[DL] question about the distinction between transitive roles and transitive closure

Uli Sattler 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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