# FullSemanticsSelfRestrictions

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