Abstract
Our focus will be on the computably enumerable (c.e.) sets and trivial, non-trivial, Friedberg, and non-Friedberg splits of the c.e. sets. Every non-computable set has a non-trivial Friedberg split. Moreover, this theorem is uniform. V. Yu. Shavrukov recently answered the question which c.e. sets have a non-trivial non-Friedberg splitting and we provide a different proof of his result. We end by showing there is no uniform splitting of all c.e. sets such that all non-computable sets are non-trivially split and, in addition, all sets with a non-trivial non-Friedberg split are split accordingly.
Draft as of August 7, 2016. We want to thank V. Yu. Shavrukov for allowing us to include his result, Theorem 3.8. Without it, this paper would look very different. This research was started while Cholak participated in the Buenos Aires Semester in Computability, Complexity and Randomness, 2013. Thanks to Rachel Epstein, Greg Igusa, Nathan Pierson, Mike Stob, and the referees for comments and suggestions. My interest in Friedberg splits was sparked in 1989 by Rod Downey. I cannot forgive him.
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Notes
- 1.
The Herrmann and Kummer Splitting Theorem appears, in a very different form, in Herrmann and Kummer [7]. This theorem appears in the only if direction of the proof of Theorem 2.4 of Herrmann and Kummer [7] starting on page 63 from the first full paragraph on that page. It is interesting enough to be isolated in its own right as a theorem.
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Cholak, P.A. (2017). On Splits of Computably Enumerable Sets. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_31
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DOI: https://doi.org/10.1007/978-3-319-50062-1_31
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