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FullSemanticsSelfRestrictions
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The OWL 2 Full semantics of Self Restrictions.
Self Restrictions
Syntax
x rdf:type owl:SelfRestriction x owl:onProperty p
Semantics
Axiomatic triples, derived from OWL 1 Full: see page on axiomatic triples.
Axiomatic triples for new vocabulary:
owl:SelfRestriction rdfs:subClassOf owl:Restriction
Main semantic condition:
IF x ∈ CEXT_I(S_I(owl:SelfRestriction)), (x,p) ∈ EXT_I(S_I(owl:onProperty)) THEN x ∈ IOR, p ∈ IOOP, CEXT_I(x) = { u ∈ IOT | (u,u) ∈ EXT_I(p) }
Comprehension principle:
IF p ∈ IOOP THEN ∃ x: x in IOR, x ∈ CEXT_I(S_I(owl:SelfRestriction)), (x,p) ∈ EXT_I(S_I(owl:onProperty))
Considerations
- The semantics here are in line with the OWL 2 DL semantics.
- The style of the semantics here is in line with the style of semantics given for restrictions in OWL 1 Full.
- The main semantic condition is not an "IFF" but an "IF" semantic condition.
- There is a comprehension principle.
- SelfRestrictionS are special owl:RestrictionS ("IOR").
- It is therefore not necessary to say "x in IOR" in the "THEN" branch of the main semantic condition.
- It is therefore not necessary to say "EXISTS x in IOR" in the "THEN" branch of the comprehension principle.
- We do it anyway for being in line with the style of presentation of the restrictions in OWL 1 Full.
- The URI 'owl:SelfRestriction' is explictly mentioned in the "IF" branch of the main semantic condition, and in the "THEN" branch of the comprehension principle.
- For the main semantic condition: Otherwise a single occurrence of "owl:onProperty" would result in the semantics given for self restrictions. This would mean that every restriction would additionally have the semantics of a self restriction. This would be a heavy bug.
- For the comprehension principle: Without owl:SelfRestriction in the "THEN" branch:
- There would be no match with the "IF" branch of the main semantic condition.
- The comprehension principle would express that for each property p there is some restriction, which restricts p. This is a weaker assertion than what is said in the actual comprehension principle above (probably useless in practice). It also doesn't match the intended meaning of a comprehension principle for self restrictions.
- This is killer semantics: Using the complete semantics proposed here will lead to inconsistency of OWL 2 Full