[DL] question about the distinction between transitive roles and transitive closure
sattler at cs.man.ac.uk
Mon May 19 10:19:29 CEST 2008
On 16 May 2008, at 16:59, Yvonne Violet Shashoua wrote:
> I have been reading through the appendix in the Description Logic
> Handbook. On page 527, they mention that AL extended with
> transitive roles is denoted by AL_R+. That's in the section
> entitled Restrictions on Role Interpretations. But in the paragraph
> right above it, they use R^c instead of R+. Then, in a separate
> section, entitled Role Constructors, on page 529, they use the same
> symbol, R+, to denote the transitive closure role constructor. I
> assume that there's a difference between extending a description
> logic with transitive roles, and making a description logic that
> includes the transitive closure role constructor.
you are right, , there is.
> But could someone clarify the distinction?
if I have transitive roles, I can say, e.g., that "hasPart" is a
transitive role (i.e., if x hasPart y and y hasPart z, then x hasPart
z). Or that "hasAncestor" is transitive.
if I have the transitive closure operator, I can apply it to a role
name to denote the *smallest* transitive role containing it, e.g., I
can use "hasDirectPart^+" and "hasParent^+"
if I have transitive roles and role hierachies (as in DLs called
SH...), I can say that a transitive role is a super-role of another
role, e.g., that "hasPart" is a transitive role and a super-role of
"hasDirectpart" or that "hasAncestor" is a transitive role and a
superrole of "hasParent". I cannot, however, enforce that, say,
"hasAncestor" is the *smallest* transitive super-role of "hasParent",
only that it is *some* transitive super-role.
In general, the transitive closure is a second order feature (due to
the "smallest transitive relation that contains another relation),
whereas transitivity and role inclusion are first order...
I hope that helps, cheers, Uli
> And is the only difference in the notation used for the name of the
> language just whether you make R+ a subscript or superscript? I
> could use some help understanding this. Thanks. ---
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