A linear order is -decidable if its universe is and the relations defined by formulas are uniformly computable. This means that there is a computable procedure which, when applied to a formula and a sequence of elements of the linear order, will determine whether or not is true in the structure. A linear order is decidable if the relations defined by all formulas are uniformly computable. These definitions suggest two questions. Are there, for each , -decidable linear orders that are (...) not -decidable? Are there linear orders that are -decidable for all but not decidable? The former was answered in the positive by Moses in 1993. Here we answer the latter, also positively. (shrink)

We show that for every computable limit ordinal α, there is a computable structure A that is $\Delta _\alpha ^0 $ categorical, but not relatively $\Delta _\alpha ^0 $ categorical (equivalently. it does not have a formally $\Sigma _\alpha ^0 $ Scott family). We also show that for every computable limit ordinal a, there is a computable structure A with an additional relation R that is intrinsically $\Sigma _\alpha ^0 $ on A. but not relatively intrinsically $\Sigma _\alpha ^0 $ (...) on A (equivalently, it is not definable by a computable $\Sigma _\alpha $ formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α. (shrink)

We study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable $\Pi _{1}^{0}$ subsets of 2ω and Kolmogorov complexity play a major role in the proof.