  We investigate intuitionistic modal logics with locally interpreted $\square$
and $\lozenge$. The basic logic LIK is stronger than constructive modal logic
WK and incomparable with intuitionistic modal logic IK. We propose an
axiomatization of LIK and some of its extensions. We propose bi-nested calculi
for LIK and these extensions, thus providing both a decision procedure and a
procedure of finite countermodel extraction.
