  In this paper from 2011 we consider $\textbf{IL}_0$, the closed fragment of
the basic interpretability logic $\textbf{IL}$. We show that we can translate
$\textbf{GL}_1$, the one variable fragment of G\"odel-L\"ob's provabilty logic
$\textbf{GL}$, into $\textbf{IL}_0$. Invoking a result on the PSPACE
completeness of $\textbf{GL}_1$ we obtain the PSPACE hardness of
$\textbf{IL}_0$.
