Deductive Closure of a Relation

Assume you have

         p(a,b), p(b,c)
         and the following two rules:
         p(x,y) -> q(x,y)
         p(x,y) & p(y,z) -> q(x,z)

Then under minimal model semantics q is the deductive closure
of p, i.e., q(a,b), q(b,c), q(a,c) are true and no other q(x,y) us true.

In FOL, this is different since you cannot exclude models in which
q(a,a) is true. That is, you cannot express deductive closure in FOL.

The difference becomes appeared when you allow negation in your
query language for facts, i.e., when you allow to query for negated
facts [2]

         -- dieter

[2] Michael Kifer, Jos de Bruijn, Harold Boley, Dieter Fensel: A Realistic 
Architecture for the Semantic Web,
to appear in the RuleML Conference Proceedings, Galway, November
2005

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Dieter Fensel, http://www.deri.org/
Tel.: +43-512-5076485/8