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Computable Reducibility for Cantor Space

We examine various versions of Borel reducibility on equivalence relations on the Cantor space 2ω , using reductions given by Turing functionals on the inputs A ∈ 2ω . In some versions, we vary the number of jumps of A which the functional is allowed to use. In others, we do not require the reduction to succeed for all elements of the Cantor space at once, but only when applied to arbitrary finite or countable subsets of 2ω . In others we allow an arbitrary oracle set in addition to the inputs. All of these versions, inspired largely by work on computable reducibility on equivalence relations on ω, combine to yield a rich set of options for evaluating the precise level of difficulty of a Borel reduction, or the reasons why a Borel reduction may fail to exist.