Reasoning OWL ontology using inverse property? - owl

Question

I'm using Protege v4.3 for making ontologies.
I have a question about OWL ontology and DL query.
For instance, in the Pizza ontology,
http://owl.cs.manchester.ac.uk/co-ode-files/ontologies/pizza.owl
I can execute the DL query
hasTopping some CheeseTopping
The result is
American, AmericanHot, Cajun,.. etc. That's OK.
Now, i tried DL query
isToppingOf some American
But the result is nothing.
Because the property isToppingOf is inverse property of hasTopping,
I expected to get the result including FourCheesesTopping, CheeseyVegetableTopping, etc. from that query(by inference). Bud it didn't.
Is there any ways automatic reasoning like that?

Answer 1

The class expression
hasTopping some CheeseTopping
is the set of individuals each of which is related to some CheeseTopping by the hasTopping property. In the Pizza ontology, where there are no individuals, you can still get class subclass results for this query because the definition of certain types of Pizzas (e.g., American) are such that any Pizza that is an American must have such a topping.
Now, the similarly-structured query
isToppingOf some American
is the set of individuals each of which is related to some American pizza by the isToppingOf property. However, the Pizza ontology defines no particular individuals, so there aren't any individuals as candidates. But what about classes that might be subclasses of this expression? For instance, you mentioned the FourCheeseTopping. Now, some particular instance of FourCheeseTopping, e.g., fourCheeseTopping23 could be a topping of some American pizza, e.g.:
fourCheeseTopping23 isToppingOf americanPizza72
However, fourCheeseTopping might not have been placed on any particular pizza yet. When we choose an arbitrary individual of type FourCheeseTopping, we can't infer that it is a topping of some American pizza, so we cannot infer that the class FourCheeseTopping is a subclass of
isToppingOf some American
because it's not the case that every instance of FourCheeseTopping must be the topping of some American pizza. For a similar case that might make the logical structure a bit clearer, consider the classes Employer and Person, and the object property employs and its inverse employedBy. We might say that every Employer must have some Person as an Employee (since otherwise they wouldn't be an employer):
Employer ⊑ employs some Person
However, since a person can be unemployed, it is not true that
Person ⊑ employedBy some Employer
even though employs and employedBy are inverses.
What you can do, though, if you want to know whether toppings of a particular type could be placed an pizza of a particular type, is to ask whether
PizzaType ⊓ ∃hasTopping.ToppingType
is equivalent to, or a subclass of, owl:Nothing. For instance, since an American pizza has only toppings of type TomatoTopping, MozzarellaTopping, and PeperoniTopping [sic], the class
American ⊓ ∃hasTopping.MixedSeafoodTopping
is equivalent to owl:Nothing:
On the other hand, since an American pizza must have a MozzarellaTopping, the class
American ⊓ ∃hasTopping.MozzarellaTopping
is equivalent to American:

Answer 2

When you ask what are the subclasses of:
isToppingOf some American
you are asking what classes contain toppings that are necessarily used on top of American pizzas. But in the pizza ontology, no such class exists. Consider cheese toppings: Are all cheese toppings on top of some American pizzas? No, some cheese toppings are on top of Italian pizzas. The same holds for all topping classes.