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

Franz Baader baader at tcs.inf.tu-dresden.de
Mon May 19 18:13:15 CEST 2008

Since I am responsible for the DL Terminology Appendix in the handbook,
I want to give some additional clarifications regarding syntax and names
for languages.

Unfortunately, there are different naming schemes for DLs, and this was
already the case when I wrote the appendix. I tried to devise a consistent
scheme that is relatively close to what was used then. The main idea was
to distinguish between:
- concept constructors (like existential restrictions, ...)
- role constructors (like composition, transitive closure, inverse, ...)
- restrictions on roles (like transitivity, subrole relations, ...)
In the naming scheme, concept constructors are appended to the name of the
familiy of DLs one considers (like the AL-family), role constructors
are indicated
by superscripts, and restrictions on roles by subscripts. More
precisely, a letter
or symbol standing for the constructor/restriction is appended/used a

For the case of transitivity vs transitive closure, the appendix proposes the
- the avialability of transitive roles is expressed by the subscript R^+
- the availability of the transitive closure role constructor is expressed by
  the superscript +

Thus, in the name of a DL transitive closure can be distinguished from
transitivity by the fact that one is a superscript and the other a subscript;
in addition, for transitive closure we only have a + (not R^+).

What my be confusing is that we write the transitive closure of the role R
as $R^+$ (but this denotes then the role itself, not the presence of the
constructor in the DL).

Unfortunately, people don't stick to this (or other) coherent naming
scheme in their research papers.

Best regards,

-Franz Baader

On Fri, May 16, 2008 at 4:59 PM, Yvonne Violet Shashoua
<yvshashoua at gmail.com> 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.  But could someone clarify
> the distinction?  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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