Working with strong reducibilities above totally $\omega$-c.e. and array computable degrees
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- by George Barmpalias, Rod Downey and Noam Greenberg PDF
- Trans. Amer. Math. Soc. 362 (2010), 777-813 Request permission
Abstract:
We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allow us to compute such sets. For example, we prove that a c.e. degree is totally $\omega$-c.e. iff every set in it is weak truth-table reducible to a hypersimple, or ranked, set. We also show that a c.e. degree is array computable iff every left-c.e. real of that degree is reducible in a computable Lipschitz way to a random left-c.e. real (an $\Omega$-number).References
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Additional Information
- George Barmpalias
- Affiliation: School of Mathematics, Statistics and Computer Science, Victoria University, P.O. Box 600, Wellington, New Zealand
- Email: George.Barmpalias@mcs.vuw.ac.nz
- Rod Downey
- Affiliation: School of Mathematics, Statistics and Computer Science, Victoria University, P.O. Box 600, Wellington, New Zealand
- MR Author ID: 59535
- Email: downey@mcs.vuw.ac.nz
- Noam Greenberg
- Affiliation: School of Mathematics, Statistics and Computer Science, Victoria University, P.O. Box 600, Wellington, New Zealand
- MR Author ID: 757288
- ORCID: 0000-0003-2917-3848
- Email: greenberg@mcs.vuw.ac.nz
- Received by editor(s): December 19, 2007
- Published electronically: September 11, 2009
- Additional Notes: We wish to thank Frank Stephan for his permission to include Theorem \ref{thm: Stephan} and its proof. All authors were supported by the Marsden fund of New Zealand.
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 777-813
- MSC (2000): Primary 03D25, 03D30
- DOI: https://doi.org/10.1090/S0002-9947-09-04910-1
- MathSciNet review: 2551506