OpenHPI 5.8 - Web Ontology Language (Part 5)

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General Role Inclusion -> Not for the Faint of Heart

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OpenHPI 5.8 - Web Ontology Language (Part 5)

  1. 1. Semantic Web TechnologiesLecture 5: Knowledge Representations II 08: Web Ontology Language (Part 5) Dr. Harald Sack Hasso Plattner Institute for IT Systems Engineering University of Potsdam Spring 2013 This file is licensed under the Creative Commons Attribution-NonCommercial 3.0 (CC BY-NC 3.0)
  2. 2. 2 Lecture 5: Knowledge Representations II Open HPI - Course: Semantic Web Technologies Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  3. 3. 3 08 OWL General Role Inclusion (OWL Part 5)Open HPI - Course: SemanticHarald Sack, Hasso-Plattner-Institut, Universität Potsdam Semantic Web Technologies , Dr. Web Technologies - Lecture 5: Knowledge Representations II
  4. 4. Not f3 or th e Fai nt of Heart 08 OWL General Role Inclusion (OWL Part 5)Open HPI - Course: SemanticHarald Sack, Hasso-Plattner-Institut, Universität Potsdam Semantic Web Technologies , Dr. Web Technologies - Lecture 5: Knowledge Representations II
  5. 5. General Role / Property Inclusion4 • Complex Roles / Properties can be constructed from simple roles / properties (R-Box) Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  6. 6. General Role / Property Inclusion4 • Complex Roles / Properties can be constructed from simple roles / properties (R-Box) • „The friends of my friends are also my friends.“ Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  7. 7. General Role / Property Inclusion4 • Complex Roles / Properties can be constructed from simple roles / properties (R-Box) • „The friends of my friends are also my friends.“ • can be expressed as SHOIN(D) transitive property Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  8. 8. General Role / Property Inclusion4 • Complex Roles / Properties can be constructed from simple roles / properties (R-Box) • „The friends of my friends are also my friends.“ • can be expressed as SHOIN(D) transitive property • But: „The foes of my friends are also my foes.“ Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  9. 9. General Role / Property Inclusion4 • Complex Roles / Properties can be constructed from simple roles / properties (R-Box) • „The friends of my friends are also my friends.“ • can be expressed as SHOIN(D) transitive property • But: „The foes of my friends are also my foes.“ • cannot be expressed as SHOIN(D) Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  10. 10. General Role / Property Inclusion4 • Complex Roles / Properties can be constructed from simple roles / properties (R-Box) • „The friends of my friends are also my friends.“ • can be expressed as SHOIN(D) transitive property • But: „The foes of my friends are also my foes.“ • cannot be expressed as SHOIN(D) hasFriend hasFoe Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  11. 11. General Role / Property Inclusion4 • Complex Roles / Properties can be constructed from simple roles / properties (R-Box) • „The friends of my friends are also my friends.“ • can be expressed as SHOIN(D) transitive property • But: „The foes of my friends are also my foes.“ • cannot be expressed as SHOIN(D) • In FOL expressed as a rule (axiom): • ∀x,y,z:hasFriend(x,y)∧hasFoe(y,z) ! hasFriendsFoe(x,z) hasFriend hasFoe Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  12. 12. General Role / Property Inclusion5 • Complex Roles / Properties can be constructed from simple roles / properties (R-Box) • SHROIQ(D) enables the construction of complex roles General Property Inclusion • R-Box expressions of the form R1ºR2ºR3º.....ºRn⊑S e.g.: hasFriend º hasFoe ⊑ hasFriendsFoe • Semantics: if (x0,x1)∈R1I,(x1,x2)∈R2I...(xn-1,xn)∈RnI , then it also holds that (x0,xn)∈SI E.g.: (x0,x1)∈hasFriendI and (x1,x2)∈hasFoeI, then it also holds (x0,x2)∈hasFriendsFoeI Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  13. 13. Expressivity of General Property Inclusion6 • With RBoxes formal Languages can be defined Example •Grammar for the (context free) language of the words ab, aabb, aaabbb, ... L ::= ab Ra º Rb ⊑ L becomes L ::= aLb Ra º L º Rb ⊑ L Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  14. 14. Expressivity of General Property Inclusion6 • With RBoxes formal Languages can be defined Example •Grammar for the (context free) language of the words ab, aabb, aaabbb, ... L ::= ab Ra º Rb ⊑ L becomes L ::= aLb Ra º L º Rb ⊑ L •∃L.⊤ ≢ ⊥ („ ∃L.⊤ necessarely non-empty“) means: „There exists a chain Ra and Rb pertaining to the language.“ •∃L1.