[DL] correct understanding of DL semantics

Ulrike Sattler sattler at cs.man.ac.uk
Thu Oct 4 11:51:33 CEST 2007

On 21 Sep 2007, at 00:36, Chuming Chen wrote:

> Dear All,
> I am new to Description Logics. I am trying to understand the  
> correct semantics
> of Description Logics. especially, the changes in semantics.
> I know DL Semantics is defined by interpretations.  An  
> interpretation I
> = (Delta^I, .^I), where  Delta^I is the domain of interpretation (a  
> non-empty set) and
> .^I is an interpretation function that maps:
> Concept (class) name A to subset of Delta^I, Role (property) name R to
> a binary relation R over Delta^I, Individual name i to an element of
> Delta^I.
> Now let's see an example, if I have concepts "Lawyer" and "Doctor",  
> and
> role "hasChild",  John is a "Lawyer" and Mary is "Doctor", John
> "hasChild" Mary. But later on in my model, Mary gets another degree  
> and
> becomes "Lawyer" also. Now Mary is both "Lawyer" and "Doctor".

What you have done is you have changed your *interpretation*: you  
have modified I, so it is no longer I but, say, I'.

> Do the
> semantics of "Laywer", "Doctor", even "hasChild" change in this case?

We would say that the *extension* of Lawyer (the set of those  
elements that are instances of Lawyer) has changed.

> Because if we treat concept as a subset of domain, adding Mary to
> "Lawyer" certainly change the set for that concept.


> If role is a subset
> of pair of elements in the domain, would that be changed too?

It can certainly be changed in a similar way: for example, we can  
think of a third interpretation, say I'', where Mary ceases to be a  
child of John or a fourth one, say I''', where Mary has a child  
called Foo or ....

> Can we still think Mary is the same Mary? What are the correct  
> understanding of semantics here?

ok, so you need to distinguish between

- an interpretation (any structure with a non-empty set and the  
mappings as you have described above)
- whether an interpretation *satisfies* an axiom (or a set of axioms  
or a TBox or an ontology) - such "legal" or "conforming"  
interpretations are often called *models* of  an axiom  (or a set of  
axioms or a TBox or an ontology)
- whether an axiom  (or a set of axioms or a TBox or an ontology)  
does have a model, i.e., is *satisfiable* or *consistent*, and
- whether an axiom A follows from a set of axioms T (or a TBox or an  
ontology): this is the case if, in all models of T, the axiom A holds  
as well. In this case we say that this axiom is *entailed* by T and  
is often written as T |= A.

The above 4 points (plus possible some other points) is what we call  
the semantics of a formalism/logic.

Cheers, Uli

> I might be missing something obvious here. But mathematically  
> speaking , the set
> has been changed. Would the semantics be changed also?
> Thank you for any comments!
> Chuming Chen
> ---
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Ulrike Sattler
sattler at cs.man.ac.uk
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