Some effects of Ash–Nerode and other decidability conditions on degree spectra

Annals of Pure and Applied Logic 55 (1):51-65 (1991)
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Abstract

With every new recursive relation R on a recursive model , we consider the images of R under all isomorphisms from to other recursive models. We call the set of Turing degrees of these images the degree spectrum of R on , and say that R is intrinsically r.e. if all the images are r.e. C. Ash and A. Nerode introduce an extra decidability condition on , expressed in terms of R. Assuming this decidability condition, they prove that R is intrinsically r.e. if and only if a natural recursive-syntactic condition is satisfied. We show that, while a recursive non-intrinsically r.e. relation may have a two element degree spectrum, a non-intrinsically r.e. relation which satisfies the Ash–Nerode decidability condition has an infinite degree spectrum. We also study several related decidability conditions and their effects on the degree spectra, including some conditions which are sufficient to obtain every r.e. degree in a spectrum

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Valentina Harizanov
George Washington University

References found in this work

Uncountable Degree Spectra.Valentina S. Harizanov - 1991 - Annals of Pure and Applied Logic 54 (3):255-263.
Foundations of Recursive Model Theory.Terrence S. Millar - 1978 - Annals of Mathematical Logic 13 (1):45.

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