Interpretation of top and bottom concepts in the scope of quantifiers in description logics

Question

What is the difference in interpretation of concepts: $\exists R.\top$, $\exists R.\bot$, $\forall R.\top$, $\forall R.\bot$?

For instance, if we assume R means hasChild, will $\exists R.\top$ denote all individuals that have a child?
And what would be the interpretation of $\exists R.\bot$? Since $(\bot)^I$ means an empty set, does that mean $\exists R.\bot$ denotes the set that has an unrealizable property, i.e. an empty set? So would it be basically equal to the bottom concept?

Will it be similar for $\forall R.\top$? Since the top concept means all elements in the interpretation will the interpretation of $\forall R.\top$ amount to it?

And how the interpretation of $\forall R.\bot$ would differ from $\exists R.\bot$? As far as I understood transition from existantial to universal quantifier doesn't change anything in this case.

Answer 1

In description logic $\top$ is the "universal" concept, i.e. the concept to which every individual in the domain belongs.

Similarly, $\bot$ is the empty concept.

$\exists R . C$ is the existential restriction of a concept $C$ by a role $R$, and is a concept.

In an interpretation $I$ with domain $D^I$, a concept $C$ is interpreted with a subset of $D$ and a role $R$ must be interpreted with a relation on $D$ (i.e. a subset of $D \times D$).

The semantical interpretation of the concept $\exists R . C$ is the set :

$\{x\in D^I \mid \text { there exists } y \text { such that } (x,y)\in R^I \text{ and } y\in C^I \}$.

Thus, the semantical interpretation of $\exists R . \top$ is the set :

$\{x\in D^I \mid \text { there exists } y \text { such that } (x,y)\in R^I \}$,

because the condition $y \in \top^I$ is always satisfied.