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.
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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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
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