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FINITARY REDUCIBILITY ON EQUIVALENCE RELATIONS
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- Journal:
- The Journal of Symbolic Logic / Volume 81 / Issue 4 / December 2016
- Published online by Cambridge University Press:
- 01 December 2016, pp. 1225-1254
- Print publication:
- December 2016
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- Article
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We introduce the notion of finitary computable reducibility on equivalence relations on the domain ω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be
${\rm{\Pi }}_{n + 2}^0$ -complete under computable reducibility, we show that, for every n, there does exist a natural equivalence relation which is 
${\rm{\Pi }}_{n + 2}^0$ -complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy.
