FullSemanticsUniversalProperty

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The OWL 2 Full semantics of the Top and Bottom Property.

The Top Property is the property which relates every existing individual to every other individual.

The Bottom Property is the property which relates no individuals at all, i.e. the property which has the empty extension.

The Top and Bottom Property exist both as data properties and object properties.

Top Object Property

Syntax

 x owl:TopObjectProperty y

Semantics

Axiomatic Triples:

 owl:TopObjectProperty rdf:type owl:ObjectProperty
 owl:TopObjectProperty rdfs:domain owl:Thing
 owl:TopObjectProperty rdfs:range owl:Thing

Main semantic condition:

 EXT_I(S_I(owl:TopObjectProperty)) = IOT×IOT 

Top Data Property

Syntax

 x owl:TopDataProperty y

Semantics

Axiomatic Triples:

 owl:TopDataProperty rdf:type owl:DatatypeProperty
 owl:TopDataProperty rdfs:domain owl:Thing
 owl:TopDataProperty rdfs:range rdfs:Literal

Main semantic condition:

 EXT_I(S_I(owl:TopDataProperty)) = IOT×LV_I 
   

Bottom Object Property

Syntax

 x owl:BottomObjectProperty y

Semantics

Axiomatic Triples:

 owl:BottomObjectProperty rdf:type owl:ObjectProperty
 owl:BottomObjectProperty rdfs:domain owl:Nothing
 owl:BottomObjectProperty rdfs:range owl:Nothing

Main semantic condition:

 EXT_I(S_I(owl:BottomObjectProperty)) = ∅ 

Bottom Data Property

Syntax

 x owl:BottomDataProperty y

Semantics

Axiomatic Triples:

 owl:BottomDataProperty rdf:type owl:DatatypeProperty
 owl:BottomDataProperty rdfs:domain owl:Nothing
 owl:BottomDataProperty rdfs:range owl:Nothing

Main semantic condition:

 EXT_I(S_I(owl:BottomDataProperty)) = ∅ 

Considerations

  • The main semantic conditions mirror the entries in the "Parts" table of OWL 1 Full.
  • The domain and range of the Top Property follow from the main semantic condition. So domain and range are redundant in principle.
  • The main semantic condition of the Bottom Property tells the same as the domain and range of the Bottom Property. So one of the two informations is redundant in principle.
  • The range of the bottom data property is currently owl:Nothing. This is technically ok in OWL Full, but looks a bit strange. A bottom datatype would be helpful here.

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