Provability and Interpretability Logics with Restricted Realizations

The provability logic of a theory $T$ is the set of modal formulas, which under any arithmetical realization are provable in $T$. We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $\Gamma$. We make an analogous modification for interpretability logics. We first study provability logics with restricted realizations and show that for various natural candidates of $T$ and restriction set $\Gamma$, the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic by capitalizing on the well-studied relationship between PRA and I$\Sigma_1$. We then study interpretability logics, obtaining upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely related to linear frames. The technique is also applied to yield the nontrivial result that IL(PRA) $\subset$ ILM