$\Pi _{1}^{0}$ Classes and Strong Degree Spectra of Relations

Journal of Symbolic Logic 72 (3):1003 - 1018 (2007)
Abstract
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
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