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FullSemanticsQCRs
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The OWL 2 Full semantics of Qualified Cardinality Restrictions (QCRs).
Note: The RDF syntax of QCRs has been changed to the current form by resolving Issue 122.
Contents
- 1 Axiomatic Triples
- 2 Exact Qualified Object Cardinality Restrictions
- 3 Exact Qualified Data Cardinality Restrictions
- 4 Min Qualified Object Cardinality Restrictions
- 5 Min Qualified Data Cardinality Restrictions
- 6 Max Qualified Object Cardinality Restrictions
- 7 Max Qualified Data Cardinality Restrictions
Axiomatic Triples
Axiomatic triples, derived from OWL 1 Full: see page on axiomatic triples.
owl:qualifiedCardinality rdf:type rdf:Property owl:qualifiedCardinality rdfs:domain owl:Restriction owl:qualifiedCardinality rdfs:range xsd:nonNegativeInteger
owl:minQualifiedCardinality rdf:type rdf:Property owl:minQualifiedCardinality rdfs:domain owl:Restriction owl:minQualifiedCardinality rdfs:range xsd:nonNegativeInteger
owl:maxQualifiedCardinality rdf:type rdf:Property owl:maxQualifiedCardinality rdfs:domain owl:Restriction owl:maxQualifiedCardinality rdfs:range xsd:nonNegativeInteger
owl:onClass rdf:type rdf:Property owl:onClass rdfs:domain owl:Restriction owl:onClass rdfs:range owl:Class
owl:onDataRange rdf:type rdf:Property owl:onDataRange rdfs:domain owl:Restriction owl:onDataRange rdfs:range rdfs:Datatype
Exact Qualified Object Cardinality Restrictions
Syntax
x rdf:type owl:Restriction x owl:onProperty p x owl:qualifiedCardinality "n"^^xsd:nonNegativeInteger x owl:onClass c
Semantics
Main semantic condition:
IF (x,n) ∈ EXT_I(S_I(owl:qualifiedCardinality)), (x,p) ∈ EXT_I(S_I(owl:onProperty)), (x,c) ∈ EXT_I(S_I(owl:onClass)) THEN x ∈ IOR, n ∈ LV_I, n is a non-negative integer, p ∈ IOOP, c ∈ IOC, CEXT_I(x) = { u | card({v | (u,v) ∈ EXT_I(p) ∧ v ∈ CEXT_I(c)}) = n }
Comprehension principle:
IF p ∈ IOOP, n ∈ LV_I, n is a non-negative integer, c ∈ IOC THEN ∃ x: x in IOR, (x,n) ∈ EXT_I(S_I(owl:qualifiedCardinality)), (x,p) ∈ EXT_I(S_I(owl:onProperty)), (x,c) ∈ EXT_I(S_I(owl:onClass))
Exact Qualified Data Cardinality Restrictions
Syntax
x rdf:type owl:Restriction x owl:onProperty p x owl:qualifiedCardinality "n"^^xsd:nonNegativeInteger x owl:onDataRange c
Semantics
Main semantic condition:
IF (x,n) ∈ EXT_I(S_I(owl:qualifiedCardinality)), (x,p) ∈ EXT_I(S_I(owl:onProperty)), (x,c) ∈ EXT_I(S_I(owl:onDataRange)) THEN x ∈ IOR, n ∈ LV_I, n is a non-negative integer, p ∈ IODP, c ∈ IDC, CEXT_I(x) = { u | card({v ∈ LV_I | (u,v) ∈ EXT_I(p) ∧ v ∈ CEXT_I(c)}) = n }
Comprehension principle:
IF p ∈ IODP, n ∈ LV_I, n is a non-negative integer, c ∈ IDC THEN ∃ x: x in IOR, (x,n) ∈ EXT_I(S_I(owl:qualifiedCardinality)), (x,p) ∈ EXT_I(S_I(owl:onProperty)), (x,c) ∈ EXT_I(S_I(owl:onDataRange))
Min Qualified Object Cardinality Restrictions
Syntax
x rdf:type owl:Restriction x owl:onProperty p x owl:minQualifiedCardinality "n"^^xsd:nonNegativeInteger x owl:onClass c
Semantics
Main semantic condition:
