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arXiv · 1806.10363

Measuring the complexity of reductions between equivalence relations

Ekaterina Fokina·Dino Rossegger·Luca San Mauro
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Abstract

Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and bi-reducibility. These spectra provide a natural way of measuring the complexity of reductions between equivalence relations. We prove that any upward closed collection of Turing degrees with a countable basis can be realised as a reducibility spectrum or as a bi-reducibility spectrum. We show also that there is a reducibility spectrum of computably enumerable equivalence relations with no countable basis and a reducibility spectrum of computably enumerable equivalence relations which is downward dense, thus has no basis.

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Ekaterina Fokina, Dino Rossegger, Luca San Mauro. 2018-06-27. Measuring the complexity of reductions between equivalence relations. https://doi.org/10.3233/com-180100

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