  In this article, we study translations between variants of defaults logics
such that the extensions of the theories that are the input and the output of
the translation are in a bijective correspondence. We assume that a translation
can introduce new variables and that the result of translating a theory can
either be produced in time polynomial in the size of the theory or its output
is polynomial in that size; we however restrict to the case in which the
original theory has extensions. This study fills a gap between two previous
pieces of work, one studying bijective translations among restrictions of
default logics, and the other one studying non-bijective translations between
default logics variants.
