Index Sets of Computable Structures

TL;DR

This paper analyzes the complexity of index sets for various computable structures, determining their classification within the arithmetical hierarchy and identifying optimal descriptions for these structures.

Contribution

It provides a detailed classification of the index set complexities for multiple structures and introduces methods to find optimal sentences describing them.

Findings

Index sets are $m$-complete $ ext{Pi}_n^0$, $d$-$ ext{Sigma}_n^0$, or $ ext{Sigma}_n^0$

Optimal sentences describing structures are identified for complexity bounds

Some proofs involve Ramsey theory to establish optimality

Abstract

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.