# [DL] can you find minimal models from the open and complete situation using Tableaux?

Uli Sattler sattler at cs.man.ac.uk
Thu Jul 14 11:12:44 CEST 2011

Hi Jun,

the answer depends on the logic - and also on the exact definition of
minimality: yours seems to be a bit unusual - have a look at recent
work by Lutz et al on the existence of a (!) canonical model,  see,
e.g., http://aaai.org/ocs/index.php/KR/KR2010/paper/view/1282 for a
starter.

Cheers, Uli

On 11 Jul 2011, at 20:07, Jun Fang wrote:

> Hi All,
>
> Given an consistent ontology O, applying the tableaux algorithm till
> it is in the final open and complete situation. There may be many
> models which can be constructed from it.
>
> A minimal model is a model that removing any element from the
> concept interpretation or role interpretation in it can make it is
> not a model.
>
> I have two question about the canonical model and the minimality:
>
> 1) Is there a method to find its minimal canonical models? If
> exists, what's the method? just need pick one from the disjunctive
> branches? (soundness)
>
> For instance, if the ontology is (C\unionD)(a), then model1: \domian
> ={a},  a \in C^I; model2: \domian ={a},  a \in D^I; model3: \domian
> ={a},  a \in C^I, a \in D^I are all its models, while model1 and
> model2 are its minimal models. For this simple example, it is easy
> to find the minimal models by just select one from the disjunctive
> branches each time. Is it also true in more complex situation.
>
> 2) Furthermore, Is there a method to find all its minimal canonical
> models? If exists, what's it? (completeness)
>
>
> If it is too complicated, we can just discuss it in simper DL
> language, such as ALC.
>
> --
> Best Regards!
>
> Jun Fang
>
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