∃L2- ≢ ⊥ for two languages L1 and L2 means: „There exists a word pertaining to L1 and also to L2.“ Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  15. 15. Expressivity of General Property Inclusion6 • With RBoxes formal Languages can be defined Example •Grammar for the (context free) language of the words ab, aabb, aaabbb, ... L ::= ab Ra º Rb ⊑ L becomes L ::= aLb Ra º L º Rb ⊑ L •∃L.⊤ ≢ ⊥ („ ∃L.⊤ necessarely non-empty“) means: „There exists a chain Ra and Rb pertaining to the language.“ •∃L1.∃L2- ≢ ⊥ for two languages L1 and L2 means: „There exists a word pertaining to L1 and also to L2.“ But from formal languages is known: Emptiness of the intersection of context free languages is not decidable Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  16. 16. Expressivity of General Property Inclusion7 • With RBoxes formal Languages can be defined Example •Grammar for the (context free) language of the words ab, aabb, aaabbb, ... L ::= ab Ra º Rb ⊑ L becomes L ::= aLb Ra º L º Rb ⊑ L •∃L.⊤ ≢ ⊥ („ ∃L.⊤ necessarely non-empty“) means: „There exists a chain Ra and Rb pertaining to the language.“ •∃L1.∃L2- ≢ ⊥ for two languages L1 and L2 means: „There exists a word pertaining to L1 and also to L2.“ OWL with general propertry inclusion in UNDECIDABLE ! Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  17. 17. Restriction to Regular RBoxes • Can property inclusion be restricted in some way to stay decidable?8 • Rboxes are like grammars for context-free languages • Intersection of context free languages is problematic • Therefore restriction to regular languages! Regular RBoxes • Property names are ordered with ≺ (strict partial Ordering). • Each RBox Inclusion must must be formed like: •R º R ⊑ R •R º S1 º S2 º S3 º...º Sn ⊑ R •R- ⊑ R •S1 º S2 º S3 º... º Sn º R ⊑ R •S1 º S2 º S3 º ... º Sn ⊑ R • Where: Si ≺ R for all i=1,2,...,n • An RBox is called regular, if such an (strict) Ordering ≺ exists. Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  18. 18. 9 Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  19. 19. Regular RBoxes - Examples9 Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  20. 20. Regular RBoxes - Examples • Example:9 hasParent º hasHusband ⊑ hasFather hasFather ⊑ hasParent Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  21. 21. Regular RBoxes - Examples • Example:9 hasParent º hasHusband ⊑ hasFather hasFather ⊑ hasParent • is not regular because regularity would enforce both hasParent ≺ hasFather and hasFather ≺ hasParent which is impossible because ≺ must be strict Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  22. 22. Restrictions for simple Properties10 • Simple properties in SHOIN(D) are properties without transitive subproperties • In SHROIQ(D) general property inclusion has to be considered Simple Properties • are all properties that • are not on the right side of a property inclusion, • are the inverse of other simple properties, • are only on the right side of property inclusions R ⊑ S, where on the left side is a simple property •Non-simple properties are properties that are directly (or indirectly) dependent of property chains (º ) Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  23. 23. Restrictions for simple Properties11 • The following expressions are permitted ONLY for simple properties: • ≤n R.C and ≥n R.C (qualified number restriction) • Irreflexive properties • Disjunctive properties •∃R.Self •¬R(a,b) •Reason: Saving Decidability Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  24. 24. General Role / Property Inclusion Summary12 • To ensure decidability the following structural restrictions must hold for SHROIQ(D): • Regularity: Restriction of potential interaction of RBox axioms • Simplicity of properties: Restrictions of how to apply properties in number restrictions • Therefore a number of restrictions arise for the overall structure of the knowledge base that have to be considered for all axioms. • Attention: The union of several SHROIQ(D) knowledge bases might violate these restrictions, although each underlying knowledge base does comply to the restrictions! Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  25. 25. OWL 2 General Property Inclusion13 • OWL enables property chaining (general property inclusion) :hasFriendsFoe a owl:ObjectProperty ; owl:PropertyChainAxiom ( :hasFriend :hasFoe ) . • Not allowed for datatype properties hasFriend hasFoe Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  26. 26. 14 09 Rules and the Semantic WebOpen HPI - Course: SemanticHarald Sack, Hasso-Plattner-Institut, Universität Potsdam Semantic Web Technologies , Dr. Web Technologies - Lecture 5: Knowledge Representations II

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