Index Sets of Computable Structures

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dc.creatorCalvert, Wesley
dc.creatorHarizanov, Valentina S.
dc.creatorKnight, Julia F.
dc.creatorMiller, Sara
dc.date2008-03-22
dc.date.accessioned2026-07-07T09:28:02Z
dc.date.available2026-07-07T09:28:02Z
dc.descriptionThe \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 $ω^{2}$, and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be $m$-complete $Π_{n}^{0}$, $d$-$Σ_{n}^{0}$, or $Σ_{n}^{0}$, 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 $Π_{n}$, $d$-$Σ_{n}$, or $Σ_{n}$) 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.
dc.identifierhttps://arxiv.org/abs/0803.3294
dc.identifierhttp://arxiv.org/abs/0803.3294
dc.identifierAlgebra and Logic 45 (2006), 306--325
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/157314
dc.subjectLogic
dc.subject03D45; 03C57
dc.titleIndex Sets of Computable Structures
dc.typetext

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