  Notions of k-asimulation and asimulation are introduced as asymmetric
counterparts to k-bisimulation and bisimulation, respectively. It is proved
that a first-order formula is equivalent to a standard translation of an
intuitionistic propositional formula iff it is invariant with respect to
k-asimulations for some k, and then that a first-order formula is equivalent to
a standard translation of an intuitionistic propositional formula iff it is
invariant with respect to asimulations. Finally, it is proved that a
first-order formula is intuitionistically equivalent to a standard translation
of an intuitionistic propositional formula iff it is invariant with respect to
asimulations between intuitionistic models.
