  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 $n$.
