Okay, again, we can already say quite a lot without being specific, without being specific

to which description logic we're referring to.

We have three kinds of tests that we're interested in, namely the consistency test, whether a

concept is satisfiable, we're defining a way, and we would not like to define things that

don't have any objects in them because we get rid of them.

A subsumption test, whether one concept assumes another, that will give us the nice graphs

and an instance test, whether an individual is an example of a concept.

Of course, we want to always ask ourselves about decidability, complexity, and what the

algorithm might be.

Let's go into an example.

We have a consistency test.

We define a man is a person that has a Y chromosome, a woman is a person that does not have a Y

chromosome, and a hermaphrodite is something that's both a man and a woman.

Now, you've probably already spotted it.

This specification is inconsistent because one of the concepts, namely hermaphrodite,

is actually empty in all models.

Why is that?

Well, because it has to have a Y chromosome and not have one.

Okay?

Now, the good thing here is we're in set description language propositional logic, so we can just

basically use a propositional satisfiability test.

Think DPLL, and that actually gives us a decision procedure for that.

A subsumption test.

We have, say, we leave out the hermaphrodite axiom definition, and we're left with these

axioms.

We can then just see that man is a subset, subconcept of person and woman as well, of

course.

Of course, we can, as always, as very often in inference, we can reduce that to a consistency

test.

If we can prove that from the axioms A implies B, then we know that if the axioms are true

and A is true and not B is inconsistent, then again, modulo DPLL or something like this,

we actually have a subsumption test.

We say A subsumes B, modulo and axiom set A. If B is a subset, the interpretation of

B is a subset of the interpretation of A for all interpretations, or if A implies B implies

A is actually valid.

In this case, person subsumes woman and man.

If you have subsumption relations among all the concepts, then you can visualize them

in a graph for inspection.

Every of these arrows is a subsumption here.

If you look at this, we have objects and persons are objects and men and women and students

and professors and children are persons, then you can define a male student as being a student

and a man and a female student and a professor.

We don't know whether they're a man or a woman.

There are something apparently totally different here.

We have children who are boys and girls.

That's what we could write down in a inset description language, but we would also start

to wonder what is a professor in relation to men and women.

Probably you would like to have professor being a sub-concept of man, union woman and

so on and so forth.

Once we have all the subsumption relations, which is nice if we have decision procedures