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We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF-reducibility introduced in  to show completeness of the isomorphism relation on many familiar classes in the context of all equivalence relations on hyperarithmetical subsets of ω.
We show that for every computable limit ordinal α, there is a computable structure that is categorical, but not relatively categorical (equivalently, it does not have a formally Scott family). We also show that for every computable limit ordinal α, there is a computable structure with an additional relation R that is intrinsically on , but not relatively intrinsically on (equivalently, it is not definable by a computable Σα formula with finitely many parameters). Earlier results in , , and  establish the same facts for computable successor ordinals α.
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