Hard Provability Logics

TL;DR

This paper characterizes various relative provability logics, proves their decidability, and introduces a framework of reductions showing how their arithmetical completeness results relate, highlighting the complexity of $ ext{PL}_{ ext{Sigma}_1}( ext{HA}, ext{N})$.

Contribution

It formalizes a general notion of reduction for provability logics and applies it to establish relationships among multiple provability logics, including their decidability and arithmetical completeness.

Findings

All studied provability logics are decidable.

The notion of reduction for provability logics is formalized and several reductions are provided.

The logic $ ext{PL}_{ ext{Sigma}_1}( ext{HA}, ext{N})$ is identified as the hardest among those considered.

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 $\mathcal{PL}_{\Sigma_1}({\sf HA},\mathbb{N})$.