# There is no classification of the decidably presentable structures

## TL;DR

This paper proves that there is no comprehensive classification for decidably presentable structures, demonstrating the complexity of their index sets even within specific algebraic categories.

## Contribution

It establishes that the index set of decidably presentable structures is $\, ext{Sigma}^1_1$-complete, showing the classification problem's high complexity across various structure types.

## Key findings

- Index set of decidable structures is $\, ext{Sigma}^1_1$-complete.
- Complexity persists even when restricted to groups, graphs, or fields.
- The result extends to $n$-decidable presentations for any $n$.

## Abstract

A computable structure $\mathcal{A}$ is decidable if, given a formula $\varphi(\bar{x})$ of elementary first-order logic, and a tuple $\bar{a} \in \mathcal{A}$, we have a decision procedure to decide whether $\varphi$ holds of $\bar{a}$. We show that there is no reasonable classification of the decidably presentable structures. Formally, we show that the index set of the computable structures with decidable presentations is $\Sigma^1_1$-complete. This result holds even if we restrict out attention to groups, graphs, or fields. We also show that the index sets of the computable structures with $n$-decidable presentations is $\Sigma^1_1$-complete for any $n$.