IF (x,n) ∈ EXT_I(S_I(owl:minQualifiedCardinality)), (x,p) ∈ EXT_I(S_I(owl:onProperty)), (x,c) ∈ EXT_I(S_I(owl:onClass)) THEN x ∈ IOR, n ∈ LV_I, n is a non-negative integer, p ∈ IOOP, c ∈ IOC, CEXT_I(x) = { u | card({v | (u,v) ∈ EXT_I(p) ∧ v ∈ CEXT_I(c)}) ≥ n }
Comprehension principle:
IF p ∈ IOOP, n ∈ LV_I, n is a non-negative integer, c ∈ IOC THEN ∃ x: x in IOR, (x,n) ∈ EXT_I(S_I(owl:minQualifiedCardinality)), (x,p) ∈ EXT_I(S_I(owl:onProperty)), (x,c) ∈ EXT_I(S_I(owl:onClass))
Min Qualified Data Cardinality Restrictions
Syntax
x rdf:type owl:Restriction x owl:onProperty p x owl:minQualifiedCardinality "n"^^xsd:nonNegativeInteger x owl:onDataRange c
Semantics
Main semantic condition:
IF (x,n) ∈ EXT_I(S_I(owl:minQualifiedCardinality)), (x,p) ∈ EXT_I(S_I(owl:onProperty)), (x,c) ∈ EXT_I(S_I(owl:onDataRange)) THEN x ∈ IOR, n ∈ LV_I, n is a non-negative integer, p ∈ IODP, c ∈ IDC, CEXT_I(x) = { u | card({v ∈ LV_I | (u,v) ∈ EXT_I(p) ∧ v ∈ CEXT_I(c)}) ≥ n }
Comprehension principle:
IF p ∈ IODP, n ∈ LV_I, n is a non-negative integer, c ∈ IDC THEN ∃ x: x in IOR, (x,n) ∈ EXT_I(S_I(owl:minQualifiedCardinality)), (x,p) ∈ EXT_I(S_I(owl:onProperty)), (x,c) ∈ EXT_I(S_I(owl:onDataRange))
Max Qualified Object Cardinality Restrictions
Syntax
x rdf:type owl:Restriction x owl:onProperty p x owl:maxQualifiedCardinality "n"^^xsd:nonNegativeInteger x owl:onClass c
Semantics
Main semantic condition:
IF (x,n) ∈ EXT_I(S_I(owl:maxQualifiedCardinality)), (x,p) ∈ EXT_I(S_I(owl:onProperty)), (x,c) ∈ EXT_I(S_I(owl:onClass)) THEN x ∈ IOR, n ∈ LV_I, n is a non-negative integer, p ∈ IOOP, c ∈ IOC, CEXT_I(x) = { u | card({v | (u,v) ∈ EXT_I(p) ∧ v ∈ CEXT_I(c)}) ≤ n }
Comprehension principle:
IF p ∈ IOOP, n ∈ LV_I, n is a non-negative integer, c ∈ IOC THEN ∃ x: x in IOR, (x,n) ∈ EXT_I(S_I(owl:maxQualifiedCardinality)), (x,p) ∈ EXT_I(S_I(owl:onProperty)), (x,c) ∈ EXT_I(S_I(owl:onClass))
Max Qualified Data Cardinality Restrictions
Syntax
x rdf:type owl:Restriction x owl:onProperty p x owl:maxQualifiedCardinality "n"^^xsd:nonNegativeInteger x owl:onDataRange c
Semantics
Main semantic condition:
IF (x,n) ∈ EXT_I(S_I(owl:maxQualifiedCardinality)), (x,p) ∈ EXT_I(S_I(owl:onProperty)), (x,c) ∈ EXT_I(S_I(owl:onDataRange)) THEN x ∈ IOR, n ∈ LV_I, n is a non-negative integer, p ∈ IODP, c ∈ IDC, CEXT_I(x) = { u | card({v ∈ LV_I | (u,v) ∈ EXT_I(p) ∧ v ∈ CEXT_I(c)}) ≤ n }
Comprehension principle:
IF p ∈ IODP, n ∈ LV_I, n is a non-negative integer, c ∈ IDC THEN ∃ x: x in IOR, (x,n) ∈ EXT_I(S_I(owl:maxQualifiedCardinality)), (x,p) ∈ EXT_I(S_I(owl:onProperty)), (x,c) ∈ EXT_I(S_I(owl:onDataRange))
Considerations
- The range of the cardinality properties has been set to xsd:nonNegativeInteger. This will lead to formal inconsistency of an ontology, which uses cardinality numbers from other number spaces such as xsd:integer. This is a well known problem.
- Checked generally for cardinality restrictions (TF Action 34): It is not a problem to have "n is a non-negative integer" in the consequent of the main semantic condition.