Is a property with exact cardinality one functional?
In the OWL-DL ontology, consider a property p with domain D and range R, where D is constrained by p to have cardinality of exactly one:
D SubClassOf p exactly 1 Thing
- (D & sqsubseteq; = 1 p.Thing)
Can we then infer that p is a functional property, since every d of type D will have exactly one value for p? If this is correct, can an intelligent person derive this knowledge?
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In OWL, a property is a function when each person has at most one value for the property. This "at the most" is important; it is allowed that something has no value for the property. (This means that a functional property in OWL is actually more like a possible partial function in mathematics.) However, if each person has exactly one property value, then they clearly have at most one property value, so the property. as you suspect will function. However, we can go through at least in a specific case to make sure that this is common and because we need to make sure that the property p here really has at most one value for each person.
Proof: Suppose property p has domain D and D is subclass = 1 p.Thing, so that each D has exactly one p value. Is this the case where each person x has at most one value for p? There are two cases:
- x is D. Then, by the constrained subclass axiom, x must have exactly one value for p, and one is less than or equal to one.
- x is not D. Then x has no value for p. If this were the case, then it would be in the region p, which is D, and this is a contradiction. Then x has zero values for p and zero is less than or equal to one.
Then any individual x is at most one value for the property p, which is the definition of the functional p. Thus, p is functional. QED
The OWL DL browser should be able to confirm this and shouldn't be difficult to verify.
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