After having looked at two, in a way extreme examples of description logics, namely propositional

logic, which was too weak to talk about individuals could only talk about sets of individuals

or concepts.

And first of all, logic, which is in a way too expressive to be interesting because it

doesn't have decidable satisfiability checking.

We'll now look at the first and in a way simplest description logic that is strong enough to

talk about individuals, but also weak enough, if you will, to admit an inference procedure

that is decidable.

So we'll first look at the syntax and semantics of it.

So we kind of look back.

So we looked at propositional logic, which is not expressive enough because we want to

define concepts like mother as women that have a child.

But we have a child here, which is akin to quantification and there are no quantifiers

in propositional logic, but this doesn't.

So we can't do that in propositional logic.

And first logic, we can do that.

We can just write down for all objects X, X is a mother, if and only if X is a woman

and there is a Y such that Y has child X.

The problem, of course, is non-termination of the algorithms.

So it's too expressive.

And the description logic paradigm we've seen is actually tries to go a middle ground, which

is weaker than first logic, but still tractable.

And the technique we're going to see is that we will only allow what you could think of

as restricted quantification, where you only quantify over variables that can be reached

by a binary relation like has child, which is in here.

And we can use that to actually make progress.

So basically, we have concepts.

We've seen this already for the set description language.

We have a top concept, which is true for all, and a bottom concept.

We have more concepts like person or woman or man or professor or student or BMW and

so on and so on.

You can make up your own next couple of thousand.

And we have binary relation like has a child, has a son, executes a computer program, has

a wheel, has a motor, all of those kind of things.

They're kind of like in first order logic.

So that allows us to define the syntax of ALC.

So we define the formula of ALC, and we'll do it by a grammar, as we've seen it for set

description logic.

And indeed, this grammar is almost identical.

We still have concepts.

We still have top and bottom.

We still have complement, intersection, union.

But we now have two new things.

We have there is an exists R and then a formula and for all R and a formula.

So we'll come back to our example of a mother being a person who has a child.

And here we do a little bit more interesting, a mother of a student, which is a person such

that there is an object that is a student and which is in the has child relation with

that person.

So this concept of exists has child student is an object that is related to something

that is a student.