Tbilisi Mathematical Journal

Undecidability of local structures of $\mathrm{s}$-degrees and $\mathrm{Q}$-degrees

Maria Libera Affatato, Thomas F. Kent, and Andrea Sorbi

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Abstract

We show that the first order theory of the $\Sigma^0_2\;$ $\mathrm{s}$-degrees is undecidable. Via isomorphism of the $\mathrm{s}$-degrees with the $\mathrm{Q}$-degrees, this also shows that the first order theory of the $\Pi^0_2\;$ $\mathrm{Q}$-degrees is undecidable. Together with a result of Nies, the proof of the undecidability of the $\Sigma^0_2\;$ $\mathrm{s}$-degrees yields a new proof of the known fact (due to Downey, LaForte and Nies) that the first order theory of the c.e. $\mathrm{Q}$-degrees is undecidable.

Article information

Source
Tbilisi Math. J., Volume 1 (2008), 15-32.

Dates
Received: 5 September 2007
Revised: 9 November 2007
Accepted: 11 December 2007
First available in Project Euclid: 12 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1528768822

Digital Object Identifier
doi:10.32513/tbilisi/1528768822

Mathematical Reviews number (MathSciNet)
MR2434435

Zentralblatt MATH identifier
1158.03028

Subjects
Primary: 03D35: Undecidability and degrees of sets of sentences
Secondary: 03D30: Other degrees and reducibilities 03D25: Recursively (computably) enumerable sets and degrees

Keywords
Computability theory $\mathrm{s}$-reducibility $\mathrm{Q}$-reducibility undecidability enumeration reducibility

Citation

Affatato, Maria Libera; Kent, Thomas F.; Sorbi, Andrea. Undecidability of local structures of $\mathrm{s}$-degrees and $\mathrm{Q}$-degrees. Tbilisi Math. J. 1 (2008), 15--32. doi:10.32513/tbilisi/1528768822. https://projecteuclid.org/euclid.tbilisi/1528768822


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