  The paper treats 4 different fragments of first-order logic induced by their
respective versions of Kripke style semantics for modal intuitionistic logic.
In order to capture these fragments, the notion of asimulation is modified and
extended to yield Van Benthem type of semantic characterization of their
respective expressive powers. It is shown, further, that this characterization
can be easily carried over to arbitrary first-order definable subclasses of
classical first-order models.
