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The notion of computable reducibility between equivalence relations on thenatural numbers provides a natural computable analogue of Borel reducibility.We investigate the computable reducibility hierarchy, comparing and contrastingit with the Borel reducibility hierarchy from descriptive set theory.Meanwhile, the notion of computable reducibility appears well suited for ananalysis of equivalence relations on the c.e.\ sets, and more specifically, onvarious classes of c.e.\ structures. This is a rich context with many naturalexamples, such as the isomorphism relation on c.e.\ graphs or on computablypresented groups. Here, our exposition extends earlier work in the literatureconcerning the classification of computable structures. An abundance of openquestions remains.

The hierarchy of equivalence relations on the natural numbers under computable reducibility