[DL] correct understanding of DL semantics
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
> Now let's see an example, if I have concepts "Lawyer" and "Doctor",
> role "hasChild", John is a "Lawyer" and Mary is "Doctor", John
> "hasChild" Mary. But later on in my model, Mary gets another degree
> 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.
> 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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sattler at cs.man.ac.uk
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