Hard Provability Logics

Abstract

Let $\mathcal{PL}({\sf T},{\sf T}')$ and $\mathcal{PL}_{\Sigma_1}({\sf T},{\sf T}')$ respectively indicates the provability logic and $\Sigma_1$-provability logic of ${\sf T}$ relative in ${\sf T}'$. In this paper we characterize the following relative provability logics: $\mathcal{PL}_{\Sigma_1}({\sf HA},\mathbb{N})$, $\mathcal{PL}_{\Sigma_1}({\sf HA},{\sf PA})$, $\mathcal{PL}_{\Sigma_1}({\sf HA}^*,\mathbb{N})$, $\mathcal{PL}_{\Sigma_1}({\sf HA}^*,{\sf PA})$, $\mathcal{PL}({\sf PA},{\sf HA})$, $\mathcal{PL}_{\Sigma_1}({\sf PA},{\sf HA})$, $\mathcal{PL}({\sf PA}^*,{\sf HA})$, $\mathcal{PL}_{\Sigma_1}({\sf PA}^*,{\sf HA})$, $\mathcal{PL}({\sf PA}^*,{\sf PA})$, $\mathcal{PL}_{\Sigma_1}({\sf PA}^*,{\sf PA})$, $\mathcal{PL}({\sf PA}^*,\mathbb{N})$, $\mathcal{PL}_{\Sigma_1}({\sf PA}^*,\mathbb{N})$ (see Table \ref{Table-Theories}). It turns out that all of these provability logics are decidable. The notion of {\em reduction} for provability logics, first informally considered in \cite{reduction}. In this paper, we formalize a generalization of this notion (\Cref{Definition-Reduction-PL}) and provide several reductions of provability logics (See diagram \ref{Diagram-full}). The interesting fact is that $\mathcal{PL}_{\Sigma_1}({\sf HA},\mathbb{N})$ is the hardest provability logic: the arithmetical completenesses of all provability logics listed above, as well as well-known provability logics like $\mathcal{PL}({\sf PA},{\sf PA})$, $\mathcal{PL}({\sf PA},\mathbb{N})$, $\mathcal{PL}_{\Sigma_1}({\sf PA},{\sf PA})$, $\mathcal{PL}_{\Sigma_1}({\sf PA},\mathbb{N})$ and $\mathcal{PL}_{\Sigma_1}({\sf HA},{\sf HA})$ are all propositionally reducible to the arithmetical completeness of $\mathcal{PL}_{\Sigma_1}({\sf HA},\mathbb{N})$.