Index Sets of Computable Structures
Wesley Calvert, Valentina S. Harizanov, Julia F. Knight, and Sara, Miller

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 is the set of indices for computable copies of . We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, -vector spaces, Archimedean real closed ordered fields, reduced Abelian -groups of length less than , and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be -complete , -, or , for various . 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 , -, or ) yields a bound on the complexity of the index set. When we show %…
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Taxonomy
TopicsComputability, Logic, AI Algorithms · semigroups and automata theory · Benford’s Law and Fraud Detection
