b , 12 �} ◮ What is the interpretation of Person ⊓ ∃ hasAge . ≤ 18 ? ( Person ⊓ ∃ hasAge . ≤ 18 ) I = Person I ∩ { x | ∃ y ∈ R such that � x , y � ∈ hasAge I ∧ y ∈ ( ≤ 18 ) I = { a , b , c , d } ∩ { x | ∃ y . � x , y � ∈ {� a , 9 � , � c , 20 � , � b , 12 �} ∧ y ≤ 18 } = { a , b , c , d } ∩ { a , b } = { a , b }

DL Knowledge Base ◮ A DL Knowledge Base is a pair KB = �T , A� , where ◮ T is a TBox ◮ containing general inclusion axioms of the form C ⊑ D , ◮ concept definitions of the form A = C ◮ primitive concept definitions of the form A ⊑ C ◮ role inclusions of the form R ⊑ P ◮ role equivalence of the form R = P ◮ A is a ABox ◮ containing assertions of the form a : C ◮ containing assertions of the form ( a , b ) : R ◮ containing (in) equality Axioms of the form a = b and a � = b ◮ An interpretation I is a model of KB , written I | = KB iff I | = T and I | = A , where ◮ I | = T ( I is a model of T ) iff I is a model of each element in T ◮ I | = A ( I is a model of A ) iff I is a model of each element in A

OWL DL as Description Logic Concept/Class constructors: Abstract Syntax DL Syntax Example Descriptions ( C ) A (URI reference) A Conference ⊤ owl:Thing ⊥ owl:Nothing intersectionOf ( C 1 C 2 . . . ) C 1 ⊓ C 2 Reference ⊓ Journal unionOf ( C 1 C 2 . . . ) C 1 ⊔ C 2 Organization ⊔ Institution complementOf ( C ) ¬ C ¬ MasterThesis oneOf ( o 1 . . . ) { o 1 , . . . } {"WISE","ISWC",...} ∃ R . C ∃ parts.InCollection restriction ( R someValuesFrom ( C )) ∀ R . C ∀ date.Date restriction ( R allValuesFrom ( C )) ∃ R . { o } ∃ date.{2005} restriction ( R hasValue ( o )) ( ≥ n R ) restriction ( R minCardinality ( n )) ( � 1 location ) ( ≤ n R ) restriction ( R maxCardinality ( n )) ( � 1 publisher ) restriction ( U someValuesFrom ( D )) ∃ U . D ∃ issue.integer restriction ( U allValuesFrom ( D )) ∀ U . D ∀ name.string restriction ( U hasValue ( v )) ∃ U . = v } ∃ series . = ” LNCS ” restriction ( U minCardinality ( n )) ( ≥ n U ) ( � 1 title ) restriction ( U maxCardinality ( n )) ( ≤ n U ) ( � 1 author ) Note: R is an abstract role, while U is a concrete property of arity two.

Axioms: Abstract Syntax DL Syntax Example Axioms Class ( A partial C 1 . . . C n ) A ⊑ C 1 ⊓ . . . ⊓ C n Human ⊑ Animal ⊓ Biped Class ( A complete C 1 . . . C n ) A = C 1 ⊓ . . . ⊓ C n Man = Human ⊓ Male EnumeratedClass ( A o 1 . . . o n ) A = { o 1 } ⊔ . . . ⊔ { o n } RGB = { r } ⊔ { g } ⊔ { b } SubClassOf ( C 1 C 2 ) C 1 ⊑ C 2 EquivalentClasses ( C 1 . . . C n ) C 1 = . . . = C n DisjointClasses ( C 1 . . . C n ) C i ⊓ C j = ⊥ , i � = j Male ⊓ Female ⊑⊥ R ⊑ R i HasDaughter ⊑ hasChild ObjectProperty ( R super ( R 1 ) . . . super ( R n ) ( ≥ 1 R ) ⊑ C i ( ≥ 1 hasChild ) ⊑ Human domain ( C 1 ) . . . domain ( C n ) ⊤ ⊑ ∀ R . C i ⊤ ⊑ ∀ hasChild . Human range ( C 1 ) . . . range ( C n ) R = P − hasChild = hasParent − [ inverseof ( P ) ] R ⊑ R − similar = similar − [ symmetric ] ⊤ ⊑ ( ≤ 1 R ) ⊤ ⊑ ( ≤ 1 hasMother ) [functional] ⊤ ⊑ ( ≤ 1 R − ) [Inversefunctional] [Transitive] ) Tr ( R ) Tr ( ancestor ) SubPropertyOf ( R 1 R 2 ) R 1 ⊑ R 2 EquivalentProperties ( R 1 . . . R n ) R 1 = . . . = R n cost = price AnnotationProperty ( S )

Abstract Syntax DL Syntax Example DatatypeProperty ( U super ( U 1 ) . . . super ( U n ) U ⊑ U i domain ( C 1 ) . . . domain ( C n ) ( ≥ 1 U ) ⊑ C i ( ≥ 1 hasAge ) ⊑ Human range ( D 1 ) . . . range ( D n ) ⊤ ⊑ ∀ U . D i ⊤ ⊑ ∀ hasAge . posInteger [ functional ]) ⊤ ⊑ ( ≤ 1 U ) ⊤ ⊑ ( ≤ 1 hasAge ) SubPropertyOf ( U 1 U 2 ) U 1 ⊑ U 2 hasName ⊑ hasFirstName EquivalentProperties ( U 1 . . . U n ) U 1 = . . . = U n Individuals Individual ( o type ( C 1 ) . . . type ( C n )) o : C i tim : Human value ( R 1 o 1 ) . . . value ( R n o n ) ( o , o i ) : R i ( tim , mary ) : hasChild value ( U 1 v 1 ) . . . value ( U n v n ) ( o , v 1 ) : U i ( tim , 14 ) : hasAge SameIndividual ( o 1 . . . o n ) o 1 = . . . = o n president _ Bush = G . W . Bush o i � = o j , i � = j john � = peter DifferentIndividuals ( o 1 . . . o n ) Symbols Object Property R (URI reference) R hasChild Datatype Property U (URI reference) U hasAge Individual o (URI reference) U tim Data Value v (RDF literal) U “International Conference on Semantic Web”