OpenHPI 5.8 - Web Ontology Language (Part 5)

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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


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08 OWL General Role Inclusion (OWL Part 5)Open HPI - Course: Semantic Web Technologies - Lecture 5: Knowledge Representations II


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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)


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•„The friends of my friends are also my friends.“


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• can be expressed as SHOIN(D) transitive property


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•But: „The foes of my friends are also my foes.“


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• cannot be expressed as SHOIN(D)


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hasFoehasFriend


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hasFoehasFriend

• In FOL expressed as a rule (axiom):• ∀x,y,z:hasFriend(x,y)∧hasFoe(y,z) ! hasFriendsFoe(x,z)



•SHROIQ(D) enables the construction of complex roles


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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


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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


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L ::= abL ::= aLb


•∃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.“


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But from formal languages is known:Emptiness of the intersection of context free languages is not decidable


L ::= abL ::= aLb





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OWL with general propertry inclusion in UNDECIDABLE !






L ::= abL ::= aLb



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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


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Regular RBoxes - Examples


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• Example:hasParent º hasHusband ⊑ hasFatherhasFather ⊑ hasParent


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• Example:hasParent º hasHusband ⊑ hasFatherhasFather ⊑ hasParent

• is not regular because regularity would enforcebothhasParent ≺ hasFatherand hasFather ≺ hasParentwhich is impossible because ≺ must be strict


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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


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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)


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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!



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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 ) .


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09 Rules and the Semantic WebOpen HPI - Course: Semantic Web Technologies - Lecture 5: Knowledge Representations II

OpenHPI 5.8 - Web Ontology Language (Part 5)

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