Okay, good morning everybody and I'm very happy that some of you

braved the weather warning. Wasn't so bad after all. Who knows what comes this afternoon.

We were talking about description logics. Description logics as languages that somehow

try to give us a sweet spot between practical expressivity, we want to talk about individuals,

without having to pay the price of undecidability. So in a way exactly what an agent representation

language should be. It's something where the agent can rely on solving entailment problems.

Is there a pit in 1.3 or something like this in the Wumpa's world? While still being able to say

things about the world succinctly and without too many encoding tricks. That's kind of what

description logics try to do. And the primary application is the semantic web which basically

means that all applications are big. One of the prime examples is DBpedia which basically scrapes

Wikipedia or by now Wikidata and extracts a couple of billion facts, assertion facts,

A-box facts of the form Emi Nöte worked in Erlangen, she was born here and died at that point there

All of those things are essentially A-box facts which is exactly what we're going to come to today.

And DBpedia also wants or tries to extract T-box things like cars or vehicles and stuff like that.

The kind of ontology language that is used for the semantic web and for many many other

knowledge representation problems very often in companies and so on is this language ALC which

is kind of hits its own sweet spot namely between complexity and teachability. It's complex enough

so that you can see what's going on but it's still small enough that we can actually teach it.

Many of the description logics that are in use for instance the ALTUDL language which is kind

of as far as we can go in a standardized setting are extensions of this ALC language.

And the ALC language is that we were talking about is essentially the stuff we've learned

for propositional logic as a set description language. We have top which is the fold domain,

bottom which is the empty concept, concept complement, intersection and union.

And we're adding two new concepts that were new is basically restricted quantification

both existential and universal. We've looked at a couple of examples and so on.

And the last thing we actually looked at was the semantics. The semantics is relatively classical.

We have a domain just like in first-order logic and we have an interpretation language that we

traditionally write with these double square brackets but it's nothing functionally different

from propositional from first-order logic and propositional logic. And indeed complement,

intersection and union do exactly and true and false do exactly the same as they always did.

We have these new things that we call roles. They are just relations on the domain.

And the interesting thing is these things like there is an RFI.

And the idea is that if we have a concept,

a set of objects that are the meaning of Phi, then the set exists Phi,

exists are Phi, are those things that are related with the meaning of R which is a relation on the

domain. All of that happens in the domain. Those are those things that have an R relation that

are one R step away from something in Phi. And the nice thing about this is we can write all

of this down essentially without variables. In first-order logic if you think about the translation,

we're going to see it, it's down there. It's essentially the translation. I always want to do

it the wrong way around. That translated to first-order logic is for all X. Well,

you have it up there. You translate it into exactly what this picture says. There is a Y

in the denotation of Phi such that the X you want to say something about has an R-or-A relation,

has an R successor in Phi. And the universal part is just basically says all of the R successors,

there's another one and another one. They all have to land in Phi. So it's a rather straightforward

semantics. It's inspired essentially by this translation or the other way around,

what they fit together. That's really what ALC semantics is. Now we have semantics which gives

us satisfiability and all of the things and if we have satisfiability we can actually do things

like subsumption, compute the hierarchy and so on. Or at least we have a specification for that.

We have defined when concept C subsumes concept D, namely when,

let's call this one E, when that is the domain. Or equivalently when the denotation of C