OpenHPI 5.8 - Web Ontology Language (Part 5)
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Transcript of OpenHPI 5.8 - Web Ontology Language (Part 5)
This file is licensed under the Creative Commons Attribution-NonCommercial 3.0 (CC BY-NC 3.0)
Dr. Harald Sack
Hasso Plattner Institute for IT Systems Engineering
University of Potsdam
Spring 2013
Semantic Web Technologies
Lecture 5: Knowledge Representations II08: Web Ontology Language (Part 5)
Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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Lecture 5: Knowledge Representations II
Open HPI - Course: Semantic Web Technologies
Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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08 OWL General Role Inclusion (OWL Part 5)Open HPI - Course: Semantic Web Technologies - Lecture 5: Knowledge Representations II
Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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08 OWL General Role Inclusion (OWL Part 5)Open HPI - Course: Semantic Web Technologies - Lecture 5: Knowledge Representations II
Not for the Faint of Heart
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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Turmbau zu Babel, Pieter Brueghel, 1563
General Role / Property Inclusion
•Complex Roles / Properties can be constructed from simple roles / properties (R-Box)
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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Turmbau zu Babel, Pieter Brueghel, 1563
General Role / Property Inclusion
•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
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Turmbau zu Babel, Pieter Brueghel, 1563
General Role / Property Inclusion
•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
4
Turmbau zu Babel, Pieter Brueghel, 1563
General Role / Property Inclusion
•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
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Turmbau zu Babel, Pieter Brueghel, 1563
General Role / Property Inclusion
•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
4
Turmbau zu Babel, Pieter Brueghel, 1563
General Role / Property Inclusion
•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)
hasFoehasFriend
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
4
Turmbau zu Babel, Pieter Brueghel, 1563
General Role / Property Inclusion
•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)
hasFoehasFriend
• In FOL expressed as a rule (axiom):• ∀x,y,z:hasFriend(x,y)∧hasFoe(y,z) ! hasFriendsFoe(x,z)
General Role / Property Inclusion
•Complex Roles / Properties can be constructed from simple roles / properties (R-Box)
•SHROIQ(D) enables the construction of complex roles
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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General Property Inclusion
• R-Box expressions of the form R1ºR2ºR3º.....ºRn⊑Se.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
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Expressivity of General Property Inclusion
• With RBoxes formal Languages can be defined
Example•Grammar for the (context free) language of the words ab, aabb, aaabbb, ...
L ::= abL ::= aLb
becomes Ra º Rb ⊑ LRa º L º Rb ⊑ L
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
6
Expressivity of General Property Inclusion
• With RBoxes formal Languages can be defined
Example•Grammar for the (context free) language of the words ab, aabb, aaabbb, ...
L ::= abL ::= aLb
becomes Ra º Rb ⊑ LRa 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
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Expressivity of General Property Inclusion
• With RBoxes formal Languages can be defined
But from formal languages is known:Emptiness of the intersection of context free languages is not decidable
Example•Grammar for the (context free) language of the words ab, aabb, aaabbb, ...
L ::= abL ::= aLb
becomes Ra º Rb ⊑ LRa 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
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OWL with general propertry inclusion in UNDECIDABLE !
Expressivity of General Property Inclusion
• With RBoxes formal Languages can be defined
Example•Grammar for the (context free) language of the words ab, aabb, aaabbb, ...
•∃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.“
L ::= abL ::= aLb
becomes Ra º Rb ⊑ LRa º L º Rb ⊑ L
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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Restriction to Regular RBoxes
• Can property inclusion be restricted in some way to stay decidable?
• 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:
•Where: Si ≺ R for all i=1,2,...,n•An RBox is called regular, if such an (strict) Ordering ≺ exists.
•R º R ⊑ R•R- ⊑ R•S1 º S2 º S3 º ... º Sn ⊑ R
•R º S1 º S2 º S3 º...º Sn ⊑ R •S1 º S2 º S3 º... º Sn º R ⊑ R
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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Regular RBoxes - Examples
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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Regular RBoxes - Examples
• Example:hasParent º hasHusband ⊑ hasFatherhasFather ⊑ hasParent
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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Regular RBoxes - Examples
• Example:hasParent º hasHusband ⊑ hasFatherhasFather ⊑ hasParent
• is not regular because regularity would enforcebothhasParent ≺ hasFatherand hasFather ≺ hasParentwhich is impossible because ≺ must be strict
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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Restrictions for simple Properties
• Simple properties in SHOIN(D) are properties without
transitive subproperties
• In SHROIQ(D) general property inclusion has to be
considered
•Non-simple properties are properties that are directly (or indirectly) dependent of property chains (º )
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
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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Restrictions for simple Properties
• The following expressions are permitted ONLY for simple properties:
•Reason: Saving Decidability
•≤n R.C and ≥n R.C (qualified number restriction)
• Irreflexive properties
•Disjunctive properties
•∃R.Self
•¬R(a,b)
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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Summary• 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!
General Role / Property Inclusion
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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OWL 2 General Property Inclusion
• OWL enables property chaining (general property inclusion)
• Not allowed for datatype properties
hasFoehasFriend
:hasFriendsFoe a owl:ObjectProperty ; owl:PropertyChainAxiom ( :hasFriend :hasFoe ) .
Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
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09 Rules and the Semantic WebOpen HPI - Course: Semantic Web Technologies - Lecture 5: Knowledge Representations II