Abstract
Let E, F be equivalence relations on ℕ. We say that E is computably reducible to F, written E ≤ F, if there is a computable function p : ℕ → ℕ such that xEy ↔ p(x) F p(y). We show that several natural \(\Sigma^0_3\) equivalence relations are in fact \(\Sigma^0_3\) complete for this reducibility. Firstly, we show that one-one equivalence of computably enumerable sets, as an equivalence relation on indices, is \(\Sigma^0_3\) complete. Thereafter, we show that this equivalence relation is below the computable isomorphism relation on computable structures from classes including predecessor trees, Boolean algebras, and metric spaces. This establishes the \(\Sigma^0_3\) completeness of these isomorphism relations.
Keywords
- Equivalence Relation
- Boolean Algebra
- Computable Function
- Computable Structure
- Computable Tree
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The first and the second authors acknowledge the generous support of the FWF through projects Elise-Richter V206, and P22430-N13. The third author is partially supported by the Marsden Fund of New Zealand under grant 09-UOA-187.
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© 2012 Springer-Verlag Berlin Heidelberg
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Fokina, E., Friedman, S., Nies, A. (2012). Equivalence Relations That Are \(\Sigma^0_3\) Complete for Computable Reducibility. In: Ong, L., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2012. Lecture Notes in Computer Science, vol 7456. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32621-9_2
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DOI: https://doi.org/10.1007/978-3-642-32621-9_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32620-2
Online ISBN: 978-3-642-32621-9
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