Counting in OWL: Flowers with Exactly Four Petals

the flower anatomy has thrown up at least one interesting little bit of OWL. This is to do with counting. Let us say a flower

hasPart some Petal

(reasonable as some flowers have only one petal). A cruciate flower has exactly four petals (and so looks like a cross). We can make a defined class in the usual way:

Class: CruciateFlower
	EquivalentTo: Flower
		that hasPart exactly 4 Petal

(My flower anatomy is not described in this way — I’ve shortened it for ease.) If I then have a class for a particular flower like:

Class: SpringCabbageFlower
	SubClassOf: Flower
		that hasPart exactly 4 Petal

this is recognised as a cruciate flower (all cabbages, such as spring cabbage and wall flowers are cruciate.). However, if we start saying things about each of the petals, such as shape, this no longer works as expected. For example:

Class: SomeFlower
	SubClassOf: Flower
		that hasPart exactly 2 (Petal that hasShape some Ovate)
		and hasPart exactly 2 (Petal that hasShape some Obovate)

isn’t classified as a cruciate flower, even though we think we’ve described it as having four petals. The reason is the "open world assumption". If we just say

hasPart exactly 4 Petal

we know that there are four petals. If, however, we say that exactly two of them are ovagte and exactly two are obovate shaped, we don’t know about petals that might be of some other shape; we might also have exactly two petals that ar lancealate and so on. So, we have to close off our description of petals by saying something like:

hasPart only ((Petal that hasShape some Obovate) or (Petal that hasShape some Ovate)) 

Having explicitly said that the "SomeFlower" flower has exactly two of one shape and two of another shape and only petals of that shape, then we know that it has exactly four petals in total. There is a little sample ontology to look at. It is worth running the reasoner both with and without the closure axiom in question — the closure axiom really slows down the reasoner.

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