Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Index Sets of Computable Structures   The \emph{index set} of a computable structure $\mathcal{A}$ is the set of
indices for computable copies of $\mathcal{A}$. We determine the complexity of
the index sets of various mathematically interesting structures, including
arbitrary finite structures, $\mathbb{Q}$-vector spaces, Archimedean real
closed ordered fields, reduced Abelian $p$-groups of length less than
$\omega^{2}$, and models of the original Ehrenfeucht theory. The index sets for
these structures all turn out to be $m$-complete $\Pi_{n}^{0}$,
$d$-$\Sigma_{n}^{0}$, or $\Sigma_{n}^{0}$, for various $n$. In each case, the
calculation involves finding an \textquotedblleft optimal\textquotedblright% \
sentence (i.e., one of simplest form) that describes the structure. The form of
the sentence (computable $\Pi_{n}$, $d$-$\Sigma_{n}$, or $\Sigma_{n}$) yields a
bound on the complexity of the index set. When we show $m$% -completeness of
the index set, we know that the sentence is optimal. For some structures, the
first sentence that comes to mind is not optimal, and another sentence of
simpler form is shown to serve the purpose. For some of the groups, this
involves Ramsey theory.

1
The categories of this paper are: cs
3
The categories of this paper are: physics
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph