Please help me to understand domain of interpretation

Question

In the literature on Description Logic, when interpretations are explained, we encounter expressions like,

$$\mathcal{I} = (\Delta^\mathcal{I}, \cdot^\mathcal{I})$$

(Actually, I am talking about, The Description Logic Handbook: Theory, Implementation and Applications, eds Franz Baader, Diego Calvanese, Deborah L. McGuinness, Daniele Nardi, Peter F. Patel-Schneider, page 52. But since the book may not be available all, this should suffice as well.)

Now, please help me understand the above expression, specifically, symbols and writing styles,

• $\Delta^\mathcal{I}$

• $\cdot^\mathcal{I}$

Please explain what is meant by $\Delta$ and $\cdot$ in this context, and how do the significance change when $\mathcal{I}$ is used as a superscript.

Explanations at beginners' level are appreciated.

Answer 1

Here is how I decode the notation in the Wikipedia link you gave. It is describing a fairly typical setting where an interpretation comprises operations that map syntactic constructs (here, individuals, concepts and role names) to values drawn from some set.

$\Delta^{\cal I}$ is a symbol denoting the domain of the interpretation: a set used for the values of individuals. In $\Delta^{\cal I}$, the superscript doesn't denote an operation on something called $\Delta$: it's just part of the symbol. The superscript ${\cal I}$ is also used as a shorthand for three operations. The dot in $\cdot^{\cal I}$ is a place-holder for the argument of the operation. If the argument is an individual $a$, $a^{\cal I}$ denotes the value of $a$ under the interpretation (an element of $\Delta^{\cal I}$). Likewise for a concept $C$ or a role name $R$, $C^{\cal I}$ and $R^{\cal I}$ denote the values of $C$ and $R$ under the interpretation (a subset of $\Delta^{\cal I}$ and a binary relation on $\Delta^{\cal I}$ respectively).