  We consider an extension of bi-intuitionistic logic with the traditional
modalities from tense logic Kt. Proof theoretically, this extension is obtained
simply by extending an existing sequent calculus for bi-intuitionistic logic
with typical inference rules for the modalities used in display logics. As it
turns out, the resulting calculus, LBiKt, seems to be more basic than most
intuitionistic tense or modal logics considered in the literature, in
particular, those studied by Ewald and Simpson, as it does not assume any a
priori relationship between the diamond and the box modal operators. We recover
Ewald's intuitionistic tense logic and Simpson's intuitionistic modal logic by
modularly extending LBiKt with additional structural rules. The calculus LBiKt
is formulated in a variant of display calculus, using a form of sequents called
nested sequents. Cut elimination is proved for LBiKt, using a technique similar
to that used in display calculi. As in display calculi, the inference rules of
LBiKt are ``shallow'' rules, in the sense that they act on top-level formulae
in a nested sequent. The calculus LBiKt is ill-suited for backward proof search
due to the presence of certain structural rules called ``display postulates''
and the contraction rules on arbitrary structures. We show that these
structural rules can be made redundant in another calculus, DBiKt, which uses
deep inference, allowing one to apply inference rules at an arbitrary depth in
a nested sequent. We prove the equivalence between LBiKt and DBiKt and outline
a proof search strategy for DBiKt. We also give a Kripke semantics and prove
that LBiKt is sound with respect to the semantics, but completeness is still an
open problem. We then discuss various extensions of LBiKt.
