FullSemanticsDatatypeRestrictions

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WORK IN PROGRESS

The OWL 2 Full semantics of Datatype Restrictions, Complements, and Enumerations.

Note: There is still some ongoing discussion on syntax and semantics of these features in OWL 1 DL. However, the situation has much stabilized over the last months, so I am going to start working on this topic now.

Datatype Restrictions

Syntax

 c rdf:type rdfs:Datatype
 c owl:onDatatype d
 c owl:withRestrictions SEQ( y_1 ... y_n )
 y_1 xsd:facet_1 v_1
 ...
 y_n xsd:facet_n v_n

Semantics

 IF
   l is a sequence of y_1, ..., y_n over R_I
   (c,d) ∈ EXT_I(S_I(owl:onDatatype)),
   (c,l) ∈ EXT_I(S_I(owl:withRestrictions)),
   (y_1, v_1) ∈ EXT_I(S_I(xsd:facet_1)),
    ...,
   (y_n, v_n) ∈ EXT_I(S_I(xsd:facet_n))
 THEN
   c, d ∈ IDC,
   v_1, ..., v_n ∈ LV_I,
   CEXT_I(c) = CEXT_I(d) ∩ FACET(xsd:facet_1,d,v_1) ∩ ... ∩ FACET(xsd:facet_n,d,v_n)

where FACET(xsd:facet, d, v) denotes the subset of LV_I, which is specified by applying the facet xsd:facet with value v to the extension CEXT_I(d) of the datatype d.

Special case: If the facet cannot be applied, then FACET(xsd:facet, d, v) is empty. [FIXME: Should ontology rather be inconsistent in such a case?]

Comprehension Principle:

 TODO

Datatype Complements

In OWL 2 DL, the complement of a datatype is defined relative to the data domain. So this semantics has to be mirrored in OWL 2 Full.

Syntax

 c rdf:type rdfs:Datatype
 c owl:datatypeComplementOf d

Semantics

Axiomatic triples:

 owl:datatypeComplementOf rdf:type rdf:Property
 owl:datatypeComplementOf rdfs:domain rdfs:Datatype
 owl:datatypeComplementOf rdfs:range rdfs:Datatype

Main semantic condition:

   (c,d) ∈ EXT_I(S_I(owl:datatypeComplementOf))
 IF AND ONLY IF
   c, d ∈ IDC,
   CEXT_I(c) = LV_I - CEXT_I(d)

Comprehension Principle:

 IF
   c ∈ IDC
 THEN ∃ x
   (x,d) ∈ EXT_I(S_I(owl:datatypeComplementOf))

Datatype Enumerations

See treetment of owl:oneOf.

Considerations

  • Question: Is it right to have ONLY-IF semantics, as for "normal" restrictions?

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