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TopIntroduction
The Description Logic 
has syntax for role subsumptions of the form:
(1) but not of the form:
(2)Such axioms, called complex role inclusion axioms, show a subsumption relation between role 
and the role composition, or chain, 
.
Given an interpretation 
, role compositions have the standard meaning:
and thus (1) has the semantics:
In general, such axioms result in undecidablity, but, decidability can be regained by enforcing restrictions on the relationships between roles. 
, for example, requires a strict partial ordering of roles. Recently, complex role inclusions with role chains on the left have been studied in 
, 
, 
and 
.
Excepting a part solution for 
(Mosurivić & Krdzăvac, 2011), no description logics that allow role chains on the right of the subsumption relation have been published.
Contribution
This paper shows that complex role inclusions on the right, that is, of the form of (2), with semantics:
are definable in terms of complex role inclusions on the left, and thus, despite not being directly permitted in the syntax, can already be expressed in
,
and, consequently, OWL 2.
1Both soundness and completeness are proved for the method. Soundness showing that the definitions given imply the desired role compositions, and completeness demonstrating that all models with the intended semantics are admitted: i.e., the definitions do not imply `hidden' restrictions on satisfying models.
A set of syntactic restrictions constrain the valid complex role inclusion axioms in 
. The definitions given here result in syntactic restrictions for complex role inclusions on the right; these restrictions are a consequence of the construction in terms of role chains on the left and do not add any further restrictions to the logic.
The restrictions are similar to the existing 
restrictions:
- •
Roles on the right of the subsumption in a complex role inclusion are not ‘simple’, 
in (2); and
- •
The global partial ordering on roles requires either 
or 
.