# FullSemanticsPropertyCharacteristics

The OWL 2 Full semantics of the new Property Characteristics.

## Reflexive Properties

### Syntax

``` x rdf:type owl:ReflexiveProperty
```

### Semantics

Axiomatic triples:

``` owl:ReflexiveProperty rdfs:subClassOf owl:ObjectProperty
```

Main semantic condition:

```   p ∈ CEXT_I(S_I(owl:ReflexiveProperty))
IF AND ONLY IF
p ∈ IOOP,
x ∈ IOT → (x,x) ∈ EXT_I(p)
```

## Irreflexive Properties

### Syntax

``` x rdf:type owl:IrreflexiveProperty
```

### Semantics

Axiomatic triples:

``` owl:IrreflexiveProperty rdfs:subClassOf owl:ObjectProperty
```

Main semantic condition:

```   p ∈ CEXT_I(S_I(owl:IrreflexiveProperty))
IF AND ONLY IF
p ∈ IOOP,
x ∈ IOT → (x,x) ∉ EXT_I(p)
```

## Asymmetric Properties

### Syntax

``` x rdf:type owl:AsymmetricProperty
```

### Semantics

Axiomatic triples:

``` owl:AsymmetricProperty rdfs:subClassOf owl:ObjectProperty
```

Main semantic condition:

```   p ∈ CEXT_I(S_I(owl:AsymmetricProperty))
IF AND ONLY IF
p ∈ IOOP,
(x,y) ∈ EXT_I(p) → (y,x) ∉ EXT_I(p)
```

## Considerations

• The semantics defined here seem to be straightforward, and there do not seem to be very different alternatives.
• All semantics specified here correspond to the respective semantics in OWL 2 DL.
• Axiomatic triples: more specific than in OWL 1 Full, where the property characteristics were only unspecified classes. Now they are subclasses of owl:ObjectProperty.
• OWL 1 Full uses IF-AND-ONLY-IF semantic conditions for all property characteristics. This means that the property characteristics are fully defined by these semantic conditions. This approach is followed here, too.
• OWL 1 Full has "p ∈ IOOP" in the "THEN" branch of the property characteristics. This means that from a typing triple it can be concluded that the subject of the triple is a property, instead of explicitly stating this in the "IF" branch. This approach is followed here, too.
• A term of the form "(x,y) ∉ EXT_I(p)" corresponds to a negative property assertion.