The closed fragment of IL is PSPACE hard

Abstract

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 $\textbf{IL}_0$.