  A Rough semiring $(T,\Delta,\nabla)$ is considered to describe a special
distributive Rough semiring known as a Rough bi-Heyting algebra. A bi-Heyting
algebra is an extension of boolean algebra and it is accomplished by weaker
notion of complements namely pseudocomplement $(^{*})$, dual pseudocomplement
$(^{+})$, relative pseudocomplement $(\rightarrow)$ and dual relative
pseudocomplement $(\leftarrow)$. In this paper, it is proved that the elements
of the Rough semiring $(T,\Delta,\nabla)$ are accomplished with the
pseudocomplement, relative pseudocomplement along with their duals. The
definition of pseudocomplement leads to the concept of Brouwerian Rough
semiring structure $(T,\Delta,\nabla,^{*},RS(\emptyset),RS(U))$ on the Rough
semiring $(T,\Delta,\nabla)$. Also it is proved
$(T,\Delta,\nabla,\rightarrow,\leftarrow,RS(\emptyset),RS(U))$ is a Rough
bi-Heyting algebra. The concepts are illustrated with the examples. As an
application, this Rough bi-Heyting algebra is used to model Rough
bi-intuitionistic logic. The syntax is defined and three types of semantics for
Rough bi-intuitionistic logic are defined and validated.
