Normal companions of intuitionistic modal logics

Previously, Došen and Božić introduced four independent intuitionistic modal logics, one for each of four types of modal operators—necessity ${\mathsf{N}}$, possibility ${\mathsf{P}}$, impossibility ${\mathsf{Im}}$, and unnecessity ${\mathsf{Un}}$. These logics are denoted $\mathsf{HK}{\mathsf{M}}$, where ${\mathsf{M}}\in\{{\mathsf{N}},{\mathsf{P}},{\mathsf{Un}},{\mathsf{Im}}\}$. Interest in treating the four types of modal operators separately is associated with just the fact that these cannot be reduced to each other over intuitionistic logic. Here we study extensions of logics $\mathsf{HK}{\mathsf{M}}$ that have normal companions. It turns out that all extensions of the logics $\mathsf{HK}{\mathsf{N}}$ and $\mathsf{HK}{\mathsf{Un}}$ possess normal companions. For the extensions of $\mathsf{HK}{\mathsf{P}}$ and $\mathsf{HK}{\mathsf{Im}}$, we obtain a criterion for the existence of normal companions, which is postulated as the presence of some modal law of double negation. Also we show how adding of this law influences expressive capacities of a logic. Of particular interest is the result saying that extensions of $\mathsf{HK}{\mathsf{P}}$ and $\mathsf{HK}{\mathsf{Im}}$ have normal companions only if they are definitionally equivalent to those of $\mathsf{HK}{\mathsf{N}}$ and $\mathsf{HK}{\mathsf{Un}}$ respectively. This result is one more example of the differences in behavior of the four types of modal operators over intuitionistic logic.